VOLUME 4,NUMBER 1 JANUARY 2006 ISSN:1548-5390 PRINT,1559-176X ONLINE

JOURNAL

OF CONCRETE AND APPLICABLE MATHEMATICS EUDOXUS PRESS,LLC

SCOPE AND PRICES OF THE JOURNAL Journal of Concrete and Applicable Mathematics A quartely international publication of Eudoxus Press,LLC Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A. ganastss@memphis.edu The main purpose of the "Journal of Concrete and Applicable Mathematics" is to publish high quality original research articles from all subareas of Non-Pure and/or Applicable Mathematics and its many real life applications, as well connections to other areas of Mathematical Sciences, as long as they are presented in a Concrete way. It welcomes also related research survey articles and book reviews.A sample list of connected mathematical areas with this publication includes and is not restricted to: Applied Analysis, Applied Functional Analysis, Probability theory, Stochastic Processes, Approximation Theory, O.D.E, P.D.E, Wavelet, Neural Networks,Difference Equations, Summability, Fractals, Special Functions, Splines, Asymptotic Analysis, Fractional Analysis, Inequalities, Moment Theory, Numerical Functional Analysis,Tomography, Asymptotic Expansions, Fourier Analysis, Applied Harmonic Analysis, Integral Equations, Signal Analysis, Numerical Analysis, Optimization, Operations Research, Linear Programming, Fuzzyness, Mathematical Finance, Stochastic Analysis, Game Theory, Math.Physics aspects, Applied Real and Complex Analysis, Computational Number Theory, Graph Theory, Combinatorics, Computer Science Math.related topics,combinations of the above, etc. In general any kind of Concretely presented Mathematics which is Applicable fits to the scope of this journal. Working Concretely and in Applicable Mathematics has become a main trend in many recent years,so we can understand better and deeper and solve the important problems of our real and scientific world. "Journal of Concrete and Applicable Mathematics" is a peer- reviewed International Quarterly Journal. We are calling for papers for possible publication. The contributor should send three copies of the contribution to the editor in-Chief typed in TEX, LATEX double spaced. [ See: Instructions to Contributors] Journal of Concrete and Applicable Mathematics(JCAAM) ISSN:1548-5390 PRINT, 1559-176X ONLINE. is published in January,April,July and October of each year by EUDOXUS PRESS,LLC, 1424 Beaver Trail Drive,Cordova,TN38016,USA, Tel.001-901-751-3553 anastassioug@yahoo.com http://www.EudoxusPress.com. Annual Subscription Current Prices:For USA and Canada,Institutional:Print $250,Electronic $220,Print and Electronic $310.Individual:Print $77,Electronic $60,Print &Electronic $110.For any other part of the world add $25 more to the above prices for Print.

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Editorial Board Associate Editors

Editor in -Chief: George Anastassiou Department of Mathematical Sciences The University Of Memphis Memphis,TN 38152,USA tel.901-678-3144,fax 901-678-2480 e-mail ganastss@memphis.edu www.msci.memphis.edu/~ganastss/jcaam Areas:Approximation Theory, Probability,Moments,Wavelet, Neural Networks,Inequalities,Fuzzyness. Associate Editors: 1) Ravi Agarwal Florida Institute of Technology Applied Mathematics Program 150 W.University Blvd. Melbourne,FL 32901,USA agarwal@fit.edu Differential Equations,Difference Equations,inequalities 2) Shair Ahmad University of Texas at San Antonio Division of Math.& Stat. San Antonio,TX 78249-0664,USA SAHMAD@utsa.edu Differential Equations,Mathematical Biology 3) Drumi D.Bainov Medical University of Sofia P.O.Box 45,1504 Sofia,Bulgaria drumibainov@yahoo.com Differential Equations,Optimal Control, Numerical Analysis,Approximation Theory 4) Carlo Bardaro Dipartimento di Matematica & Informatica Universita' di Perugia Via Vanvitelli 1 06123 Perugia,ITALY tel.+390755855034, +390755853822, fax +390755855024 bardaro@unipg.it , bardaro@dipmat.unipg.it Functional Analysis and Approximation Th.,

19) Rupert Lasser Institut fur Biomathematik & Biomertie,GSF -National Research Center for environment and health Ingolstaedter landstr.1 D-85764 Neuherberg,Germany lasser@gsf.de Orthogonal Polynomials,Fourier Analysis, Mathematical Biology 20) Alexandru Lupas University of Sibiu Faculty of Sciences Department of Mathematics Str.I.Ratiu nr.7 2400-Sibiu,Romania lupas@ulbsibiu.ro Classical Analysis,Inequalities, Special Functions,Umbral Calculus, Approximation Th.,Numerical Analysis and Methods 21) Ram N.Mohapatra Department of Mathematics University of Central Florida Orlando,FL 32816-1364 tel.407-823-5080 ramm@mail.ucf.edu Real and Complex analysis,Approximation Th., Fourier Analysis, Fuzzy Sets and Systems 22) Rainer Nagel Arbeitsbereich Funktionalanalysis Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tuebingen Germany tel.49-7071-2973242 fax 49-7071-294322 rana@fa.uni-tuebingen.de evolution equations,semigroups,spectral th., positivity 23) Panos M.Pardalos Center for Appl. Optimization University of Florida 303 Weil Hall P.O.Box 116595 Gainesville,FL 32611-6595

Summability,Signal Analysis,Integral Equations, Measure Th.,Real Analysis 5) Francoise Bastin Institute of Mathematics University of Liege 4000 Liege BELGIUM f.bastin@ulg.ac.be Functional Analysis,Wavelets 6) Paul L.Butzer RWTH Aachen Lehrstuhl A fur Mathematik D-52056 Aachen Germany tel.0049/241/80-94627 office, 0049/241/72833 home, fax 0049/241/80-92212 Butzer@rwth-aachen.de Approximation Th.,Sampling Th.,Signals, Semigroups of Operators,Fourier Analysis 7) Yeol Je Cho Department of Mathematics Education College of Education Gyeongsang National University Chinju 660-701 KOREA tel.055-751-5673 Office, 055-755-3644 home, fax 055-751-6117 yjcho@nongae.gsnu.ac.kr Nonlinear operator Th.,Inequalities, Geometry of Banach Spaces 8) Sever S.Dragomir School of Communications and Informatics Victoria University of Technology PO Box 14428 Melbourne City M.C Victoria 8001,Australia tel 61 3 9688 4437,fax 61 3 9688 4050 sever.dragomir@vu.edu.au, sever@sci.vu.edu.au Math.Analysis,Inequalities,Approximation Th., Numerical Analysis, Geometry of Banach Spaces, Information Th. and Coding 9) A.M.Fink Department of Mathematics Iowa State University Ames,IA 50011-0001,USA

tel.352-392-9011 pardalos@ufl.edu Optimization,Operations Research 24) Svetlozar T.Rachev Dept.of Statistics and Applied Probability Program University of California,Santa Barbara CA 93106-3110,USA tel.805-893-4869 rachev@pstat.ucsb.edu AND Chair of Econometrics and Statistics School of Economics and Business Engineering University of Karlsruhe Kollegium am Schloss,Bau II,20.12,R210 Postfach 6980,D-76128,Karlsruhe,Germany tel.011-49-721-608-7535 rachev@lsoe.uni-karlsruhe.de Mathematical and Empirical Finance, Applied Probability, Statistics and Econometrics 25) Paolo Emilio Ricci Universita' degli Studi di Roma "La Sapienza" Dipartimento di Matematica-Istituto "G.Castelnuovo" P.le A.Moro,2-00185 Roma,ITALY tel.++39 0649913201,fax ++39 0644701007 riccip@uniroma1.it,Paoloemilio.Ricci@uniroma1.itOrthogonal Polynomials and Special functions, Numerical Analysis, Transforms,Operational Calculus, Differential and Difference equations 26) Cecil C.Rousseau Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA tel.901-678-2490,fax 901-678-2480 ccrousse@memphis.edu Combinatorics,Graph Th., Asymptotic Approximations, Applications to Physics 27) Tomasz Rychlik Institute of Mathematics Polish Academy of Sciences Chopina 12,87100 Torun, Poland T.Rychlik@impan.gov.pl Mathematical Statistics,Probabilistic Inequalities 28) Bl. Sendov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria

tel.515-294-8150 fink@math.iastate.edu Inequalities,Ordinary Differential Equations 10) Sorin Gal Department of Mathematics University of Oradea Str.Armatei Romane 5 3700 Oradea,Romania galso@uoradea.ro Approximation Th.,Fuzzyness,Complex Analysis 11) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis, Memphis,TN 38152,USA tel.901-678-2484 jgoldste@memphis.edu Partial Differential Equations, Semigroups of Operators 12) Heiner H.Gonska Department of Mathematics University of Duisburg Duisburg,D-47048 Germany tel.0049-203-379-3542 office gonska@informatik.uni-duisburg.de Approximation Th.,Computer Aided Geometric Design 13) Dmitry Khavinson Department of Mathematical Sciences University of Arkansas Fayetteville,AR 72701,USA tel.(479)575-6331,fax(479)575-8630 dmitry@uark.edu Potential Th.,Complex Analysis,Holomorphic PDE,Approximation Th.,Function Th. 14) Virginia S.Kiryakova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria virginia@diogenes.bg Special Functions,Integral Transforms, Fractional Calculus 15) Hans-Bernd Knoop Institute of Mathematics Gerhard Mercator University D-47048 Duisburg Germany tel.0049-203-379-2676

bsendov@bas.bg Approximation Th.,Geometry of Polynomials, Image Compression 29) Igor Shevchuk Faculty of Mathematics and Mechanics National Taras Shevchenko University of Kyiv 252017 Kyiv UKRAINE shevchuk@univ.kiev.ua Approximation Theory 30) H.M.Srivastava Department of Mathematics and Statistics University of Victoria Victoria,British Columbia V8W 3P4 Canada tel.250-721-7455 office,250-477-6960 home, fax 250-721-8962 harimsri@math.uvic.ca Real and Complex Analysis,Fractional Calculus and Appl., Integral Equations and Transforms,Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th. 31) Ferenc Szidarovszky Dept.Systems and Industrial Engineering The University of Arizona Engineering Building,111 PO.Box 210020 Tucson,AZ 85721-0020,USA szidar@sie.arizona.edu Numerical Methods,Game Th.,Dynamic Systems, Multicriteria Decision making, Conflict Resolution,Applications in Economics and Natural Resources Management 32) Gancho Tachev Dept.of Mathematics Univ.of Architecture,Civil Eng. and Geodesy 1 Hr.Smirnenski blvd BG-1421 Sofia,Bulgaria Approximation Theory 33) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock Germany manfred.tasche@mathematik.uni-rostock.de Approximation Th.,Wavelet,Fourier Analysis, Numerical Methods,Signal Processing,

knoop@math.uni-duisburg.de Approximation Theory,Interpolation 16) Jerry Koliha Dept. of Mathematics & Statistics University of Melbourne VIC 3010,Melbourne Australia koliha@unimelb.edu.au Inequalities,Operator Theory, Matrix Analysis,Generalized Inverses 17) Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston,RI 02881,USA kulenm@math.uri.edu Differential and Difference Equations 18) Gerassimos Ladas Department of Mathematics University of Rhode Island Kingston,RI 02881,USA gladas@math.uri.edu Differential and Difference Equations

Image Processing,Harmonic Analysis 34) Chris P.Tsokos Department of Mathematics University of South Florida 4202 E.Fowler Ave.,PHY 114 Tampa,FL 33620-5700,USA profcpt@math.usf.edu,profcpt@chuma1.cas.usf.edu Stochastic Systems,Biomathematics, Environmental Systems,Reliability Th. 35) Lutz Volkmann Lehrstuhl II fuer Mathematik RWTH-Aachen Templergraben 55 D-52062 Aachen Germany volkm@math2.rwth-aachen.de Complex Analysis,Combinatorics,Graph Theory

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G ½

M.QAFSAOUI32

Exponential operators and solution

of pseudo-classical evolution problems

Caterina Cassisa, Paolo E. Ricci,

Universita di Roma “La Sapienza”, Dipartimento di Matematica, P.le A. Moro, 2 00185Roma, Italia - e-mail: cassisa@uniroma1.it, riccip@uniroma1.it

Ilia Tavkhelidze

“I.N. Vekua” Institute of Applied Mathematics, Tbilisi State University2, University Street – 380043 – Tbilisi (Georgia) - e-mail: iliko@viam.hepi.edu.ge

Abstract

We use the multi-dimensional polynomials considered by Hermite, and subse-quently studied by P. Appell and J. Kampe de Feriet, in order to obtain explicitsolutions of pseudo-classical PDE problems in the half-plane y > 0. We considersystems of PDE, including some problems with degeneration on the x-axis.

2000 Mathematics Subject Classification. 33C45, 44A45, 35G15.Key words and phrases. Hermite-Kampe de Feriet polynomials, Operational calculus,Pseudo-hyperbolic and Pseudo-circular functions, Boundary value problems.

1 Introduction

In preceding articles [3], [4], [5], by using an operatorial approach, we showed thatthe solution of many classical or pseudo-classical boudary value problems in the halfplane for PDE, (with constant coefficients and analytic boundary, or initial, data) canbe expressed in terms of the higher order Hermite-Kampe de Feriet (shortly H-KdF)polynomials. The relevant results were extended to the multi-dimensional case in [13],[14].

In this article we extend the results of our article [5] to the pseudo-classical casetoo.

We start recalling properties of the H-KdF polynomials and extending technicaltools considered in [3], [4] to the case under examination.

33JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.1,33-45,2006,COPYRIGHT 2006 EUDOXUSPRESS,LLC

2 Hermite-Kampe de Feriet polynomials

2.1 Definitions

We recall the definitions of the H-KdF polynomials [1], [10], [16], in the two-dimensionalcase.

Put D := ddx

, and consider the shift operator

eyDf(x) = f(x + y) =∞∑

n=0

yn

n!f (n)(x), (2.1)

(see e.g. [17], p. 171), the second equation being meaningful for analytic functions.Note that,

• if f(x) = xm, then eyDxm = (x + y)m;

• if f(x) =∞∑

m=0

amxm, then eyDf(x) =∞∑

m=0

am(x + y)m.

Definition 2.1 The Hermite polynomials in two variables H(1)m (x, y) are then defined

byH(1)

m (x, y) := (x + y)m. (2.2)

Consequently,

• if f(x) =∞∑

m=0

amxm, then

eyDf(x) =∞∑

m=0

amH(1)m (x, y). (2.3)

Consider now the exponential containing the second derivative, defined for an an-alytic function f , as follows:

eyD2

f(x) =∞∑

n=0

yn

n!f (2n)(x). (2.4)

Note that

• if f(x) = xm, then for n = 0, 1, . . . ,[

m2

]we can write:

D2nxm = m(m− 1) · · · (m− 2n + 1)xm−2n = m!(m−2n)!

xm−2n and therefore

eyD2

xm =

[m2 ]∑

n=0

yn

n!

m!

(m− 2n)!xm−2n (2.5)

C.CASSISA ET AL34

• if f(x) =∞∑

m=0

amxm, then eyD2f(x) =

∞∑

m=0

amH(2)m (x, y).

Definition 2.2 The H-KdF polynomials in two variables H(2)m (x, y) are then defined

by

H(2)m (x, y) := m!

[m2 ]∑

n=0

ynxm−2n

n!(m− 2n)!(2.6)

Considering, in general, the exponential raised to the j-th derivative we have:

eyDj

f(x) =∞∑

n=0

yn

n!f (jn)(x) (2.7)

and therefore:

• if f(x) = xm, then for n = 0, 1, . . . ,[

mj

]it follows:

Djnxm = m(m− 1) · · · (m− jn + 1)xm−jn = m!(m−jn)!

xm−jn so that

eyDj

xm =

[mj ]∑

n=0

yn

n!

m!

(m− jn)!xm−jn (2.8)

• if f(x) =∞∑

m=0

amxm, then eyDjf(x) =

∞∑

m=0

amH(j)m (x, y),

Definition 2.3 The H-KdF polynomials in two variables H(j)m (x, y) are then defined

by

H(j)m (x, y) := m!

[mj ]∑

n=0

ynxm−jn

n!(m− jn)!(2.9)

2.2 Properties

In a number of articles by G. Dattoli et al., (see e.g. [6], [7], [8]), by using the so calledmonomiality principle, the following properties for the two-variable H-KdF polynomialsH(j)

m (x, y), j ≥ 2 have been recovered (the case when j = 1 reducing to resultsabout simple powers).

• Operational definition

H(j)m (x, y) = ey ∂j

∂xj xm =

(x + jy

∂j−1

∂xj−1

)m

(1). (2.10)

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 35

• Generating function∞∑

n=0

H(j)n (x, y)

tn

n!= ext+ytj . (2.11)

In the case when j = 2, (see [18]), the H-KdF polynomials H(2)m (x, y) admit the

following

• Integral representation

H(2)m (x, y) =

1

2√

πy

∫ +∞

−∞ξme−

(x−ξ)2

4y dξ, (2.12)

which is a particular case of the so called Gauss-Weierstrass (or Poisson) transform.For the case when j > 2 see [11], [12].

3 A decomposition theorem

In [15], in a general form, the following theorem was shown:

Theorem 3.1 Fix the integer r ≥ 2, and denote by

ωh,r := exp

(2πih

r

), (h = 0, 1, . . . , r − 1), (3.1)

the complex roots of unity.Consider an analytic function of the complex variable z, defined in a circular neigh-borhood of the origin, by means of the series expansion

ϕ(z) =∞∑

n=0

anzn, (3.2)

and the series

Π[h,r]ϕ(z) = ϕh(z; r) :=∞∑

n=0

arn+hzrn+h, (3.3)

called the components of ϕ with respect to the cyclic group of order r, then therepresentation formula

ϕh(z; r) =1

r

r−1∑

j=0

ϕ(z ωj,r)

ωjh,r

(3.4)

holds true.

C.CASSISA ET AL36

For r = 2 and h = 0 or h = 1 the functions (3.3) reduce to the even or oddcomponents, respectively, of the function ϕ(z). In general, the ϕh(z; r) functions arecharacterized by the symmetry property

ϕh(z ω1,r; r) = ωh,r ϕh(z; r) (h = 0, 1, . . . , r − 1) (3.5)

with respect to the roots of unity (see also [2]).In particular, assuming ϕ(z) := ez, the well known decomposition of the ex-

ponential function in terms of the so called pseudo-hyperbolic functions appear (see[9]). For shortness, omitting in the sequel the label r, i.e. assuming ωh := ωh,r,ϕh(z) := ϕh(z; r), and so on, we can write:

fh(z) := Π[h,r]ez =

∞∑

n=0

zrn+h

(rn + h)!, (3.6)

so that the exponential is decomposed as the sum:

ez =r−1∑

h=0

fh(z), (3.7)

and the following properties hold true:

f0(0) = 1; fh(0) = 0 (if h 6= 0), (3.8)

fh(ω1z) = ωh fh(z), (symmetry property), (3.9)

Dzfh(z) = fh−1(z), (differentiation rule), (3.10)

where the indices are assumed to be congruent (mod r), so that, ∀h, the pseudo-hyperbolic functions are solutions of the differential equation:

w(r)(z)− w(z) = 0. (3.11)

The pseudo-circular functions are obtained by introducing any complex r-th rootσ0 of the number −1, and putting:

gh(z) := σ−h0 fh(σ0 z). (3.12)

The pseudo-circular functions are given by the series expansions:

gh(z) =∞∑

n=0

(−1)n zrn+h

(rn + h)!, (3.13)

and satisfy the following properties:

g0(0) = 1; gh(0) = 0 (if h 6= 0), (3.14)

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 37

gh(ω1z) = ωh gh(z), (symmetry property), (3.15)

Dzg0(z) = −gr−1(z), Dzgh(z) = gh−1(z), (if h 6= 0), (3.16)

(differentiation rule)

where the indices are assumed to be congruent (mod r), so that ∀h, the pseudo-circularfunctions are solutions of the differential equation:

w(r)(z) + w(z) = 0. (3.17)

For further properties, generalizing the ordinary trigonometrical rules, see [15].

4 Pseudo-hyperbolic or pseudo-circular functions of thederivative operator

The properties of the pseudo-hyperbolic or pseudo-circular functions of the derivativeoperator (see [15]), can be easily extended to the operational case. We give in thefollowing a list of the relevant results, which can be deduced in the same way as in thefunctional case, considering the commutative property of the powers of D.

The pseudo-hyperbolic functions of the derivative operator are defined by the seriesexpansions:

fh(Dj) := Π[h,r]e

Dj

=∞∑

n=0

Dj(rn+h)

(rn + h)!, (h = 0, 1, . . . , r − 1) (4.1)

so that the exponential is decomposed as the sum:

eDj

=r−1∑

h=0

fh(Dj), (4.2)

and the following properties hold true:

fh(ω1Dj) = ωh fh(D

j), (symmetry property), (4.3)

Considering two independent variables x and y and denoting by Dx and Dy therelevant differentiations, it follows:

Dyfh(yDjx) = Dj

xfh−1(yDjx), (differentiation rule), (4.4)

where the indices are assumed to be congruent (mod r), so that, ∀h, the pseudo-hyperbolic functions are solutions of the abstract differential equation:

Dryfh(yDj

x)−Drjx fh(yDj

x) = 0. (4.5)

C.CASSISA ET AL38

The pseudo-circular functions of the derivative operator are given by the seriesexpansions:

gh(Dj) =

∞∑

n=0

(−1)n Dj(rn+h)

(rn + h)!, (4.6)

and satisfy the following properties:

gh(ω1Dj) = ωh gh(D

j), (symmetry property), (4.7)

Dyg0(yDjx) = −Dj

xgr−1(yDjx), Dygh(yDj

x) = Djxgh−1(yDj

x), (if h 6= 0), (4.8)

(differentiation rule),

where the indices are assumed to be congruent (mod r), so that ∀h, the pseudo-circularfunctions are solutions of the abstract differential equation:

Drygh(yDj

x) + Drjx gh(yDj

x) = 0. (4.9)

4.1 Connections with the H-KdF polynomials

Fixing the integer r, we consider now the action of the pseudo-hyperbolic and pseudo-circular functions on analytic functions, showing relations with the H-KdF. polynomi-als.

Theorem 4.1 For any real number `, for any h=0,1, . . . , r-1, and for any positiveinteger j, denoting by x and y (y > 0) independent variables, and by D := Dx,the action of the pseudo-hyperbolic function fh(y

`Dj) on the power xm (m ∈ N), isgiven by

fh

(y`Dj

)xm = m!

[m−jhjr ]∑

n=0

y`(rn+h)xm−j(rn+h)

(rn + h)!(m− j(rn + h))!= Π[h,r]

y`

[H(j)

m (x, y`)], (4.10)

where Π[h,r]y`

[K(x, y`)

], (h = 0, 1, . . . , r − 1), denote the components of the function

K with respect to the cyclic group of order r, and relevant to the variable y`,(assuming x as a parameter).

Then, if q(x) =∞∑

m=0

amxm, we can write

fh

(y`Dj

)q(x) =

∞∑

m=0

am Π[h,r]y`

[H(j)

m (x, y`)]

= (4.11)

=∞∑

m=0

m! am

[m−jhjr ]∑

n=0

y`(rn+h)xm−j(rn+h)

(rn + h)!(m− j(rn + h))!.

For the pseudo-circular functions, using the above notations, we have:

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 39

Theorem 4.2 For any real number `, for any h=0,1, . . . , r-1, and for any positiveinteger j, denoting by x and y (y > 0) independent variables, and by D := Dx,the action of the pseudo-circular function gh(y

`Dj) on the power xm (m ∈ N), isgiven by

gh

(y`Dj

)xm = m!

[m−jhjr ]∑

n=0

(−1)n y`(rn+h)xm−j(rn+h)

(rn + h)!(m− j(rn + h))!=

1

σh0

Π[h,r]y`

[H(j)

m (x, σ0y`)

],

(4.12)

and furthermore, if q(x) =∞∑

m=0

amxm, then

gh

(y`Dj

)q(x) =

1

σh0

∞∑

m=0

am Π[h,r]y`

[H(j)

m (x, σ0y`)

]= (4.13)

=∞∑

m=0

m! am

[m−jhjr ]∑

n=0

(−1)h y`(rn+h)xm−j(rn+h)

(rn + h)!(m− j(rn + h))!,

so that, if the considered coefficients am and variables x, y are real, the resultingexpansions are also real.

5 Convergence results

In this section we recall an uniform estimate, with respect to j, for the convergenceof series involving the H-KdF polynomials H(j)

n (x, y`), for every real ` ≥ 1:

∞∑

n=0

an H(j)n (x, y`). (5.1)

Theorem 5.1 For every real ` ≥ 1, j ≥ 2, −∞ < x < +∞, −∞ < y < +∞,n = 0, 1, 2, . . ., the following estimate holds true:

|H(j)n (x, y`)| ≤ n! exp

|x|+ |y`|

, (5.2)

The proof is derived by the same method used in the book of Widder [18], p. 166,for the case ` = 1 j = 2 (see [5]).

In the present article, we limit ourselves to consider, as boundary (or initial) data,

analytic functions f(x) =∞∑

n=0

anxn for which the coefficients an tend to zero sufficiently

fast, in order to guarantee the convergence of the series expansion (5.1). To this aim,we only use the following theorem (see [5])

C.CASSISA ET AL40

Theorem 5.2 Suppose there exists a number α > 1, such that the coefficients an

satisfy the following estimate:

|an| = O(

1

nαn!

), (5.3)

then, for every real ` ≥ 1 and integer j, the series expansion (5.1) is absolutely anduniformly convergent in every bounded region of the (x, y) plane.

Remark 5.1 The condition (5.3) include analytic functions with polynomial growthat infinity, but not the exponential function ex, whereas, in the case ` = 1 j = 2,the book of Widder [18] include all functions belonging to the so called Huygens classH0 however, by the point of view of the Applied Analysis, the considered conditionsare sufficient to cover all realistic situations.

When the function f(x) =∞∑

n=0

anxn is decomposed with respect to the cyclic group

of order r, the same estimate of Theorem 5.2 is sufficient to guarantee the convergenceof the relevant expansions, according to the property:

∣∣∣∣∣ Π[h,r]

( ∞∑

n=0

|an| xn

) ∣∣∣∣∣ ≤∣∣∣∣∣∞∑

n=0

|an|xn

∣∣∣∣∣ , (h = 0, 1, . . . , r − 1).

6 Systems solved in terms of pseudo-hyperbolic operators

For introducing our subject in a more friendly way, we start considering the particularcase when r = 3.

Consider the system:

∂Sh

∂y=

∂Sh−1

∂x(h = 0, 1, 2) in the half plane y > 0,

S0(x, 0) = q0(x)

∂S1

∂y(x, 0) = q′0(x)

∂2S2

∂y2(x, 0) = q′′0(x) ,

(6.1)

where q0(x) is a given analytic function. We assume that the considered indices arecongruent mod. 3, so that S3 ≡ S0.

By using the same technique developed in our article [5], we find

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 41

Theorem 6.1 The operational solution of system (6.1) is given by the following func-tions:

S0(x, y) = f0 (yD) q0(x) ,

S1(x, y) = f1 (yD) q0(x) ,

S2(x, y) = f2 (yD) q0(x) .

(6.2)

Proof. A straightforward computation gives:

∂S0

∂y= f2(yD)Dq0(x) = Dxf2(yD)q0(x) =

∂S2

∂x,

∂S1

∂y= f0(yD)Dq0(x) = Dxf0(yD)q0(x) =

∂S0

∂x,

∂S2

∂y= f1(yD)Dq0(x) = Dxf1(yD)q0(x) =

∂S1

∂x,

and the boundary conditions are trivially satisfied.

Remark 6.1 Note that the given boundary conditions (6.1) can be essentially derivedfrom the first one: S0(x, 0) = q0(x), by extending the considered equations on theboundary y = 0.

Remark 6.2 Note that the system (6.1) has connections with the similar one consid-ered in [4] (eqs. (7.1)-(7.2)), but there the boundary conditions were given by threeindependent functions, whereas in (6.1) only the function q0 appears.

A more general problem which can be solved is as follows:

∂Sh

∂y=

∂Sh−1

∂x(h = 0, 1, 2) in the half plane y > 0,

S0(x, 0) = q0(x)

S1(x, 0) = q1(x)

S2(x, 0) = q2(x) ,

(6.3)

where qh(x) (h = 0, 1, 2) are given analytic functions, assuming again that theconsidered indices are congruent mod. 3.

In this case, we find

Theorem 6.2 The operational solution of system (6.3) is given by the following func-tions:

S0(x, y) = f0 (yD) q0(x) + f1 (yD) q2(x) + f2 (yD) q1(x) ,

S1(x, y) = f0 (yD) q1(x) + f1 (yD) q0(x) + f2 (yD) q2(x) ,

S2(x, y) = f0 (yD) q2(x) + f1 (yD) q1(x) + f2 (yD) q0(x) .

(6.4)

C.CASSISA ET AL42

6.1 The case when r is an arbitrary integer

For any fixed integral r, consider the system

∂Sk

∂y=

∂Sk−1

∂x, (k = 0, 1, . . . , r − 1) in the half plane y > 0,

Sk(x, 0) = qk(x) ,

(6.5)

where qk(x), (k = 0, 1, . . . , r − 1), are given analytic functions, assuming that theconsidered indices are congruent mod. r, so that Sr ≡ S0.

By using the same technique, we find the general result:

Theorem 6.3 The operational solution of the system (6.5) is given by the followingr-tuples of functions:

Sk(x, y) =r−1∑

h=0

fh(yD) qr−h+k(x) , (k = 0, . . . , r − 1) . (6.6)

7 Higher order pseudo-hyperbolic systems

For any real number ` ≥ 1 and integer j ≥ 1, we consider now the system:

∂Sk

∂y= `y`−1∂jSk−1

∂xj, (k = 0, 1, . . . , r − 1) in the half plane y > 0,

Sk(x, 0) = qk(x) ,

(7.1)

where the qk(x) are given analytic functions. We assume again that k = 0, 1, . . . , r−1and that the considered indices are congruent mod. r, so that Sr ≡ S0.

Note that equations in system (7.1), assuming ` > 1, degenerate on the boundarywhen y → 0.

Theorem 7.1 The operational solution of system (7.1) is given by the following r-tuples of functions:

Sk(x, y) =r−1∑

h=0

fh

(y`Dj

)qr+k−h(x) , (k = 0, . . . , r − 1) . (7.2)

Remark 7.1 Theorem 7.1 generalizes some results already obtained in [4]. More pre-cisely, if ` = 1, j = 1, r = 3 we find eqs. (7.1)-(7.2), if ` = 1, j = 1 we find eqs.(8.1)-(8.2) in ref. [4]. Furthermore, if ` = 1, j = s, we recover (9.1)-(9.2) of the samereference.

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 43

8 Pseudo-circular systems

For any fixed integral r, consider, in the half-plane y > 0, the system

∂S0

∂y+ `y`−1∂jSr−1

∂xj= 0

∂Sk

∂y− `y`−1∂jSk−1

∂xj= 0 (k = 1, 2, . . . , r − 1),

Sk(x, 0) = qk(x) (k = 0, 1, . . . , r − 1),

(8.1)

where the qk(x) are given analytic functions. We assume again that k = 0, 1, . . . , r−1and that the considered indices are congruent mod. r.

By using same methods as in the preceding section, but relevant to pseudo-circularoperators, we find the result:

Theorem 8.1 The operational solution of system (8.1) is given by the following r-tuples of functions:

Sk(x, y) =k∑

h=0

gh

(y`Dj

)qr+k−h(x)−

r−1∑

h=k+1

gh

(y`Dj

)qr+k−h(x),

(k = 0, . . . , r − 1)

(8.2)

Remark 8.1 Theorem 8.1 generalizes some results already obtained in [4]. More pre-cisely, assuming ` = 1, j = 1, r = 3 we find eqs. (10.1)-(10.2) in ref. [4]. Furthermore,if ` = 1, j = s, we recover (11.1)-(11.2) of the same reference.

AcknowledgementsThis article was concluded during the visit of Dr. Ilia Tavkhelidze, partially supportedby the visiting professors program of the Ateneo Roma “La Sapienza”.

References

[1] P. Appell, J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques.Polynomes d’Hermite, Gauthier-Villars, Paris, 1926.

[2] Y. Ben Cheikh, Decomposition of some complex functions with respect to thecyclic group of order n, Appl. Math. Inform., 4, 30–53, (1999).

[3] C. Cassisa, P.E. Ricci, I. Tavkhelidze, Operational identities for circular and hy-perbolic functions and their generalizations, Georgian Math. J., 10, 45–56, (2003).

C.CASSISA ET AL44

[4] C. Cassisa, P.E. Ricci, I. Tavkhelidze, An operatorial approach to solutions ofBVP in the half-plane, J. Concr. Appl. Math., 1, 37–62, (2003).

[5] C. Cassisa, P.E. Ricci, I. Tavkhelidze, Exponential operators for solving evolutionproblems with degeneration, J. Appl. Funct. Anal., (to appear).

[6] G. Dattoli, A. Torre, P.L. Ottaviani, L. Vazquez, Evolution operator equations:integration with algebraic and finite difference methods. Application to physicalproblems in classical and quantum mechanics, La Rivista del Nuovo Cimento, 2,(1997).

[7] G. Dattoli, A. Torre, C. Carpanese, Operational rules and arbitrary order Hermitegenerating functions, J. Math. Anal. Appl., 227, 98–111, (1998).

[8] G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials,Atti Acc. Sc. Torino, 132, 1–7, (1998).

[9] A. Erdely, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher TrascendentalFunctions, McGraw-Hill, New York, 1953.

[10] H.W. Gould, A.T. Hopper, Operational formulas connected with two generaliza-tions of Hermite Polynomials, Duke Math. J., 29, 51–62, (1962).

[11] D.T. Haimo and C. Markett, A reprentation theory for solutions of a higher orderheat equation, I, J. Math. Anal. Appl. 168, 89–107, (1992).

[12] D.T. Haimo and C. Markett, A reprentation theory for solutions of a higher orderheat equation, II, J. Math. Anal. Appl. 168, 289–305, (1992).

[13] G. Maroscia, P.E. Ricci, Hermite-Kampe de Feriet polynomials and solutions ofBoundary Value Problems in the half-space, J. Concr. Appl. Math., (to appear).

[14] G. Maroscia, P.E. Ricci, Explicit solutions of multidimensional pseudo-classicalBVP in the half-space, Math. Comput. Modelling, 40 (2004), 667–698.

[15] P.E. Ricci, Le Funzioni Pseudo-Iperboliche e Pseudo-trigonometriche, Pubbl. Ist.Mat. Appl. Fac. Ingegneria Univ. Stud. Roma, 12, 37–49, (1978).

[16] H.M. Srivastava, H.L. Manocha, A treatise on generating functions, Wiley, NewYork, 1984.

[17] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York,1971.

[18] D.V. Widder, The Heat Equation, Academic Press, New York, 1975.

EXPONENTIAL OPERATORS...EVOLUTION PROBLEMS 45

46

Newton sum rules of the zeros of some associated

orthogonal q-polynomials

C.G.Kokologiannaki1, P.D.Siafarikas2 and I.D.Stabolas3,4

Department of Mathematics, University of Patras, 26500 Patras, Greece

e-mails:1chrykok@math.upatras.gr, 2panos@math.upatras.gr, 3stabol@math.upatras.gr

Abstract

By using a functional analytic method we evaluate the Newton sum rules ofthe zeros of some classes of associated orthogonal q-polynomials, in terms of thecoefficients of the three-term recurrence relation, which they satisfy. As particularcases we obtain the Newton sum rules of some associated orthogonal polynomialsfound recently.

Keywords: Newton sum rules, associated orthogonal q-polynomials.

MSC2000: 33C47, 33D45.

1 Introduction

In the mathematical physics literature the moments of the distribution of zeros of

polynomials have received a great deal of attention [8]. In order to know the distribution

of zeros of polynomials, it is desirable to learn as much as possible about the density of

their zeros. This can be done by studying the moments around the origin of the density

of zeros of polynomials from which valuable information can be derived [5].

The evaluation of the moments of zero distribution or equivalently the Newton sum

rules of the zeros of classical orthogonal polynomials have been studied in [2, 7, 8, 9, 11,

12, 18, 21, 23, 28], of semiclassical orthogonal polynomials in [22, 32], of associated and

co-recursive associated orthogonal polynomials in [18, 24, 29], of the scaled co-recursive

associated orthogonal polynomials in [16, 25], of quasiorthogonal polynomials of the clas-

sical class in [34] and of relativistic polynomials in [26].

Lately there has been an increasing interest in the orthogonal q-polynomials due

to their applications in various areas of mathematics and physics (see [1] and the re-

ferences there in). The classical orthogonal q-polynomials were introduced by Hahn

4Partially supported by the Greek State Scholarships Foundation (I.K.Y.)

47JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.1,47-65,2006,COPYRIGHT 2006 EUDOXUSPRESS,LLC

in 1949 [10] when in analogy to the classical orthogonal polynomials he was interested

in finding all the orthogonal polynomial sequences, the q-derivatives of which, defined

by Dqf(x) =f(x)− f(qx)

(1− q)x, are also orthogonal. Hahn obtained the first results as q-

hypergeometric series, after solving a Sturm-Liouville type equation in q-differences. In

1985, Andrews and Askey [4] continued the work of Hahn from the hypergeometric point

of view. They showed that all the classical orthogonal polynomials can be obtained as

limit cases of the q-Racah or the Askey-Wilson polynomials which are defined in terms

of basic hypergeometric series 4φ3. For more information regarding the q-polynomials

see [3, 19]. Recently various results were derived in [20, 27, 30] concerning the mono-

tonicity and convexity properties, and various inequalities regarding the zeros of some

classes of associated orthogonal q-polynomials. We recall that the associated orthogonal

q-polynomials are obtained from the orthogonal q-polynomials after replacing n by n + c

in the coefficients of the recurrence relation that they satisfy. In [1] the authors study

the discrete density of zeros (i.e the number of zeros per unit of zero interval) of the

q-polynomials and its asymptotic limit.

As far as the classical orthogonal polynomials are concerned, it is known that they

satisfy a second order differential equation while the corresponding associated and the co-

recursive associated ones satisfy a fourth order differential equation [29, 33]. This property

has been be used in [6, 7, 8, 13, 22, 23, 28, 29, 33, 34] to evaluate the moments of zero

distribution of these polynomials, in terms of the coefficients of the differential equation

that they satisfy. However, the orthogonal q-polynomials do not satisfy a differential

equation but a q-difference equation of the following form

σ(x)DqD1/qy(x) + τ(x)Dqy(x) + λq,ny(x) = 0,

where σ(x) and τ(x) are polynomials of at most second and first degree respectively.

In the case of the associated orthogonal q-polynomials, a functional analytic method

based on the three term recurrence relation that they satisfy, can be used to determine the

Newton sum rulesN−1∑n=0

λkn(c|q), k = 1, 2 . . . , of the zeros λn(c|q) of the polynomials under

consideration. This method has been introduced in [16] to obtain the Newton sum rules

of the scaled co-recursive associated polynomials. In the present paper using this method

we give the explicit expressions for the Newton sum rulesN−1∑n=0

λkn(c|q), k = 1, 2, 3, 4 of the

zeros of some families of associated orthogonal q-polynomials. We mention that by use

of this method it is possible, although tedious, to calculate the Newton sum rules for any

desired k.

KOKOLOGIANNAKI ET AL48

In section 2 we describe briefly the method we use and in section 3 we determine

the Newton sum rules for the zeros of the associated q-Laguerre, associated q-Charlier,

associated continuous q-Ultraspherical, q-Lommel, associated q-Meixner, and associated

Al-Salam-Carlitz I, II polynomials. From these, if we let q → 1−, we obtain as particular

cases the Newton sum rules for the zeros of the associated Laguerre, associated Charlier,

associated Ultraspherical and associated Meixner polynomials found recently [2, 12, 16,

18, 24, 25]. Also, from the Newton sum rules for the zeros of the q-Lommel polynomials,

we obtain the corresponding results for the zeros of q-Bessel functions and from them the

known Rayleigh sums [31] for the zeros of Bessel functions.

2 Preliminaries

A sequence Qn(x|q)∞n=0 of orthogonal q-polynomials satisfies a three term recurrence

relation of the form

αn(q)Qn+1(x|q) + βn(q)Qn−1(x|q) + bn(q)Qn(x|q) = xQn(x|q), n = 0, 1, 2, ...

Q−1(x|q) = 0, Q0(x|q) = 1,

with αn(q), βn(q), bn(q) real sequences and αn−1(q)βn(q) > 0. We will always assume that

0 < q < 1.

If we set Pn(x|q) = UnQn(x|q), where Un =√

αn−1(q)βn(q)

Un−1, U−1 = 0, U0 = 1, we

obtain the corresponding q-orthonormal polynomials Pn(x|q) which satisfy the recurrence

relation

an(q)Pn+1(x|q) + an−1(q)Pn−1(x|q) + bn(q)Pn(x|q) = xPn(x|q), (2.1)

P−1(x|q) = 0, P0(x|q) = 1,

where an(q) =√

αn(q)βn+1(q). Obviously the polynomials Pn(x|q) and Qn(x|q) have the

same zeros.

The q-associated orthonormal polynomials P(c)n (x|q) are obtained after replacing n by

n + c, for arbitrary real c ≥ 0 or c > −1 in the coefficients an(q) and bn(q) of (2.1), i.e.

they satisfy the following recurrence relation:

an+c(q)P(c)n+1(x|q) + an+c−1(q)P

(c)n−1(x|q) + bn+c(q)P

(c)n (x|q) = xP (c)

n (x|q), (2.2)

P(c)−1 (x|q) = 0, P

(c)0 (x|q) = 1.

The functional analytic method that we use is, briefly, the following:

NEWTON SUM RULES OF THE ZEROS... 49

Let en, n = 0, 1, ..., N−1 be an orthonormal base in a finite dimensional Hilbert space

HN , V the truncated shift operator V en = en+1, n = 0, 1, ..., N − 2 and V eN−1 = 0 and

V ∗ its adjoint V ∗en = en−1, n = 1, 2, ..., N − 1, V ∗e0 = 0. Also let A(c)(q), B(c)(q) be the

diagonal operators A(c)(q)en = an+c(q)en and B(c)(q)en = bn+c(q)en, n = 0, 1, ..., N − 1.

According to [14, 15] the zeros λn(c|q) of the polynomials P(c)n (x|q) defined by (2.2)

are the eigenvalues of the operator

T (c)(q) = A(c)(q)V ∗ + V A(c)(q) + B(c)(q), (2.3)

i.e.

(A(c)(q)V ∗ + V A(c)(q) + B(c)(q))xn(c|q) =λn(c|q)xn(c|q),‖xn(c|q)‖ = 1, n = 0, 1, ..., N − 1,

(2.4)

and vise versa.

In the following, we use an important result in the operator theory, for a symmetric

operator M in a finite dimensional real Hilbert space, e.g. the space HN : “The sumN−1∑n=0

(Men, en) is independent of the orthonormal base en, n = 0, 1, ...N − 1.” As in [16] it

can be proved that, if xn(c|q), n = 0, 1, ..., N − 1, is the complete orthonormal system of

eigenvectors of the operator T (c)(q) in HN , then

N−1∑n=0

((T (c))k(q)en,en) =N−1∑n=0

((T (c))k(q)xn(c|q), xn(c|q)) =N−1∑n=0

λkn(c|q). (2.5)

From (2.5), we obtain for k = 1, 2, 3, 4, correspondingly

N−1∑n=0

λn(c|q) =N−1∑n=0

bn+c(q), (2.6)

N−1∑n=0

λ2n(c|q) = 2

N−2∑n=0

a2n+c(q) +

N−1∑n=0

b2n+c(q), (2.7)

N−1∑n=0

λ3n(c|q) = 3

N−2∑n=0

a2n+c(q)bn+c(q) + 3

N−2∑n=0

a2n+c(q)bn+c+1(q) +

N−1∑n=0

b3n+c(q), (2.8)

andN−1∑n=0

λ4n(c|q) =

N−1∑n=0

b4n+c(q) + 2

N−2∑n=0

a4n+c(q) + 4

N−3∑n=0

a2n+c(q)a

2n+c+1(q)+

+ 4N−2∑n=0

a2n+c(q)b

2n+c(q) + 4

N−2∑n=0

a2n+c(q)b

2n+c+1(q) + 4

N−2∑n=0

a2n+c(q)bn+c(q)bn+c+1(q).

(2.9)

KOKOLOGIANNAKI ET AL50

3 Main results

3.1 Associated q-Laguerre polynomials

The associated q-Laguerre polynomials L(c)n (x; a|q) satisfy the recurrence relation (2.2)

with

an+c(q) =

√(1− qn+c+1)(1− qn+c+a+1)

q4n+4c+2a+3, bn+c(q) =

1− qn+c+1 + q − qn+c+a+1

q2n+2c+a+1,

for 0 < q < 1. From (2.6)-(2.9) we obtain, after some manipulations, the relations:

N−1∑n=0

λn(c|q) =(1 + q)q−2c−a−1

N−1∑n=0

q−2n − (1 + qa)q−c−a

N−1∑n=0

q−n =

=1− q2N − (1 + qa)qc+N(1− qN)

q2c+2N+a−1(1− q),

(3.1)

N−1∑n=0

λ2n(c|q) = 2q−4c−2a−3

N−2∑n=0

q−4n − 2(1 + qa)q−3c−2a−2

N−2∑n=0

q−3n+

+ 2q−2c−a−1

N−2∑n=0

q−2n + (1 + q)2q−4c−2a−2

N−1∑n=0

q−4n+

+ (1 + qa)2q−2c−2a

N−1∑n=0

q−2n − 2(1 + q)(1 + qa)q−3c−2a−1

N−1∑n=0

q−3n =

=(−q − 2q2 + 2q4N + 2q1+c+N + 2q2+c+N + 2q1+a+c+N + 2q2+a+c+N−− q1+2c+2N − 2q1+a+2c+2N − 2q2+a+2c+2N − q1+2a+2c+2N + q1+4N−− 2qc+4N − 2q1+c+4N − 2qa+c+4N − 2q1+a+c+4N + q1+2c+4N+

+2qa+2c+4N + 2q1+a+2c+4N + q1+2a+2c+4N)/((q2 − 1)q2a+4c+4N−1

),

(3.2)

N−1∑n=0

λ3n(c|q) = −(3 + 3q + 3q2 + q3)

q6c+3a (1− q3)+

(1 + 3q + 3q2 + 3q3)

q6N+6c+3a−3(1− q3)+

+3 (1 + q) (1 + qa)

q5c+3a(1− q)− 3 (1 + q) (1 + qa)

q5N+5c+3a−3(1− q)+

+3 (1 + 2qa + q2a + q1+a)

q4N+4c+3a−3(1− q)− 3 (q + qa + 2q1+a + q1+2a)

q4c+3a(1− q)−

− (1 + 3qa + 3q2a + q3a + 3q1+a + 3q2+a + 3q1+2a + 3q2+2a)

q3N+3c+3a−3 (1− q3)+

+(q2 + 3qa + 3q2a + 3q1+a + 3q2+a + 3q1+2a + 3q2+2a + q2+3a)

q3c+3a−1 (1− q3),

(3.3)

NEWTON SUM RULES OF THE ZEROS... 51

and

N−1∑n=0

λ4n(c|q) =− (4 + 4q + 8q2 + 8q3 + 6q4 + 4q5 + q6)

q8c+4a+2 (1− q4)+

+(1 + 4q + 6q2 + 8q3 + 8q4 + 4q5 + 4q6)

q8N+8c+4a−4(1− q4)−

4 (1 + 2q + q2 + q3) (1 + qa)

q7N+7c+4a−4(1− q)+

4 (1 + q + 2q2 + q3) (1 + qa)

q7c+4a+2(1− q)+

4 (1 + qa) (q2 + qa + 2q1+a + 2q2+a + q2+2a)

q5c+4a+1(1− q)−

4 (1 + 3qa + 3q2a + q3a + 2q1+a + q2+a + 2q1+2a + q2+2a)

q5N+5c+4a−4(1− q)+

2 (3 + 6q + 4q2 + 2q3 + 6qa + 3q2a + 14q1+a + 12q2+a)

q6N+6c+4a−4 (1− q2)+

2 (6q3+a + 2q4+a + 6q1+2a + 4q2+2a + 2q3+2a)

q6N+6c+4a−4 (1− q2)−

2 (2q + 4q2 + 6q3 + 3q4 + 2qa + 6q1+a + 12q2+a)

q6c+4a+2 (1− q2)−

2 (14q3+a + 6q4+a + 2q1+2a + 4q2+2a + 6q3+2a + 3q4+2a)

q6c+4a+2 (1− q2)+

(1 + 4qa + 6q2a + 4q3a + q4a + 4q1+a + 4q2+a + 4q3+a + 8q1+2a)

q4N+4c+4a−4 (1− q4)+

(10q2+2a + 8q3+2a + 4q4+2a + 4q1+3a + 4q2+3a + 4q3+3a)

q4N+4c+4a−4 (1− q4)−

(q4 + 4q2a + 4q1+a + 4q2+a + 4q3+a + 4q4+a + 8q1+2a + 10q2+2a)

q4c+4a (1− q4)−

(8q3+2a + 6q4+2a + 4q1+3a + 4q2+3a + 4q3+3a + 4q4+3a + q4+4a)

q4c+4a (1− q4)

(3.4)

respectively.

Remark 3.1.1. If we divide (3.1) by 1 − q, (3.2) by (1 − q)2, (3.3) by (1 − q)3, (3.4) by

(1− q)4 and let q → 1−, we obtain the Newton sum rules for the zeros of the associated

Laguerre polynomials, which have been found in [16, 24]. Also, for c = 0 we obtain the

Newton sum rules for the Laguerre polynomials which were given in [2].

KOKOLOGIANNAKI ET AL52

3.2 Associated q-Charlier polynomials

The associated q-Charlier polynomials C(c)n (q−x; a|q) and the associated q-Laguerre

polynomials L(c)n (x; a|q) are related in the following way

L(c)n (x; a|q) =

C(c)n (−x;−q−a|q)

(q; q)n

.

Hence, if λLn(a, c|q) and λC

n (a, c|q) are the zeros of the associated q-Laguerre and asso-

ciated q-Charlier polynomials respectively, it holds

N−1∑n=0

(λL

n(− ln(−a)

ln q, c|q)

)k

=N−1∑n=0

(−q−λCn (a,c|q))k, k = 1, 2, . . . . (3.5)

By using relation (3.5) we can find the following sumsN−1∑n=0

ykn(c; a|q), were yn(c; a|q) =

q−λCn (c;a|q) − 1, for the associated q-Charlier polynomials. From these, by setting α(1− q)

instead of a, dividing by (1 − q)k and taking the limit q → 1− we get the Newton sum

rules for the associated Charlier polynomials, which for k = 1, 2, 3, 4 are the following:

N−1∑n=0

λn(α, c) =1

2N (N + 2c + 2α− 1) , (3.6)

N−1∑n=0

λ2n(α, c) =α2N +

N (1 + 6 (c− 1) c− 3N + 6cN + 2N2)

6+

+ 2α ((N − 1) N + c (2N − 1)) ,

(3.7)

N−1∑n=0

λ3n(α, c) =α3N +

N (−1 + 2c + N) (2c2 + 2c (−1 + N) + (−1 + N) N)

4+

+ 3α((−1 + N)2N + c2 (−2 + 3N) + c (−1 + N) (−1 + 3N)

)+

+3α2 (3 (−1 + N) N + c (−4 + 6N))

2,

(3.8)

and

NEWTON SUM RULES OF THE ZEROS... 53

N−1∑n=0

λ4n(α, c) = α4N +

((c− 1)2c2 − 1

30

)N + (c− 1) c (2c− 1) N2+

+(1 + 6 (c− 1) c) N3

3+

(c− 1

2

)N4 +

N5

5+ 4α3 (2 (N − 1) N + c (4N − 3)) +

+ 4α((N − 1)3N + c3 (4N − 3) + c(N − 1)2 (4N − 1) + c2

(3− 9N + 6N2

))+

+ 2α2((−1 + N) N (−7 + 6N) + c2 (−17 + 18N) + 2c (3 + N (−13 + 9N))

)

(3.9)

Remark 3.2.1. For c = 0, from relations (3.6)-(3.9) we obtain the Newton sum rules for

the zeros of Charlier polynomials, which were also given in [2].

3.3 Associated continuous q-Ultraspherical polynomials

The associated continuous q-Ultraspherical polynomials satisfy the recurrence relation

(2.2) with

an+c(q) =1

2

√(1− qn+c+1)(1− λ2qn+c)

(1− λqn+c)(1− λqn+c+1)bn+c(q) = 0.

From (2.6) and (2.8) we have

N−1∑n=0

λn(c|q) =N−1∑n=0

λ3n(c|q) = 0.

From (2.7) and (2.9), using the relation

(1− qn+c+1)(1− λ2qn+c)

(1− λqn+c)(1− λqn+c+1)= 1 +

qn+c(1− λ)(λ− q)

(1− λqn+c)(1− λqn+c+1)=

= 1 +(1− λ)(λ− q)

λ(1− q)

[1

1− λqn+c− 1

1− λqn+c+1

],

(3.10)

after some manipulations we obtain:

N−1∑n=0

λ2n(c|q) =

1

2

N−1∑n=0

(1− qn+c+1)(1− λ2qn+c)

(1− λqn+c)(1− λqn+c+1)=

=N − 1

2+

(1− λ) (λ− q)

2λ(1− q)

[1

1− λqc− 1

1− λqc+N−1

] (3.11)

KOKOLOGIANNAKI ET AL54

and

N−1∑n=0

λ4n(c|q) = 2

1

42

N−2∑n=0

1 +

(1− λ)(λ− q)

λ(1− q)

[1

1− λqn+c− 1

1− λqn+c+1

]2

+

+41

42

N−3∑n=0

1 +

(1− λ)(λ− q)

λ(1− q)

[1

1− λqn+c− 1

1− λqn+c+1

]

1 +

(1− λ)(λ− q)

λ(1− q)

[1

1− λqn+c+1− 1

1− λqn+c+2

]=

=3N − 5

8+

(1− λ)2 (λ− q)2

8λ2(1− q)2×

[(1

1− λqc+N−2− 1

1− λqc+N−1

)2

+1

(1− λqc)2− 1

(1− λqc+N−2)2

]

+(1− λ) (λ− q)

4λ(1− q)

[1

1− λqc+1− 2

1− λqc+N−1+

2

1− λqc− 1

1− λqc+N−2

]

+(1− λ)2 (λ− q)2

4λ2(1− q)2(1− q2)

[q

1− λqc+1− 1

1− λqc+

1

1− λqc+N−2− q

1− λqc+N−1

].

(3.12)

Remark 3.3.1. Setting in (3.11) and (3.12) qλ instead of λ and taking the limit q → 1−,

we find the Newton sum rules for the associated Ultraspherical polynomials, which can

also be obtained, as special cases (a = b = λ− 1/2) from the corresponding formulas for

the associated Jacobi polynomials which are given in [16, 24, 25].

Remark 3.3.2. The associated continuous q-Legendre polynomials P(c)n (x|q) are related to

the associated continuous q-Ultraspherical polynomials C(c)n (x; λ|q) in the following way

C(c)n (x; q

12 |q) = q−

14nP (c)

n (x|q).

Therefore, the Newton sum rules for the zeros of the associated continuous q-Legendre

polynomials can be obtained from the corresponding Newton sum rules for the zeros of the

associated continuous q-Ultraspherical polynomials by setting λ = q12 . Moreover, taking

the limit q → 1− we get the Newton sum rules for the zeros of the associated Legendre

polynomials, which for c = 0 (Legendre polynomials) were also given in [2].

Remark 3.3.3. The continuous associated q-Hermite polynomials can been obtained from

the continuous associated q-Ultraspherical polynomials, by setting λ = 0. Thus the

Newton sum rules for the zeros of the continuous associated q-Hermite polynomials are

obtained from (3.11) and (3.12) for λ = 0. From these, by setting x√

1−q2

instead of x and

letting q → 1− we obtain the Newton sum rules for the associated Hermite polynomials

NEWTON SUM RULES OF THE ZEROS... 55

found in [12, 16, 18, 24]. Moreover, for c = 0 we find the Newton sum rules of the zeros

of Hermite polynomials which were also given in [2].

3.4 q-Lommel polynomials

The q-Lommel polynomials satisfy the recurrence relation (2.2) with

an(ν|q) =1

2

√qν+n

(1− qν+n+1)(1− qν+n), bn(ν|q) = 0.

From (2.6) and (2.8) we obtain

N−1∑n=0

λn(ν|q) =N−1∑n=0

λ3n(ν|q) = 0.

Also, from (2.7), using the relation

qn+ν

(1− qn+ν+1)(1− qν+n)=

1

1− q

[1

1− qν+n− 1

1− qν+n+1

], (3.13)

we obtainN−1∑n=0

λ2n(ν|q) =

1

2(1− q)

[1

1− qν− 1

1− qν+N−1

]=

qν(1− qN−1)

2(1− q)(1− qν)(1− qν+N−1). (3.14)

Also, from (2.9) using (3.13) and the relations

1

(1− qν+n+1)(1− qν+n+2)=

1

1− q

[1

1− qν+n+1− q

1− qν+n+2

],

1

(1− qν+n)(1− qν+n+2)=

1

1− q2

[1

1− qν+n− q2

1− qν+n+2

],

after some manipulations we obtain

N−1∑n=0

λ4n(ν|q) =

1

23

q2ν+2(N−2)

(1− qν+N−1)2(1− qν+N−2)2+

+1

23

N−3∑n=0

[q2(ν+n)

(1− qν+n+1)2(1− qν+n)2+

2q2(ν+n)+1

(1− qν+n)(1− qν+n+1)2(1− qν+n+2)

]=

=1

23

q2ν+2(N−2)

(1− qν+N−1)2(1− qν+N−2)2+

+1

23

[1

(1− qν)2− 1

(1− qν+N−2)2+

2

1− q2

(1

1− qν+N−2− 1

1− qν

)+

+2q

1− q2

(1

1− qν+1− 1

1− qν+N−1

)]1

(1− q)2

(3.15)

KOKOLOGIANNAKI ET AL56

Remark 3.4.1. It is known [17] that when N →∞ then λn(ν|q) → ± 1jν−1,n(q)

, where jν,n(q)

are the zeros of the q-Bessel function. Thus, from (3.14) and (3.15) for N →∞ we obtain

∞∑n=0

1

j2ν,n(q)

=qν+1

4(1− q)(1− qν+1), (3.16)

∞∑n=0

1

j4ν,n(q)

=1

23

[1

(1− qν+1)2− 2

(1 + q)(1− qν+1)(1− qν+2)+

1− q

1 + q

]1

(1− q)2. (3.17)

If we multiply (3.16) by (1 − q)2, (3.17) by (1 − q)4 and let q → 1−, we obtain the

known [31] Rayleigh sums for the zeros jν,n of Bessel functions.

3.5 Associated q-Meixner polynomials

The associated q-Meixner polynomials satisfy the recurrence relation (2.2) for

x = q−x − 1 and

an(r, b, c|q) =

√r(1− qn+c+1)(r + qn+c+1)(1− bqn+c+1)

q4n+4c+3,

bn(r, b, c|q) = q−n−c + r[1 + q − (1 + b)qn+c+1]q−2n−2c−1 − 1,

where c > −1, r > 0 and b < q−c−1. Setting yn(r, b, c|q) = q−λn(r,b,c|q) − 1 and after some

manipulations, from (2.6) - (2.8) we obtain

N−1∑n=0

yn(r, b, c|q) =N−1∑n=0

bn(r, b, c|q) =N−1∑n=0

(−1) + q−c

N−1∑n=0

q−n+

+ rq−2c−1

N−1∑n=0

q−2n + rq−2c

N−1∑n=0

q−2n − r(1 + b)q−c

N−1∑n=0

q−n =

=−N +q−c−N+1

1− q[(1− qN)(1− r − rb) + rq−c−N(1− q2N)],

(3.18)

N−1∑n=0

y2n(r, b, c|q) = 2

N−2∑n=0

a2n(r, b, c|q) +

N−1∑n=0

b2n(r, b, c|q) = 2r2q−4c−3

N−2∑n=0

q−4n+

+ 2r(1− r − rb)q−3c−2

N−2∑n=0

q−3n − 2r(1 + b− rb)q−2c−1

N−2∑n=0

q−2n + 2rbq−c

N−2∑n=0

q−n+

NEWTON SUM RULES OF THE ZEROS... 57

+N−1∑n=0

1 + r2(1 + q)2q−4c−2

N−1∑n=0

q−4n + 2r(1 + q)(1− r − rb)q−3c−1

N−1∑n=0

q−3n

+ (1− r − rb)2q−2c

N−1∑n=0

q−2n − 2r(1 + q)q−2c−1

N−1∑n=0

q−2n + 2(1− r − rb)q−c

N−1∑n=0

q−n =

= N − q−4c+1(2 + q)r2

1− q2+

q−4N−4c+2(1 + 2q)r2

1− q2− 2q−3c+1r(1− r − rb)

1− q+

+2q−3N−3c+2r(1− r − rb)

1− q+

2q−c+1(1− r − 2rb)

1− q−

− 2q−N−c+1(1− r − rb− rbq)

1− q−

− q−2c+1(q − 4r − 2br − 4qr − 2bqr + 2br2 + qr2 + 2bqr2 + b2r2q)

1− q2+

+q−2N−2c+1(q − 2(1 + q)(1 + q + bq)r + q((1 + b)2 + 2bq)r2)

1− q2,

(3.19)

and

N−1∑n=0

y3n(r, b, c|q) = A + Br + Γr2 + ∆r3, (3.20)

where

A =−N − 3q1−c(1− q−N)

1− q+

3q2−2c(1− q−2N)

(1− q2)− q3−3c(1− q−3N)

(1− q3),

B =3q1−4c−4N

1− qq2 − q4N + qc+N

[(3 + b) q3N − q (2 + q + bq)

]+ q3(c+N)

[−1 + qN+

+ b(−1− 2q + 3qN

)]+ q2(c+N)

[1− 3 (1 + b) q2N + q (2 + b (2 + q))

],

Γ =3 (1 + q)

(−1 + q) q5c− 3q3−5c−5N (1 + q)

−1 + q+

3 (b + 3q + 5bq + b2q)

(−1 + q) q3c−

− 3q1−2c (3b + b2 + q + 3bq + 2b2q)

(−1 + q) (1 + q)− 3q2−3c−3N (2 + 2b + q + 3bq + b2q + bq2)

−1 + q−

− 3 (1 + b + 5q + 3bq + 3q2 + 2bq2)

(−1 + q) q4c (1 + q)+

+3q2−2c−2N (1 + 2b + b2 + 3bq + b2q + bq2 + b2q2)

(−1 + q) (1 + q)+

+3q2−4c−4N (1 + 4q + 2bq + 3q2 + 3bq2 + q3 + bq3)

(−1 + q) (1 + q),

KOKOLOGIANNAKI ET AL58

and

∆ =−3 (1 + b) (1 + q)

(−1 + q) q5c+

3 (1 + b) q3−5c−5N (1 + q)

−1 + q− 3q3−4c−4N (1 + 2b + b2 + bq)

−1 + q+

+3 (b + q + 2bq + b2q)

(−1 + q) q4c− (1 + b) q1−3c (3b + 3bq + q2 + 2bq2 + b2q2)

(−1 + q) (1 + q + q2)+

+q3−3c−3N (1 + 3b + 3b2 + b3 + 3bq + 3b2q + 3bq2 + 3b2q2)

(−1 + q) (1 + q + q2)+

+3 + 3q + 3q2 + q3

(−1 + q) q6c (1 + q + q2)− q3−6c−6N (1 + 3q + 3q2 + 3q3)

(−1 + q) (1 + q + q2).

Remark 3.5.1. Setting qβ−1, andµ

1− µinstead of b and r respectively and taking the limit

q → 1− in (3.18) - (3.20), we obtain the following Newton sum rules for the zeros of the

associated Meixner polynomials.

N−1∑n=0

λn(β, µ, c) =N (1−N − (N − 1 + 2 β) µ− 2 c (1 + µ))

2 (µ− 1), (3.21)

N−1∑n=0

λ2n(β, µ, c) =

1

6(−1 + µ)2

[N

(1 + 6 (−1 + c) c− 3N + 6cN + 2N2

)

+ 2(6c2 (2N − 1) + 6c (2N − 1) (N + β − 1) + (N − 1) N (4N + 6β − 5)

)µ

+N(1 + 6c2 + 2N2 + 6 (−1 + β) β + 6c (−1 + N + 2β) + N (−3 + 6β)

)µ2

],

(3.22)

N−1∑n=0

λ3n(β, µ, c) =

1

4(1− µ)3

[(N(2c + N − 1)(2c2 + 2c(N − 1) + (N − 1)N)+

3(4c3(3N − 2) + 2c2(3N − 2)(3N + 2β − 3) + (N − 1)2N(3N + 4β − 4)+

c(N − 1)(2− 2β + 3N(2N + 2β − 3)))µ + 3(4c3(−2 + 3N) + 2c2(−2 + 3N)

× (3N + 4β − 3) + (N − 1)N(4 + N(3N − 7)− 10β + 8Nβ + 6β2)+

2c(6N3 + 6β − 2− 4β2 + 3N2(4β − 5) + N(11 + 6(β − 3)β)))µ2+

N(2c + N + 2β − 1)(2c2 + N2 + 2(β − 1)β+

N(2β − 1) + 2c(N + 2β − 1))µ3)].

(3.23)

Also, from the sumN−1∑n=0

(q−λn(r,b,c|q) − 1)4, which can be obtained in a similar way, but

which we omit due to its complexity and lack of space, we get the following Newton sum

rule of the fourth power of the zeros of the associated Meixner polynomials

NEWTON SUM RULES OF THE ZEROS... 59

N−1∑n=0

λ4n(β, µ, c) =

1

30(−1 + µ)4A + Bµ + Γµ2 + ∆µ3 + Eµ4, (3.24)

where

A =N(−1 + 30c4 + 60c3 (−1 + N) + 30c(−1 + N)2N+

30c2 (−1 + N) (−1 + 2N) + N2 (10 + 3N (−5 + 2N))),

B =4[30c4 (−3 + 4N) + 30c3 (−3 + 4N) (−2 + 2N + β) +

(−1 + N) N(−31 + 89N − 81N2 + 24N3 + 30(−1 + N)2β

)+

30c(−1 + N)2 (1− β + 2N (−3 + 2N + 2β)) +

30c2 (−1 + N) (4− 3β + N (−13 + 8N + 6β))],

Γ =610c4 (−17 + 18N) + 20c3 (−17 + 18N) (−1 + N + β) +

(−1 + N) N [−79 + N (181 + N (−139 + 36N)) + 150β+

10N (−23 + 9N) β + 10 (−7 + 6N) β2]+ 10c2

[−25 + 36N3+

(40− 17β) β + 6N (−4 + β) (−4 + 3β) + 3N2 (−35 + 18β)]+

20c (−1 + N + β) (−4 + 3β + N (22− 13β + N (−26 + 9N + 9β))),

∆ =430c4 (−3 + 4N) + 30c3 (−3 + 4N) (−2 + 2N + 3β) +

30c[(−1 + N)2 (

1− 6N + 4N2)

+ 2 (−1 + N)(2− 9N + 6N2

)β+

6 (−1 + N) (−1 + 2N) β2 + (−3 + 4N) β3]+ 30c2

[8N3 − (2− 3β)2+

3N2 (−7 + 6β) + N (17 + 6β (−5 + 2β))]+ (−1 + N) N×

[−31 + 24N3 + 9N2 (−9 + 10β) +

30β (4 + β (−5 + 2β)) + N (89 + 30β (−7 + 4β))],and

E = N[−1 + 10N2 − 60cN2 + 60c2N2 − 15N3 + 30cN3 + 6N4 − 60N2β + 120cN2β+

30N3β + 60N2β2 + 30(−1 + c + β)2(c + β)2+

30N (−1 + c + β) (c + β) (−1 + 2c + 2β)] .

Moreover for c = 0 from (3.21)-(3.24), we obtain the Newton sum rules for the Meixner

polynomials. The first three of them were also given in [2].

KOKOLOGIANNAKI ET AL60

Remark 3.5.2. The associated q-Charlier polynomials can be obtained from the associated

q-Meixner polynomials for b = 0 and r = a. Thus from (3.18)-(3.20) we immediately find

the corresponding formulas for the associated q-Charlier polynomials. Moreover, setting

a = α(1 − q) and taking the limit q → 1− we obtain the Newton sum rules for the

associated Charlier polynomials (see (3.6)-(3.9)).

Remark 3.5.3. The associated Al-Salam-Carlitz II polynomials can be obtained from the

associated q-Meixner polynomials by setting b = −t−1a and taking the limit t → 0+.

Then from (3.18)-(3.20), after some manipulations, we find

N−1∑n=0

λn(a, c|q) =a + 1

qc

N−1∑n=0

q−n =(a + 1)(1− qN)

qc+N−1(1− q), (3.25)

N−1∑n=0

λ2n(a, c|q) =

2a

q2c+1

N−2∑n=0

1− qn+c+1

q2n+

(a + 1)2

q2c

N−1∑n=0

q−2n =

=2a(1− q2N−2)− 2a(1 + q)(1− qN−1)qc+N+1 + (a + 1)2(1− q2N)q

q2c+2N−3(1− q2),

(3.26)

N−1∑n=0

λ3n(a, c|q) =− 3a(1 + a)

(1− q)q2N+2c−3+

(1 + a)((1 + a)2 + 3aq(1 + q))

q3N+3c−3(1− q3)−

− (1 + a)(3a(1 + q + qc+1) + q2(1 + a)2 − 3aqc(1 + q)2)

q3c−1(1− q3).

(3.27)

Also, from the formula ofN−1∑n=0

(q−λn(r,b,c|q) − 1)4, we find

N−1∑n=0

λ4n(a, c|q) =

(1 + a)4q4−4c−4N(1− q4N

)

1− q4− 2aq−4c−4N

1− q4−aq6 − 2aq8+

+ q4N(2a + aq2

)+ 2(1 + a)2 (

1 + q + q2) (−q5 + q1+4N

)+

+ aq2(c+N)(q + q3

) (q2N (2 + q)− q3 (1 + 2q)

)+ 2qc+N (1 + q)×

× (1 + q2

) (−q3N(a + (1 + a)2q

)+ q4 (1 + a (2 + a + q))

).

(3.28)

3.6 Associated Al-Salam Carlitz I polynomials

The associated Al-Salam Carlitz I polynomials satisfy the recurrence relation (2.2)

with

NEWTON SUM RULES OF THE ZEROS... 61

an+c(a|q) =√−a(1− qn+c+1)qn+c, bn+c(a|q) = (1 + a)qn+c, a < 0.

From equations (2.6)-(2.9) we find

N−1∑n=0

λn(a, c|q) =qc(1 + a)(1− qN)

1− q,

N−1∑n=0

λ2n(a, c|q) =

q2c(1 + a)2(1− q2N)

1− q2+

2aq2c+1(1− q2N−2)

1− q2− 2aqc(1− qN−1)

1− q,

N−1∑n=0

λ3n(a, c|q) = −3a(1 + a)q2c−2(q2 − q2N + q3N+c)

1− q−

− q3N+3c(a2 − a + 1)(a + 1)

1− q3+

(1 + a)q3c [(1 + a)2 + 3aq(1 + q)]

1− q3,

and

N−1∑n=0

λ4n(a, c|q) =

4aq3N+3c−4 (a + q + 2aq + a2q)

1− q− 2a2 q2N+2c−3 (2 + q)

1− q2−

− q4(N+c−1)[4a2 + 4a(1 + a)2q + 2a (2 + a) (1 + 2a) q2 + 4a(1 + a)2q3 + (1 + a)4q4

]

1− q4+

+q2c [q2c + a4q2c − 4aqc (1 + q) (1 + q2) (1− qc)− 4a3qc (1 + q) (1 + q2) (1− qc)]

1− q4−

− q2c2a2 (1 + q2) (1− qc) [1 + 2q − qc (3 + 2q (2 + q))]1− q4

.

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NEWTON SUM RULES OF THE ZEROS... 65

66

Kondurar Theorem and Ito formula

in Riesz spaces

A. Boccuto - D. Candeloro - E. Kubinska

Department of Mathematics and Computer Sciences,

via Vanvitelli 1, I-06123 Perugia (Italy)

e-mail:boccuto@dipmat.unipg.it, boccuto@yahoo.it;

Department of Mathematics and Computer Sciences,

via Vanvitelli 1, I-06123 Perugia (Italy)

e-mail:candelor@dipmat.unipg.it;

Nowy Sacz Business School- National-Louis University,

ul. Zielona 27, PL-33300 Nowy Sacz (Poland), and

Department of Mathematics, Silesian University,

ul. Bankowa 14, PL-40007 Katowice (Poland)

e-mail: kubinska@wsb-nlu.edu.pl

Abstract

In this paper we formulate a version of the Kondurar theorem and

a generalization of the Ito formula for functions taking values in Riesz

spaces with respect to a convergence, satisfying suitable axioms. When

the involved space is the space of measurable functions, both convergence

1

67JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.1,67-90,2006,COPYRIGHT 2006 EUDOXUSPRESS,LLC

almost everywhere and convergence in probability are included. Finally

we present some comments and possible applications of the Ito formula.

2000 AMS Subject Classification: 28B15, 28B05, 28B10, 46G10.

Key words: Riesz spaces, convergence, Riemann-Stieltjes integration, Kon-

durar theorem, Ito formula.

1 Introduction

In this paper we establish the Kondurar theorem and the Ito formula in Riesz

spaces.

The classical (scalar) version of the Kondurar theorem ensures the ex-

istence of the Riemann-Stieltjes integral∫ b

af dg, as soon as f and g are

Holder-continuous, of orders α and β respectively, with α+ β > 1.

Here we chose [a, b] = [0, 1], and fixed the Riesz space setting as a product

triple (R1, R2, R) such that f is R1-valued, g is R2-valued, and the integral

takes values in R.

More generally Riemann-Stieltjes integration is investigated also with re-

spect to an interval function q in the place of g: we obtain meaningful exten-

sions of the Kondurar theorem, and deduce a version of the Ito formula for in-

tegrands in a Riesz space. The classical stochastic integrals (Ito, Stratonovich,

Backward) and the classical Ito formula are included.

Finally, we present some examples in order to illustrate the possible appli-

cations.

Our warmest thanks to the referees for their helpful suggestions.

A.BOCCUTO ET AL68

2 Preliminaries

Definition 2.1 Throughout the paper, R will denote a Dedekind complete

Riesz space. An axiomatic notion of convergence in R is defined by choosing

a suitable order ideal N (in the sense of [6], p. 93) on the Riesz product

space IRIN . More precisely, this ideal N is assumed to satisfy the following

additional conditions:

a) Every element of N is a bounded sequence.

b) Every sequence in R whose terms eventually vanish belongs to N .

c) Every sequence of the form(

un

)n, with u ∈ R and u ≥ 0, belongs to N .

By means of the ideal N , it is possible to define the concept of convergence

for nets in the following manner. Let (T,≥) be any directed set. Given any

bounded net Ψ : T → R, we say that it N -converges to an element y of R

(or that it admits y as N -limit) if there exists a sequence (sn)n of elements

of T such that the sequence (pn)n defined by pn = supt∈T,t≥sn|ψ(t) − y|

belongs to N . The ideal N then coincides with the space of all bounded

sequences in R which converge to zero. For N -convergence all usual properties

of limits hold; in particular, uniqueness of the limit and Cauchy criterion for

convergence: a bounded net Ψ : T → R converges if and only if there exists

a sequence (sn)n of elements of T such that the sequence (qn)n defined by

qn = sups,t∈T,s∧t≥sn|Ψ(s)−Ψ(t)| belongs to N . Moreover, for a fixed directed

set T , the space of all bounded nets of the form Ψ : T → R for which the

limit exists is a Riesz subspace of RT , and the operator which assigns to each

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 69

convergent net the corresponding limit is linear and order-preserving.

Now, let us denote by J the family of all (nontrivial) closed subintervals

of the interval [0, 1], and by D the family of all finite decompositions of [0, 1],

D = 0 = t0 < t1 < . . . < tn−1 < tn = 1, or also D = I1, . . . , In, where

Ii = [ti−1, ti] for each i = 1, . . . , n.

Given any decomposition D = I1, . . . , In, the mesh of D is the number

δ(D) := max|Ii| : i = 1, . . . , n, where |I| as usual denotes the length of

the interval I. Then the set D can be endowed with the mesh ordering:

D1 ≥ D2 iff δ(D1) ≤ δ(D2); this makes D a directed set, and, unless otherwise

specified, convergence of any net Ψ : D → R will be always related to this

order relation. However, besides this order relation, a weaker ordering will be

helpful: for any two decompositions, D1, D2, we say that D2 is finer than D1,

and write D2 D1, if every interval from D2 is entirely contained in some

interval from D1. Moreover, we shall say that a decomposition D is rational if

all its endpoints are rational, and that a decomposition D is equidistributed if

all intervals in D have the same length. If D is equidistributed, and consists

of 2n intervals, we say that D is dyadic of order n. Equidistributed and dyadic

decompositions of any interval [a, b] are similarly defined. In general, the

family of all decompositions of any interval [a, b] ⊂ [0, 1] will be denoted by

D[a,b].

From now on, we shall assume that an ideal N has been fixed, according

with Definition 2.1, and consequently all limits we shall introduce (unless oth-

erwise specified) are related to N -convergence, without a particular notation.

A.BOCCUTO ET AL70

Definition 2.2 Let q : J → R be any interval function. We say that q is

integrable, if the net S(D) =∑I∈D

q(I) is convergent to some element Y :=∫q.

A very useful tool to prove integrability is the Cauchy property. We first

introduce a notation: for every interval function q : J → R, we set

OB(q)(I) = sup

∣∣∣∣∣∑J∈D

q(J)−∑

H∈D′

q(H)

∣∣∣∣∣ : D,D′ ∈ DI

.

Now we have the following:

Theorem 2.3 Assume that q : J → R is any interval function. The following

three conditions are equivalent:

(i) q is integrable;

(ii) (rn)n ∈ N , where

rn := sup|S(D)− S(D0)| : D,D0 ∈ D, δ(D0) ≤

1n,D D0

;

(iii)∫OB(q) = 0.

Proof. It is easy to prove that (i) implies (ii). To prove the implication (ii)⇒

(iii), one can proceed in a similar fashion as in the proof of Theorem 1.4 in [4].

So we only prove that (iii) implies (i). By supposition, we have (κn)n ∈ N ,

where κn = sup

∑I∈D

OB(q)(I) : δ(D) ≤ 1n

. Let us fix n, and choose two

decompositions, D0 and D, such that δ(D0) ≤1n

and D D0. From

S(D0)− S(D) =∑I∈D0

q(I)− ∑J∈D,J⊂I

q(J)

we obtain |S(D0)− S(D)| ≤ κn. From this one can easily deduce that

|S(D1)− S(D2)| ≤ 2κn, (1)

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 71

as soon as δ(D1) and δ(D2) are less than1n

. Now define

Mn = supS(D) : δ(D) ≤ 1

n

, mn = inf

S(D) : δ(D) ≤ 1

n

.

Of course, mn ≤Mn. Moreover, thanks to (1), it follows easily thatMn−mn ≤

2κn. Then infk∈IN Mk = supk∈IN mk, and now we prove that the common value

L is the integral: L =∫q. For every integer n, and each decomposition D0

with δ(D0) ≤1n

, we have:

s(D0)− L ≤ s(D0)−mn ≤Mn −mn ≤ 2κn;

L− s(D0) ≤Mn − s(D0) ≤Mn −mn ≤ 2κn.

Therefore |L− s(D0)| ≤ 2κn, and this completes the proof, by arbitrariness of

D0. 2

Our next goal is to prove that, if an interval function is integrable in [a, b],

then it is integrable in any subinterval, and the integral is an additive function.

Theorem 2.4 Let q : J → R be an integrable function. Then, for every

subinterval J ⊂ [0, 1], the function qJ is integrable, where qJ is defined as

qJ(I) = q(I ∩ J), as soon as I ∩ J is nondegenerate, and 0 otherwise. More-

over, if J1, J2 is any decomposition of some interval J ⊂ [0, 1], we have∫Jq =

∫J1

q +∫

J2

q. (Here, as usual,∫

Jq means the integral of qJ)

Proof. Fix any interval J ⊂ [0, 1], and, for every positive integer n, define

σn(J) := sup

∣∣∣∣∣∣∑

I∈D′J

q(I)−∑

H∈D′′J

q(H)

∣∣∣∣∣∣ : D′J , D

′′J ∈ DJ , δ(D′

J) ≤ 1n, δ(D′′

J) ≤ 1n

.It is easy to deduce that σn(J) ≤ σn([0, 1]). As (σn([0, 1]))n ∈ N , the same

holds for (σn(J))n. Therefore, by Theorem 2.3, it follows that qJ is integrable.

A.BOCCUTO ET AL72

To prove additivity, fix an interval J = [a, b], an integer n > 0, and a point

c ∈]a, b[. Now, choose any decomposition D0 of [a, b], such that δ(D0) ≤1n

,

and such that c is one of its intermediate points; next, denote by D′ the

decomposition consisting of the intervals from D0 that are contained in [a, c]

and by D′′ the decomposition consisting of the remaining intervals from D0.

Clearly we have ∣∣∣∣∣∫

Jq −

∫[a,c]

q −∫

[c,b]q

∣∣∣∣∣ =∣∣∣∣∣∣∫

Jq −

∑I∈D0

q(I) +∑I∈D′

q(I)−∫

[a,c]q +

∑I∈D′′

q(I)−∫

[c,b]q

∣∣∣∣∣∣ ≤ γn,

where (γn)n is a suitable element from N . By arbitrariness of n, we can infer

that∣∣∣∣∣∫

Jq −

∫[a,c]

q −∫

[c,b]q

∣∣∣∣∣ = 0, i.e. additivity of the integral. 2

A further consequence of the previous result is the following.

Theorem 2.5 Let q : J → R be any integrable function, and let us denote by

ψ its integral function, i.e. ψ(I) =∫

Iq, I ∈ J . Then the function |q−ψ| has

null integral.

Proof. Let us denote by Z the function Z = |q − ψ|. We must show that the

sequence (βn)n belongs to N , where βn = sup

∑I∈D

Z(I) : δ(D) ≤ 1n

. For

every n > 0, let us set σn := sup

∑I∈D

OB(q)(I) : δ(D) ≤ 1n

. Fix now an

integer n > 0, and a decomposition D0, with δ(D0) ≤1n

. For every element

I ∈ D0, we have q(I)−ψ(I) = limD∈DI(q(I)−S(D)) (intended as N -limit of

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 73

a net, depending on I), and therefore

∑I∈D0

Z(I) =∑I∈D0

| limD∈DI

(q(I)− S(D) )| ≤∑I∈D0

OB(q)(I) ≤ σn

according with all properties of convergence. But (σn)n ∈ N by Theorem 2.3,

hence also (βn)n ∈ N , and the proof is finished. 2

We now introduce some structural assumptions, which will be needed later.

Assumptions 2.6 Let R1, R2, R be three Dedekind complete Riesz spaces.

We say that (R1, R2, R) is a product triple if there exists a map · : R1×R2 → R,

which we will call product, such that

2.6.1) (r1 + s1) · r2 = r1 · r2 + s1 · r2,

2.6.2) r1 · (r2 + s2) = r1 · r2 + r1 · s2,

2.6.3) [r1 ≥ s1, r2 ≥ 0] ⇒ [r1 · r2 ≥ s1 · r2],

2.6.4) [r1 ≥ 0, r2 ≥ s2] ⇒ [r1 · r2 ≥ r1 · s2] for all rj , sj ∈ Rj , j = 1, 2;

2.6.5) if (aλ)λ∈Λ is any net in R2 and b ∈ R1, then [aλ ↓ 0, b ≥ 0] ⇒ [b ·aλ ↓ 0];

2.6.6) if (aλ)λ∈Λ is any net in R1 and b ∈ R2, then [aλ ↓ 0, b ≥ 0] ⇒ [aλ ·b ↓ 0].

A Dedekind complete Riesz space R is called an algebra if (R,R,R) is a product

triple.

Let now (R1, R2, R) be a product triple of Riesz spaces. Given a bounded

function g : [0, 1] → R2, we can associate with g its jump function ∆(g):

A.BOCCUTO ET AL74

∆(g)([a, b]) := g(b)− g(a). Moreover, the interval function

ω(g)(I) = sup|∆(g)([u, v])| : [u, v] ⊂ I

is called the oscillation of g in the interval I.

Definition 2.7 Let f : [0, 1] → R1 and q : J → R2 be two functions. In

order to properly define the Riemann-Stieltjes integral of f with respect to q,

we need a slight complication of the directed set D. In fact, we shall denote

by D the set of all pairs (D, τ), where D runs in D and τ is a mapping which

associates to every element I ∈ D an arbitrary point τI ∈ I. The set D will be

ordered according with the usual mesh ordering on D, i.e. (D, τ) ≤ (D1, τ1) iff

δ(D1) ≤ δ(D). Given an element (D, τ) ∈ D, we call Riemann-Stieltjes sum

of f with respect to q (and denote it by S(D, τ)) the following quantity:

S(D, τ) =∑I∈D

f(τI) q(I).

Definition 2.8 Assume that f : [0, 1] → R1 and q : J → R2 are two func-

tions. We say that f is Riemann-Stieltjes integrable with respect to q, if the

net (S(D, τ))(D,τ)∈D

is N -convergent, i.e. if there exists an element Y ∈ R,

such that the sequence (πn)n belongs to N , where

πn = sup|Y − S(D, τ)| : (D, τ) ∈ D, δ(D) ≤ 1

n

, n ∈ IN.

If this is the case, the element Y ∈ R is called the Riemann-Stieltjes integral

and denoted by Y := (RS)∫ 1

0f dq. When q = ∆(g), for some suitable func-

tion g : [0, 1] → R2, integrability of f with respect to q will be expressed by

saying that f is Riemann-Stieltjes integrable w.r.t. g, and its integral will be

denoted by (RS)∫ 1

0f dg.

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 75

3 The Kondurar theorem in Riesz spaces

Definition 3.1 Let f : [0, 1] → R1 and q : J → R2 be two fixed functions.

We say that f and q satisfy assumption (C) if the interval function φ : J → R,

defined as φ(I) = ω(f)(I)|q(I)|, is integrable, and its integral is 0.

The next result is easy.

Proposition 3.2 Let f : [0, 1] → R1 and q : J → R2 be two bounded func-

tions, satisfying assumption (C). Then the Riemann-Stieltjes integral

(RS)∫ 1

0f dq exists in R if and only if the interval function Q(I) = f(aI)q(I)

is integrable, where aI denotes the left endpoint of I: if this is the case, the

two integrals coincide.

In the next theorems, the concept of Holder-continuous function will be crucial:

Definition 3.3 Let R be an arbitrary Riesz space. We say that an interval

function q is Holder-continuous (of order γ) if there exist a unit u ∈ R, that

is an element R 3 u ≥ 0, u 6= 0, and a real constant γ > 0, such that

|q(I)| ≤ |I|γu for all I ⊂ [0, 1]. As usual, a function g : [0, 1] → R is said to

be Holder-continuous of order γ if the function ∆(g) is.

We shall assume in addition that the interval function q is additive, i.e. q([a, b]) =

q([a, c])+q([c, b]) as soon as 0 ≤ a < c < b ≤ 1. This means also that q = ∆(g)

for some suitable bounded function g : [0, 1] → R2, but we shall mantain our

notation, without mentioning g.

The next step is the following result.

A.BOCCUTO ET AL76

Proposition 3.4 Assume that f : [0, 1] → R1 and q : J → R2 are two

Holder-continuous functions, for which assumption (C) is satisfied. Suppose

also that q is additive, and that (ρn)n ∈ N , where

ρn = sup

∣∣∣∣∣∣∑J∈D

f(aJ)q(J)−∑I∈D0

f(aI)q(I)

∣∣∣∣∣∣ :

D 3 D0, D rational, δ(D0) ≤1n,D D0

for every n ∈ IN . Then, there exists in R the (RS)-integral of f w.r.t. q.

Proof. Since assumption (C) is satisfied, from Proposition 3.2 it’s enough to

prove that the R-valued interval function Q(I) := f(aI)q(I) is integrable. As

usual, for every decomposition D of [0, 1], we write S(D) =∑

I∈D Q(I) =∑I∈D f(aI) q(I). By using Holder-continuity and boundedness of f and q

(which is a simple consequence), one can obtain the following key tool, which

is rather technical, but not difficult.

Key Tool: There exists a unit w ∈ R such that, for every n ∈ IN and

every D ∈ D with δ(D) <1n

, one can find a rational decomposition D0 such

that δ(D0) <1n

and

|S(D)− S(D0)| ≤w

n. (2)

Let us now turn to the proof that Q is integrable. We first observe that

sup|S(D1)− S(D2)| : D1 and D2 rational , δ(D1) <

1n, δ(D2) <

1n

≤ 2ρn

for every n; hence, using the key tool above, we see also that

sup|S(D)− S(D′)| : D ∈ D, D′ ∈ D, δ(D) <

1n, δ(D′) <

1n

≤ 2ρn + 2

w

n

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 77

for every n. Now, the assertion follows from Theorem 2.3. 2

Unless otherwise specified, we always assume that q is additive, that f and

q are Holder-continuous of order α and β respectively, and related units u1

and u2 respectively. Moreover, we shall require that γ := α+ β > 1.

Lemma 3.5 Let [a, b] be any sub-interval of [0, 1], and let us denote by c its

midpoint. Then

|Q([a, b])− (Q([a, c]) +Q([c, b]))| ≤ u1 u2

(b− a

2

)γ

.

Proof. We have:

|Q([a, b])− (Q([a, c]) +Q([c, b])| = |f(a)q([a, b])− f(a)q([a, c])

− f(c)q([c, b])| = |f(a)q([c, b])− f(c)q([c, b])|

≤ |f(c)− f(a)|q([c, b])| ≤ u1 u2

(b− a

2

)α+β

. 2

Proposition 3.6 Let [a, b] be any subinterval of [0, 1], and assume that D is a

dyadic decomposition of [a, b], of order n. Then there exists a positive element

v ∈ R, independent of a, b and n, such that |Q([a, b])−∑I∈D

Q(I)| ≤ v |b− a|γ .

Proof. Let us consider the dyadic decompositions of [a, b] of order 1, 2, . . . , n

and denote them by D1, D2, . . . , Dn = D respectively. Then we have

|Q([a, b])−∑I∈D

Q(I)| ≤ |Q([a, b])−∑I∈D1

Q(I)|+n∑

i=2

|∑

J∈Di

Q(J)−∑

I∈Di−1

Q(I)|.

Thanks to Lemma 3.5, the first summand in the right-hand side is less than

u1 u2

(b− a

2

)γ

. By the same Lemma, for every index i from 2 to n, we have

A.BOCCUTO ET AL78

|∑

J∈Di

Q(J)−∑

I∈Di−1

Q(I)| ≤ u1 u22i−1

(b− a

2i

)γ

, hence

|Q([a, b])−∑I∈D

Q(I)| ≤ u1 u2

(b− a

2

)γ

+n∑

i=2

u1 u22i−1

(b− a

2i

)γ

=12u1 u2(b− a)γ

n∑i=1

(1

2γ−1

)i

≤ 12u1 u2(b− a)γ

∞∑i=1

(1

2γ−1

)i

=1

2(2γ−1 − 1)(b− a)γ u1 u2.

The assertion follows, by choosing v =1

2(2γ−1 − 1)u1 u2. 2

Proposition 3.7 Let [a, b] be any subinterval of [0, 1], and assume that D is

an equidistributed decomposition of [a, b], consisting of N elements. Then

|Q([a, b])−∑I∈D

Q(I)| ≤ r(b− a)γ

for some unit r ∈ R, which does not depend on N, D and [a, b].

Proof. Of course, if N = 2h for some integer h, the result follows from

Proposition 3.6. So, let us assume that N is not a power of 2, and write N

in dyadic expansion, N = e020 + e121 + . . . + eh2h, where (e0, e1, . . . , eh) is a

suitable element of 0, 1h+1, and h = [log2N ]. Clearly then, eh = 1. Now let

us define, for j = 1, 2, . . . , h:

t0 := a, th+1 := b, and tj := a+j∑

i=1

ei−1 2i−1 b− a

N.

Let us denote by D0 the decomposition of [a, b] whose intermediate points are

t0, t1, . . . , th+1. From Proposition 3.6, it follows:

|S(D0)− S(D)| ≤∑I∈D0

|Q(I)−∑

J∈D, J⊂I

Q(J)| ≤∑I∈D0

v|I|γ ≤

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 79

≤ v

h∑i=0

(2i b− a

N

)γ

≤ v(b− a)γ 2(h+1)γ

Nγ≤ v(b− a)γ2γ .

Thus, setting v1 := 2γv, we get |S(D0)− S(D)| ≤ v1(b− a)γ .

Let us now evaluate |Q([a, b])− S(D0)|. We find

|Q([a, b])− S(D0)| ≤ |(f(a)− f(t1))q([t1, t2])+

+(f(a)− f(t2))q([t2, t3]) + . . .+ (f(a)− f(th))q([th, b])| ≤

≤h∑

j=1

|f(a)− f(tj)| |q([tj , tj+1])| ≤h∑

j=1

u1 u2(tj − a)α(tj+1 − tj)β.

For all j = 1, 2, . . . , h we have easily tj − a ≤ 2j b−aN , hence

|Q([a, b])− S(D0)| ≤h∑

j=1

u1 u2

(2j b− a

N

)α(2j b− a

N

)β

≤ u1 u22γ(b− a)γ .

Thus, setting v2 := u1 u22γ , we get |Q([a, b])−S(D0)| ≤ v2(b−a)γ , and finally

we obtain the assertion by setting r = v1 + v2. 2

We are now ready for the main result.

Theorem 3.8 (The Kondurar Theorem) Let f : [0, 1] → R1 and q : J → R2

be Holder-continuous of order α and β respectively, with γ = α + β > 1, and

suppose that q is additive. Then f is (RS)-integrable w.r.t. q.

Proof. We first observe that the real valued interval function W (I) := |I|γ

is integrable, and its integral is null. This immediately implies that f and

q satisfy assumption (C). Let r be the unit in R given by Proposition 3.7,

and set bn :=2rn

. Fix n and choose any rational decomposition D0 such that

δ(D0) ≤1n

and∑J∈D

|J |γ ≤ 1n

for every decomposition D, finer than D0. If D

is any rational decomposition, finer than D0, then there exists an integer N

A.BOCCUTO ET AL80

such that the equidistributed decomposition D∗, consisting of N subintervals,

is finer than D. Of course, D∗ is also finer than D0, so we have, by Proposition

3.7:

|S(D∗)− S(D0)| ≤∑I∈D0

r|I|γ ≤ 1nr and |S(D∗)− S(D)| ≤

∑J∈D

r|J |γ ≤ 1nr.

Therefore, by axioms of 2.1, we obtain:

(sup |S(D)− S(D0)| : D,D0 are rational, δ(D0) ≤ 1/n, D D0)n ∈ N ,

and the theorem is proved, thanks to Proposition 3.4. 2

More concretely, we have the following version of the previous theorem.

Theorem 3.9 Let f : [0, 1] → R1 and g : [0, 1] → R2 be two Holder-

continuous functions, of order α and β respectively, and assume that α+β > 1.

Then f is (RS)-integrable with respect to g.

However, we can obtain a more general result.

Theorem 3.10 Assume that f : [0, 1] → R1 is any Holder-continuous func-

tion of order α, and q : J → R2 is any integrable interval function (not

necessarily additive). If the integral function ψ(I) =∫

Iq is Holder-continuous

of order β, and α+ β > 1, then f is (RS)-integrable with respect to q and to

ψ, and the two integrals coincide.

Proof. In 2.4 we proved that ψ is an additive function; hence, if it is Holder-

continuous of order β, and α+β > 1, we can deduce that f is (RS)-integrable

with respect to ψ, thanks to 3.8. Moreover, we also know from 2.5 that

|q − ψ| has null integral. This fact, together with boundedness of f , implies

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 81

that∫

Jf d(q − ψ) = 0, for every interval J ⊂ [0, 1]. Now the conclusion is

obvious. 2

Remark 3.11 Assume that W is the standard Brownian Motion, on a prob-

ability space (X,B, P ), and g : [0, 1] → L0(X,B, P ) is the function which

associates to each t ∈ [0, 1] the random variable Wt ∈ L0. We can endow

the Riesz space L0 with the convergence in probability (which satisfies all as-

sumptions in Definition 2.1); thus the interval function q(I) = (∆(g)(I))2 is

well-known (from the literature) to be integrable, and its integral function is

ψ(I) = |I|. So, Theorem 3.10 implies integrability with respect to q = ∆(g)2

of every function f which is Holder continuous (of any order), and also of more

general continuous functions (but we shall not discuss this here).

Remark 3.12 The previous existence theorems remain true, if the defini-

tion 2.8 is modified, replacing the value f(τI) with any element θI such that

inff(x) : x ∈ I ≤ θI ≤ supf(x) : x ∈ I. This mainly rests on Proposition

3.2 and on the fact that |θI − f(aI)| ≤ ω(f)(I).

4 The Ito formula in Riesz spaces

In [4] we can find some results about the Ito formula for Riesz space-valued

functions of one variable. Here we deal with functions of two variables; from

now on we shall assume that our Riesz spaceR is an algebra, hence the involved

functions and their integrals will take values in R.

Definition 4.1 Let R be an algebra, and f : [0, 1]×R→ R a fixed function.

A.BOCCUTO ET AL82

We say that f satisfies Taylor’s formula of order 1 if there exist two R-valued

functions, defined on [0, 1] × R and denoted by∂f

∂t,∂f

∂x, such that for every

(t, x) ∈ [0, 1]×R, h ∈ IR and k ∈ R, such that t+ h ∈ [0, 1], we have:

f(t+ h, x+ k) = f(t, x) +(h∂f

∂t+ k

∂f

∂x

)(t, x) + (|h|+ |k|) B(t, x, h, k),

where B is a suitable R-valued function, defined on all points of the type

(t, x, h, k) with t+ h ∈ [0, 1], and bounded on bounded sets.

Moreover, we say that f satisfies Taylor’s formula of order 2 if f satis-

fies Taylor’s formula of the first order, and moreover there exist three more

functions, denoted by∂2f

∂t2,∂2f

∂t∂x,∂2f

∂x2, such that for every (t, x) in [0, 1]×R,

h ∈ IR and k ∈ R, (t+ h ∈ [0, 1]), we have

f(t+ h, x+ k)− f(t, x) = h∂f

∂t(t, x) + k

∂f

∂x(t, x)

+12

[h2∂

2f

∂t2(t, x) + 2hk

∂2f

∂t∂x(t, x) + k2∂

2f

∂x2(t, x)

](3)

+[h2 + |hk|+ |k|2

]B(t, x, h, k),

where B is a suitable function, bounded on bounded sets.

We now introduce a ”weaker” concept of integrability for Riesz space-valued

functions. As above, N will always denote any fixed ideal as in 2.1.

Definitions 4.2 Let F : [0, 1] × R → R and g : [0, 1] → R be two functions.

Following the Stratonovich approach ([8]), for every λ ∈ [0, 1] we say that

F (·, g(·)) is 〈λ〉-integrable with respect to g (or to q(I) := ∆(g)(I)) if there

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 83

exists an element Yλ ∈ R such that

(sup

∣∣∣∣∣Yλ −∑I∈D

F (λui + (1− λ)vi, λg(ui) + (1− λ)g(vi))∆(g) (I)

∣∣∣∣∣ :δ(D) ≤ 1

n

)n

∈ N ,

where D = ([ui, vi]) : i = 1, . . . , N is the involved decomposition of [0, 1].

We denote the 〈λ〉-integral of a function F with respect to g by the symbol

〈λ〉∫ 1

0F dg.

Similarly, we say that the function F (·, g(·)) is 〈λ〉-integrable with respect to

q(I) := (∆(g)(I))2 if there exists an element Jλ ∈ R such that

(sup

∣∣∣∣∣Jλ −∑I∈D

F (λui + (1− λ)vi, λg(ui) + (1− λ)g(vi)) (∆(g)(I))2∣∣∣∣∣ :

δ(D) ≤ 1n

)n

∈ N ,

where D = ([ui, vi]) : i = 1, . . . , N is the involved decomposition of [0, 1].

In this case, we write 〈λ〉∫ 1

0F (s, g(s)) (dg)2(s) := Jλ(F ). We now prove the

following:

Proposition 4.3 Let R be an algebra, and f : [0, 1]×R→ R, (t, x) 7→ f(t, x),

satisfy Taylor’s formula of order 1; suppose also that g : [0, 1] → R is Holder-

continuous of order β, where β >13. If f is 〈1〉-integrable with respect to

q(I) := (∆(g)(I))2, i.e. there exists the integral J1(f), then for every λ ∈ [0, 1]

the integral Jλ(f) exists in R, and Jλ(f) = J1(f).

Proof. Let D = [ui, vi] , i = 1, . . . , n be a decomposition of the interval [0, 1]

and λ ∈ [0, 1]. By hypotheses, we have:

A.BOCCUTO ET AL84

∑ni=1 f (λui + (1− λ)vi, λg (ui) + (1− λ)g(vi))·q([ui, vi]) =

∑ni=1 f (ui, g (ui))·

q([ui, vi])+

+∑n

i=1 [f (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))− f(ui, g(ui))]·q([ui, vi]) =∑ni=1

[f (ui, g (ui)) + (1− λ) (vi − ui) ∂f

∂t (ui, g (ui)) +

(1− λ) (∆(g)([ui, vi]))∂f∂x (ui, g(ui))

]· q([ui, vi]) +

∑ni=1B([ui, vi])|vi − ui|τ =

V1 + V2 + V3 + V4, where B : I → R is a suitable interval function, bounded

on bounded sets (with bound independent on λ), and τ > 1 by virtue of

Holder-continuity of g (we denote the consecutive terms of the above sum by

V1, V2, V3, V4). The first part V1 tends to J1(f) as δ(D) tends to 0. Now we

can conclude, observing that the expressions V2, V3, V4 are negligible because

we can present them in the formn∑

i=1

B0([ui, vi]) |vi − ui|ζ , where B0 : J → R

is bounded on bounded sets and ζ > 1. 2

We now prove our generalization of the Ito formula in the context of Riesz

spaces:

Theorem 4.4 Let R be an algebra, F : [0, 1] × R → R satisfy Taylor’s for-

mula of order 2 and∂F

∂t: [0, 1]×R→ R,

∂2F

∂x2: [0, 1]×R→ R, satisfy Tay-

lor’s formula of order 1. Assume that g : [0, 1] → R is Holder-continuous of

order β > 13 . Assume also that q(I) = (∆(g)(I))2 is integrable and that its

integral function is Holder-continuous of order γ > 1− β. Then

1) for every λ ∈ [0, 1] the integral Jλ := 〈λ〉∫ 1

0

∂2F

∂x2(s, g(s)) (dg)2(s) exists

in R, and is independent on λ;

2) for every λ ∈ [0, 1] the integral 〈λ〉∫ 1

0

∂F

∂x(s, g(s)) dg(s) exists in R;

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 85

3) the following formula holds:

F (1, g (1))− F (0, g (0)) =

=∫ 1

0

∂F

∂t(s, g (s)) ds+ 〈λ〉

∫ 1

0

∂F

∂x(s, g (s)) dg (s) +

12

(2λ− 1) J,

where J denotes any of the integrals Jλ above.

Proof. The function t 7→ ∂2F

∂x2(t, g(t)) is Holder-continuous of order β, because

of Taylor’s formula and of Holder-continuity of g. Thus, using Theorem 3.10,

we deduce that the function t 7→ ∂2F

∂x2(t, g(t)) is (RS)-integrable with respect

to q, and thus by virtue of Proposition 4.3 (where f =∂2F

∂x2), for every λ ∈ [0, 1]

the integral J := 〈λ〉∫ 1

0

∂2F

∂x2(s, g(s)) (dg)2(s) exists in R and is independent

on λ. This shows the assertion 1).

Let now D = [ui, vi] , i = 1, . . . , n be a decomposition of [0, 1] and fix λ ∈

[0, 1]. We have:

F (1, g (1))− F (0, g (0)) =∑ni=1 [F (vi, g (vi))− F (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))]−∑ni=1 [F (ui, g (ui))− F (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))]. Now we shall

apply Taylor’s formula of order 2:

F (1, g (1))− F (0, g (0)) =

∑n

i=1∂F∂t (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · [λ (vi − ui)]+∑n

i=1∂F∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · [λ (∆(g)([ui, vi]))] +∑n

i=1∂2F∂t∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))

·[λ2 (vi − ui) (∆(g)([ui, vi]))

]+∑n

i=1∂2F∂t2

(λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·[

12λ

2 (vi − ui)2]+

A.BOCCUTO ET AL86

∑ni=1

∂2F∂x2 (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·

[12λ

2 q([ui, vi])]+∑n

i=1B1 ([ui, vi]) |vi − ui|γ−

∑n

i=1∂F∂t (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · [(λ− 1) (vi − ui)]+∑n

i=1∂F∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · [(λ− 1) (∆(g)([ui, vi]))] +∑n

i=1∂2F∂t∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))

·[(1− λ)2 (vi − ui) (∆(g)([ui, vi]))

]+∑n

i=1∂2F∂t2

(λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·[

12 (1− λ)2 (vi − ui)

2]+∑n

i=1∂2F∂x2 (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·

[12 (1− λ)2 q([ui, vi])

]+∑n

i=1B2 ([ui, vi]) |vi − ui|γ,

where B1, B2 : I → R are suitable interval functions, bounded on bounded

sets (with bound independent on λ), and γ > 1. By collecting the similar

terms we obtain:

F (1, g (1))− F (0, g (0)) =∑ni=1

∂F∂t (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · (vi − ui) +∑n

i=1∂F∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · (∆(g)([ui, vi]))+∑n

i=1∂2F∂t∂x (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi))

· [(2λ− 1) (vi − ui) (∆(g)([ui, vi]))] +∑ni=1

∂2F∂t2

(λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·[

12 (2λ− 1) (vi − ui)

2]+∑n

i=1∂2F∂x2 (λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) ·

[12 (2λ− 1) q([ui, vi])

]+∑n

i=1 (B1 ([ui, vi])−B2 ([ui, vi])) |vi − ui|γ =

S1 +S2 +S3 +S4 +S5 +S6 (where S1, S2, . . . , S6 denote the consecutive terms

of the above sum, and the expressions S3, S4, S6 are negligible, as previously

observed).

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 87

Moreover, since∂F

∂tsatisfies Taylor’s formula of order 1, we have:

S1 =∑n

i=1∂F∂t (ui, g (ui)) · (vi − ui) +W, where W as usual is negligible.

The first summand, and hence S1, tends to∫ 1

0

∂F

∂t(s, g (s)) ds as n tends to

+∞ (the function∂F

∂t(·, g (·)) is Holder-continuous, so

∂F

∂t(·, g (·)) is (RS)-

integrable w.r.t. dt). Now,

S5 =12

(2λ− 1)n∑

i=1

∂2F

∂x2(λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) q([ui, vi])

tends to12

(2λ− 1) J, as already observed. Hence it follows that

S2 =n∑

i=1

∂F

∂x(λui + (1− λ) vi, λg (ui) + (1− λ) g (vi)) · (∆(g)([ui, vi])) (4)

converges, and its limit is the 〈λ〉-integral of∂F

∂x(·, g(·)) with respect to g, that

is the assertion 2). The formula in assertion 3) follows now easily. 2

Remark 4.5

I) Sometimes, the 〈λ〉-integral exists, without assumptions on the function g.

For example, let F (t, x) = x2, and λ =12: so we are looking for the integral

〈12〉∫ 1

02g(s)dg(s). By definition, this integral is the limit of

2∑I∈D

[g(ui+1)− g(ui)][g(ui+1) + g(ui)

2

](5)

as δ(D) → 0, where D = [ui, ui+1] : i = 0, ..., n. But the quantity (5) always

coincides with g2(1)− g2(0), so we obtain 〈12〉∫ 1

02g(s)dg(s) = g2(1)− g2(0).

II) On the other hand, once we have a function g satisfying the hypotheses of

Theorem 4.4, at least in particular spaces R it is possible to find many other

functions g with the same properties. For example, let R = L0(X,B, P ) be as

in the Remark 3.11 and assume that g : [0, 1] → R is the standard Brownian

A.BOCCUTO ET AL88

Motion. Now, choose any C2-function φ : IR → IR and define, for every

t ∈ [0, 1]: g(t)(x) = φ(g(t)(x)) for almost all x ∈ X. We shall see that ∆(g)2

is integrable, and its integral function is Lipschitz. Indeed, if 0 ≤ u < v ≤ 1,

we get (a.e.)

(g(v)− g(u)) (x) = φ (g(v)(x))−φ (g(u)(x)) = (g(v)(x)− g(u)(x))φ′(g(τ)(x)),

where τ is a suitable point in [u, v], depending also on x. So we have

[g(v)− g(u)]2 (x) = [g(v)(x)− g(u)(x)]2(φ′(g(τ)(x))

)2. (6)

(We remark here that (φ′(g(τ)(x)))2 is an element of R, between the extrema

of the function t 7→ f(t) = φ′(g(t))2 in the interval [u, v]). Now consider the

function t 7→ f(t) = φ′(g(t))2, for all t ∈ [0, 1]. This is a Holder-continuous

function, hence Riemann-Stieltjes integrable with respect to q = ∆(g)2 by The-

orem 3.10. From this, (6) and the Remark 3.12, we deduce that (∆(g)(I))2

is integrable, and its integral function coincides with the Riemann integral∫Iφ′(g(t))2 dt (which can be computed pathwise), and this is clearly a Lips-

chitz function of interval.

References

[1] A. BOCCUTO, Abstract Integration in Riesz Spaces, Tatra Mountains Math.

Publ. 5 (1995), 107-124.

[2] A. BOCCUTO, Differential and Integral Calculus in Riesz Spaces, Tatra Moun-

tains Math. Publ. 14 (1998), 293-323.

[3] A. BOCCUTO, A Perron-type integral of order 2 for Riesz spaces, Math. Slovaca

51 (2001), 185-204.

KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES 89

[4] D. CANDELORO, Riemann-Stieltjes Integration in Riesz Spaces, Rend. Mat.

(Roma) 16 (1996), 563-585.

[5] V. KONDURAR, Sur l’integrale de Stieltjes, Mat. Sbornik 2 (1937), 361-366.

[6] W. A. J. LUXEMBURG & A. C. ZAANEN, Riesz Spaces, I, (1971), North-

Holland Publ. Co.

[7] Z. SCHUSS, Theory and Applications of Stochastic Differential Equations,

(1980), John Wiley & Sons, New York.

[8] R. L. STRATONOVICH, A new representation for stochastic integrals and equa-

tions, Siam J. Control 4 (1966), 362-371.

[9] B. Z. VULIKH, Introduction to the theory of partially ordered spaces, (1967),

Wolters - Noordhoff Sci. Publ., Groningen.

[10] A. C. ZAANEN, Riesz spaces, II, (1983), North-Holland Publishing Co.

A.BOCCUTO ET AL90

Bounds On Triangular Discrimination,Harmonic Mean and Symmetric Chi-square

Divergences

Inder Jeet Taneja

Departamento de MatematicaUniversidade Federal de Santa Catarina

88.040-900 Florianopolis, SC, Brazil.e-mail: taneja@mtm.ufsc.br

http://www.mtm.ufsc.br/∼taneja

Abstract

There are many information and divergence measures exist in the literature oninformation theory and statistics. The most famous among them are Kullback-Leiber [15] relative information and Jeffreys [14] J-divergence. The measures likeBhattacharya distance, Hellinger discrimination, Chi-square divergence, triangulardiscrimination and harmonic mean divergence are also famous in the literature onstatistics. In this paper we have obtained bounds on triangular discrimination andsymmetric chi-square divergence in terms of relative information of type s usingCsiszar’s f-divergence. A relationship among triangular discrimination and har-monic mean divergence is also given.

Key words: Relative information of type s; Harmonic mean divergence; Triangu-lar discrimination; Symmetric Chi-square divergence; Csiszar’s f-divergence; Informationinequalities.

1 Introduction

Let

Γn =

P = (p1, p2, ..., pn)

∣∣∣∣∣pi > 0,n∑

i=1

pi = 1

, n > 2,

be the set of all complete finite discrete probability distributions. For all P, Q ∈ Γn, thefollowing measures are well known in the literature on information theory and statistics:

91JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.1,91-111,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

• Bhattacharya Distance (Bhattacharya [2])

B(P ||Q) =n∑

i=1

√piqi. (1)

• Hellinger discrimination (Hellinger [13])

h(P ||Q) = 1−B(P ||Q) =1

2

n∑i=1

(√

pi −√qi)2. (2)

• χ2−Divergence (Pearson [18])

χ2(P ||Q) =n∑

i=1

(pi − qi)2

qi

=n∑

i=1

p2i

qi

− 1. (3)

• Relative Information (Kullback and Leibler [15])

K(P ||Q) =n∑

i=1

pi ln(pi

qi

). (4)

The above four measures can be obtained as particular or limiting case of the relativeinformation of type s. This measure is given by

• Relative Information of Type s

Φs(P ||Q) =

Ks(P ||Q) = [s(s− 1)]−1

[n∑

i=1

psiq

1−si − 1

], s 6= 0, 1

K(Q||P ) =n∑

i=1

qi ln(

qi

pi

), s = 0

K(P ||Q) =n∑

i=1

pi ln(

pi

qi

), s = 1

. (5)

The measure (5) admits the following interesting particular cases:

(i) Φ−1(P ||Q) = 12χ2(Q||P ).

(ii) Φ0(P ||Q) = K(Q||P ).

(iii) Φ1/2(P ||Q) = 4 [1−B(P ||Q)] = 4h(P ||Q).

(iv) Φ1(P ||Q) = K(P ||Q).

I.TANEJA92

(v) Φ2(P ||Q) = 12χ2(P ||Q).

Thus we observe that Φ2(P ||Q) = Φ−1(Q||P ) and Φ1(P ||Q) = Φ0(Q||P ).

For more studies on the measure (5) refer to Liese and Vajda [16], Vajda [27], Taneja[19], [20], [22] and Cerone et al. [4].

Recently Taneja [23] and Taneja and Kumar [25] studied the (5) and obtained boundsin terms of the measures (1)-(4). Here we shall extend the our study for the other mea-sures known in the literature as triangular discrimination, harmonic mean divergence andsymmetric chi-square divergence.

The triangular discrimination is given by

∆(P ||Q) =n∑

i=1

(pi − qi)2

pi + qi

. (6)

After simplification, we can write

∆(P ||Q) = 2 [1−W (P ||Q)] , (7)

where

W (P ||Q) =n∑

i=1

2piqi

pi + qi

, (8)

is the well known harmonic mean divergence.

We observe that the measures (3) and (4) are not symmetric with respect to probabilitydistributions. The symmetric version of the measure (4) famous as Jeffreys-Kullback-Leiber J-divergence is given by

J(P ||Q) = K(P ||Q) + K(Q||P ). (9)

Let us consider the symmetric chi-square divergence given by

Ψ(P ||Q) = χ2(P ||Q) + χ2(Q||P ) =n∑

i=1

(pi − qi)2(pi + qi)

piqi

. (10)

Dragomir [12] studied the measure (10) and obtained interesting result relating it totriangular discrimination and J-divergence.

Some studies on the measures (6) and (8) can be seen in Dragomir [10], [11] andTopsøe [26]. Recently, Taneja [23] and Taneja and Kumar [25] studied the measure (5)and obtained bounds in terms of the measures (1)-(4). Similar kind of bounds on themeasure (9) are recently obtained by Taneja [24]. In this paper, we shall extend theour study for triangular discrimination and symmetric chi-square divergence. In order toobtain bounds on these measures we make use of Csiszar’s [5] f-divergence.

BOUNDS ON TRIANGULAR DISCRIMINATION... 93

2 Csiszar’s f−Divergence and Its Particular Cases

Given a convex function f : [0,∞) → R, the f−divergence measure introduced by Csiszar[5] is given by

Cf (P ||Q) =n∑

i=1

qif

(pi

qi

), (11)

where P,Q ∈ Γn.

It is well known in the literature [5] that if f is convex and normalized, i.e., f(1) = 0,then the Csiszar’s function Cf (P ||Q) is nonnegative and convex in the pair of probabilitydistribution (P,Q) ∈ Γn × Γn.

Here below we shall give the measures (6) and (10) being examples of the measure (11).

Example 2.1. (Triangular discrimination). Let us consider

f∆(x) =(x− 1)2

x + 1, x ∈ (0,∞) (12)

in (11), we have

Cf (P ||Q) = ∆(P ||Q) =n∑

i=1

(pi − qi)2

pi + qi

,

where ∆(P ||Q) is as given by (6).Moreover,

f ′∆(x) =(x− 1)(x + 3)

(x + 1)2(13)

and

f ′′∆(x) =8

(x + 1)3. (14)

Thus we have f ′′∆(x) > 0 for all x > 0, and hence, f∆(x) is strictly convex for allx > 0. Also, we have f∆(1) = 0. In view of this we can say that the triangular discrimi-nation is nonnegative and convex in the pair of probability distributions (P, Q) ∈ Γn×Γn.

Example 2.2. (Symmetric chi-square divergence). Let us consider

fΨ(x) =(x− 1)2(x + 1)

x, x ∈ (0,∞) (15)

in (11), we have

Cf (P ||Q) = Ψ(P ||Q) =n∑

i=1

(pi − qi)2(pi + qi)

piqi

,

where Ψ (P ||Q) is as given by (10).

I.TANEJA94

Moreover,

f ′Ψ(x) =(x− 1)(2x2 + x + 1)

x2(16)

and

f ′′Ψ(x) =2(x3 + 1)

x3. (17)

Thus we have f ′′Ψ(x) > 0 for all x > 0, and hence, fΨ(x) is strictly convex for allx > 0. Also, we have fΨ(1) = 0. In view of this we can say that the symmetric chi-squaredivergence is nonnegative and convex in the pair of probability distributions (P, Q) ∈Γn × Γn.

3 Csiszar’s f−Divergence and Relative Information

of Type s

During past years Dragomir done a lot of work giving bounds on Csiszar’s f−divergence.Here below we shall summarize the some his results [6], [7], [9].

Theorem 3.1. Let f : R+ → [0,∞) be differentiable convex and normalized i.e., f(1) = 0.If P, Q ∈ Γn, then we have

0 6 Cf (P ||Q) 6 ρCf(P ||Q), (18)

where ρCf(P ||Q) is given by

ρCf(P ||Q) = Cf ′

(P 2

Q||P

)− Cf ′(P ||Q) =

n∑i=1

(pi − qi)f′(

pi

qi

). (19)

If P, Q ∈ Γn are such that

0 < r 6 pi

qi

6 R < ∞, ∀i ∈ 1, 2, ..., n,

for some r and R with 0 < r 6 1 6 R < ∞, then we have the following inequalities:

0 6 Cf (P ||Q) 6 αCf(r, R), (20)

0 6 Cf (P ||Q) 6 βCf(r,R) (21)

and

0 6 βCf(r, R)− Cf (P ||Q) (22)

6 γCf(r, R)

[(R− 1)(1− r)− χ2(P ||Q)

]6 αCf

(r, R),

BOUNDS ON TRIANGULAR DISCRIMINATION... 95

where

αCf(r, R) =

1

4(R− r)2 γCf

(r,R), (23)

βCf(r, R) =

(R− 1)f(r) + (1− r)f(R)

R− r(24)

and

γCf(r, R) =

f ′(R)− f ′(r)R− r

. (25)

The following proposition is due to Taneja [23] and Taneja and Kumar [25] and is aconsequence of the above theorem.

Proposition 3.1. Let P,Q ∈ Γn and s ∈ R, then we have

0 6 Φs(P ||Q) 6 ρΦs(P ||Q), (26)

where

ρΦs(P ||Q) = Cφ′s

(P 2

Q||P

)− Cφ′s (P ||Q) (27)

=

(s− 1)−1n∑

i=1

(pi − qi)(

pi

qi

)s−1

, s 6= 1

n∑i=1

(pi − qi) ln(

pi

qi

), s = 1

.

If there exists r, R (0 < r 6 1 6 R < ∞) such that

0 < r 6 pi

qi

6 R < ∞, ∀i ∈ 1, 2, ..., n,

then we have the following inequalities

0 6 Φs(P ||Q) 6 αΦs(r, R), (28)

0 6 Φs(P ||Q) 6 βΦs(r, R) (29)

and

0 6 βΦs(r, R)− Φs(P ||Q) (30)

6 γΦs(r, R)[(R− 1)(1− r)− χ2(P ||Q)

]6 αΦs(r, R),

where

αΦs(r, R) =1

4(R− r)2 γΦs(r,R), (31)

βΦs(r, R) =

(R−1)(rs−1)+(1−r)(Rs−1)(R−r)s(s−1)

, s 6= 0, 1(R−1) ln 1

r+(1−r) ln 1

R

(R−r), s = 0

(R−1)r ln r+(1−r)R ln R(R−r)

, s = 1

(32)

I.TANEJA96

and

γΦs(r, R) =

Rs−1−rs−1

(R−r)(s−1), s 6= 1

ln R−ln rR−r

, s = 1, (33)

We can also write γΦs(r, R) as follows

γΦs(r, R) =

Ls−2

s−2(r, R), s 6= 1

L−1−1(r,R) s = 1

, (34)

where Lp(a, b) is the famous (Bullen, Mitrinovic and Vasic [3]) p-logarithmic power meangiven by

Lp(a, b) =

[bp+1−ap+1

(p+1)(b−a)

] 1p, p 6= −1, 0

b−aln b−ln a

, p = −1

1e

[bb

aa

] 1b−a

, p = 0

, (35)

for all p ∈ R, a 6= b.The expression (27) admits the following particular cases:

(i) ρΦ−1(P ||Q) = 3 Φ3(Q||P )− 12χ2(Q||P ).

(ii) ρΦ0(P ||Q) = χ2(Q||P ).

(iii) ρΦ1(P ||Q) = J(P ||Q).

(iv) ρΦ1/2(P ||Q) = 2

n∑i=1

(qi − pi)√

qi

pi

(v) ρΦ2(P ||Q) = χ2(P ||Q)

The expression (32) admits the following particular cases:

(i) βΦ−1(P ||Q) = (R−1)(1−r)2rR

.

(ii) βΦ0(r, R) =(R−1) ln 1

r+(1−r) ln 1

R

R−r.

(iii) βΦ1(r, R) = (R−1)r ln r+(1−r)R ln RR−r

.

BOUNDS ON TRIANGULAR DISCRIMINATION... 97

(iv) βΦ1/2(P ||Q) = 4(

√R−1)(1−√r)√

R+√

r.

(v) βΦ2(P ||Q) = (R−1)(1−r)2

.

The following theorem is due to Taneja [23] and Taneja and Kumar [25].

Theorem 3.2. Let f : I ⊂ R+ → [0,∞) the generating mapping is normalized, i.e.,f(1) = 0 and satisfy the assumptions:

(i) f is twice differentiable on (r, R), where 0 6 r 6 1 6 R 6 ∞;(ii) there exists real constants m,M such that m < M and

m 6 x2−sf ′′(x) 6 M, ∀x ∈ (r, R), s ∈ R. (36)

If P, Q ∈ Γn are discrete probability distributions satisfying the assumption

0 < r 6 pi

qi

6 R < ∞,

then we have the inequalities:

mΦs(P ||Q) 6 Cf (P ||Q) 6 MΦs(P ||Q), (37)

m [ρΦs(P ||Q)− Φs(P ||Q)] (38)

6 ρCf(P ||Q)− Cf (P ||Q)

6 M [ρΦs(P ||Q)− Φs(P ||Q)]

and

m [βΦs(r, R)− Φs(P ||Q)] (39)

6 βCf(r,R)− Cf (P ||Q)

6 M [βΦs(r,R)− Φs(P ||Q)] ,

where Cf (P ||Q), Φs(P ||Q), ρCf(P ||Q), ρΦs(P ||Q), βCf

(r, R) and βΦs(r, R) are as given by(11), (5), (19), (27), (24) and (32) respectively.

The above theorem unifies some of the results studied by Dragomir [8], [10], [11].

In the papers Taneja [23] and Taneja and Kumar [25] considered the particular valuesof s and Φs by taking s = −1, s = 0, s = 1

2, s = 1 and s = 2. The aim here is to

obtain results by taking different values of f given by examples 2.1-2.2, and then obtainparticular cases for different values of s.

Remark 3.1. If is it not specified, from now onwards, it is understood that, if there arer, R then 0 < r 6 pi

qi6 R < ∞, ∀i ∈ 1, 2, ..., n, with 0 < r 6 1 6 R < ∞ where

P = (p1, p2, ...., pn) ∈ Γn and P = (q1, q2, ...., qn) ∈ Γn.

I.TANEJA98

4 Triangular Discrimination and Inequalities

In this section, we shall give bounds on triangular discrimination based on the Theorems3.1 and 3.2.

Theorem 4.1. For all P, Q ∈ Γn, we have the following inequalities

0 6 ∆(P ||Q) 6 ρ∆(P ||Q), (40)

where

ρ∆(P ||Q) =n∑

i=1

(pi − qi

pi + qi

)2

(pi + 3qi). (41)

If there exists r, R (0 < r 6 1 6 R < ∞) such that

0 < r 6 pi

qi

6 R < ∞, ∀i ∈ 1, 2, ..., n,

then we have the following inequalities:

0 6 ∆(P ||Q) 6 α∆(r, R), (42)

0 6 ∆(P ||Q) 6 β∆(r, R) (43)

and

0 6 β∆(r, R)−∆(P ||Q) (44)

6 γ∆(r, R)[(R− 1)(1− r)− χ2(P ||Q)

]6 α∆(r, R),

where

α∆(r, R) =1

4(R− r)

[(R− 1)(R + 3)

(R + 1)2+

(1− r)(r + 3)

(r + 1)2

], (45)

β∆(r, R) =2(R− 1)(1− r)

(R + 1)(1 + r)(46)

and

γ∆(r, R) = (R− r)−1

[(R− 1)(R + 3)

(R + 1)2+

(1− r)(r + 3)

(r + 1)2

]. (47)

Proof. Follows from the Theorem 3.1 by considering f by f∆ and making necessary cal-culations.

Theorem 4.2. Let P, Q ∈ Γn and s ∈ R. Let there exists r, R (0 < r 6 1 6 R < ∞)such that 0 < r 6 pi

qi6 R < ∞, ∀i ∈ 1, 2, ..., n.

BOUNDS ON TRIANGULAR DISCRIMINATION... 99

(a) For s 6 −1, we have the following inequalities:

8r2−s

(r + 1)3Φs(P ||Q) 6 ∆(P ||Q) 6 8R2−s

(R + 1)3Φs(P ||Q), (48)

8r2−s

(r + 1)3[ρΦs(P ||Q)− Φs(P ||Q)] (49)

6 ∆∗(P ||Q) 6 8R2−s

(R + 1)3[ρΦs(P ||Q)− Φs(P ||Q)]

and

8r2−s

(r + 1)3[βΦs(r, R)− Φs(P ||Q)] (50)

6 β∆(r, R)−∆(P ||Q)

6 8R2−s

(R + 1)3[βΦs(r,R)− Φs(P ||Q)] .

(b) For s > 2, we have the following inequalities:

8R2−s

(R + 1)3Φs(P ||Q) 6 ∆(P ||Q) 6 8r2−s

(r + 1)3Φs(P ||Q), (51)

8R2−s

(R + 1)3[ρΦs(P ||Q)− Φs(P ||Q)] (52)

6 ∆∗(P ||Q) 6 8r2−s

(r + 1)3[ρΦs(P ||Q)− Φs(P ||Q)]

and

R1−s

(R + 1)2[βΦs(r, R)− Φs(P ||Q)] (53)

6 β∆(r, R)−∆(P ||Q)

6 r1−s

(r + 1)2[βΦs(r,R)− Φs(P ||Q)] ,

where

∆∗(P ||Q) = ρ∆(P ||Q)−∆(P ||Q) = 2n∑

i=1

qi

(pi − qi

pi + qi

)2

. (54)

Proof. Let us consider

g∆(x) = x2−sf ′′∆(x) =8x2−s

(x + 1)3, x ∈ (0,∞), (55)

I.TANEJA100

where f ′′∆(x) is as given by (14).We have

g′∆(x) = −8x1−s [(s + 1)x + (s− 2)]

(x + 1)4

> 0, s 6 −1

6 0, s > 2. (56)

In view of (56), we conclude the followings:

m = infx∈[r,R]

g(x) = minx∈[r,R]

g(x) =

8r2−s

(r+1)3, s 6 −1

8R2−s

(R+1)3, s > 2

(57)

and

M = supx∈[r,R]

g(x) = maxx∈[r,R]

g(x) =

8R2−s

(R+1)3, s 6 −1

8r2−s

(r+1)3, s > 2

. (58)

¿From (57) and (58) and Theorem 3.2, we have the required proof.

The following propositions are the particular cases of the above theorem.

Proposition 4.1. We have the following bounds in terms of χ2−divergence:

4r3

(r + 1)3χ2(Q||P ) 6 ∆(P ||Q) 6 4R3

(R + 1)3χ2(Q||P ), (59)

8r3

(r + 1)3

[3 Φ3(Q||P )− χ2(Q||P )

](60)

6 ∆∗(P ||Q) 6 8R3

(R + 1)3

[3 Φ3(Q||P )− χ2(Q||P )

]

and

4r3

(r + 1)3

[(R− 1)(1− r)

rR− χ2(Q||P )

](61)

6 2(R− 1)(1− r)

(R + 1)(1 + r)−∆(P ||Q)

6 4R3

(R + 1)3

[(R− 1)(1− r)

rR− χ2(Q||P )

].

Proof. Take s = −1 in (48), (49) and (50) we get respectively (59), (60) and (61).

Proposition 4.2. We have the following bounds in terms of χ2−divergence:

4

(R + 1)3χ2(P ||Q) 6 ∆(P ||Q) 6 4

(r + 1)3χ2(P ||Q), (62)

4

(R + 1)3χ2(P ||Q) 6 ∆∗(P ||Q) 6 4

(r + 1)3χ2(P ||Q) (63)

BOUNDS ON TRIANGULAR DISCRIMINATION... 101

and

4

(R + 1)3

[(R− 1)(1− r)− χ2(P ||Q)

](64)

6 2(R− 1)(1− r)

(R + 1)(1 + r)−∆(P ||Q)

6 4

(r + 1)3

[(R− 1)(1− r)− χ2(P ||Q)

].

Proof. Take s = 2 in (51), (52) and (53) we get respectively (62), (63) and (64).

We observe that the Theorem 4.2 is not valid for s = 0, 12

and 1. These particularvalues of s we shall do separately. In these cases, we don’t have inequalities on both sidesas in the case of Propositions 4.1 and 4.2.

Proposition 4.3. The following inequalities hold:

0 6 ∆(P ||Q) 6 32

27K(Q||P ), (65)

0 6 ∆∗(P ||Q) 6 32

27

[χ2(Q||P )−K(Q||P )

](66)

and

0 6 32

27K(Q||P )−∆(P ||Q) (67)

6 32

27

(R− 1) ln 1r

+ (1− r) ln 1R

R− r− 2(R− 1)(1− r)

(R + 1)(1 + r)

Proof. For s = 0 in (55), we have

g∆(x) =8x2

(x + 1)3. (68)

This gives

g′∆(x) = −8x(x− 2)

(x + 1)4

> 0, x 6 2

6 0, x > 2. (69)

Thus we conclude that the function gW (x) given by (68) is increasing in x ∈ (0, 2) anddecreasing in x ∈ (2,∞), and hence

M = supx∈(0,∞)

g∆(x) = maxx∈(0,∞)

g∆(x) = g∆(2) =32

27. (70)

Now (70) together with (37), (38) and (39) give respectively (65), (66) and (67).

I.TANEJA102

Proposition 4.4. The following inequalities hold:

0 6 ∆(P ||Q) 6 4 h(P ||Q), (71)

0 6 ∆∗(P ||Q) 6 2n∑

i=1

(qi − pi)

√qi

pi

− 4 h(P ||Q) (72)

and

0 6 4 h(P ||Q)−∆(P ||Q) (73)

6 4(√

R− 1)(1−√r)√R +

√r

− 2(R− 1)(1− r)

(R + 1)(1 + r).

Proof. For s = 12

in (55), we have

g∆(x) =8x3/2

(x + 1)3. (74)

This gives

g′∆(x) = −12√

x(x− 1)

(x + 1)4

> 0, x 6 1

6 0, x > 1. (75)

Thus we conclude that the function g∆(x) given by (74) is increasing in x ∈ (0, 1) anddecreasing in x ∈ (1,∞), and hence

M = supx∈(0,∞)

g∆(x) = maxx∈(0,∞)

g∆(x) = g∆(1) = 1. (76)

Now (76) together with (37), (38) and (39) give respectively (71), (72) and (73).

Proposition 4.5. We have following inequalities:

0 6 ∆(P ||Q) 6 32

27K(P ||Q), (77)

0 6 ∆∗(P ||Q) 6 32

27K(Q||P ) (78)

and

0 6 32

27K(P ||Q)−∆(P ||Q) (79)

6 32

27

(R− 1)r ln r + (1− r)R ln R

R− r− 2(R− 1)(1− r)

(R + 1)(1 + r).

BOUNDS ON TRIANGULAR DISCRIMINATION... 103

Proof. For s = 1 in (55), we have

g∆(x) =8x

(x + 1)3. (80)

This gives

g′W (x) = −8(2x− 1)

(x + 1)4=

> 0, x 6 1

2

6 0, x > 12

. (81)

Thus we conclude that the function g∆(x) given by (80) is increasing in x ∈ (0, 12) and

decreasing in x ∈ (12,∞), and hence

M = supx∈(0,∞)

g∆(x) = maxx∈(0,∞)

g∆(x) = g(1

2) =

32

27. (82)

Now (82) together with (37), (38) and (39) give respectively (77), (78) and (79).

Remark 4.1. In view of relation (7) and Propositions 4.1-4.5, we have the followingmain bounds on harmonic mean divergence:

2r3

(r + 1)3χ2(Q||P ) 6 1−W (P ||Q) 6 2R3

(R + 1)3χ2(Q||P ), (83)

2

(R + 1)3χ2(P ||Q) 6 1−W (P ||Q) 6 2

(r + 1)3χ2(P ||Q), (84)

0 6 1−W (P ||Q) 6 16

27K(Q||P ), (85)

0 6 1−W (P ||Q) 6 2 h(P ||Q) (86)

and

0 6 1−W (P ||Q) 6 16

27K(P ||Q). (87)

The inequalities (84) were also studied by Dragomir [8]. The inequalities (87) canbe seen in Dragomir [10]. The inequalities (71) can be seen been in Dragomir [11] andTopsφe [26]. The inequalities (77) can be in and Dragomir [10].

5 Symmetric Chi-square Divergence and Inequalities

In this section, we shall give bounds on symmetric chi-square divergence based on theTheorems 3.1 and 3.2.

Theorem 5.1. For all P, Q ∈ Γn, we have the following inequalities:

0 6 Ψ(P ||Q) 6 ρΨ(P ||Q), (88)

I.TANEJA104

where

ρΨ(P ||Q) = Ψ(P ||Q) +n∑

i=1

(pi − qi)2(p2

i + q2i )

p2i qi

. (89)

If there exists r, R (0 < r 6 1 6 R < ∞) such that

0 < r 6 pi

qi

6 R < ∞, ∀i ∈ 1, 2, ..., n,

then we have the following inequalities:

0 6 Ψ(P ||Q) 6 αΨ(r, R), (90)

0 6 Ψ(P ||Q) 6 βΨ(r, R) (91)

and

0 6 βΨ(r, R)−Ψ(P ||Q) (92)

6 γΨ(r, R)[(R− 1)(1− r)− χ2(P ||Q)

]6 αΨ(r, R),

where

αΨ(r,R) =1

4(R− r)2

[2L−1

2 (r,R)− L−11 (r, R)

], (93)

βΨ(r,R) = (R− 1)(1− r)(R + r) (94)

and

γΨ(r,R) = 2L−12 (r,R)− L−1

1 (r, R). (95)

Proof. Follows from Theorem 3.1 by considering f by fΨ and making necessary calcula-tions.

Theorem 5.2. Let P,Q ∈ Γn and s ∈ R. Let there exists r, R (0 6 r 6 1 6 R 6 ∞)such that 0 < r 6 pi

qi6 R < ∞, ∀i ∈ 1, 2, ..., n.

(a) For s 6 −1, we have the following inequalities:

2(r3 + 1)

r1+sΦs(P ||Q) 6 Ψ(P ||Q) 6 2(R3 + 1)

R1+sΦs(P ||Q), (96)

2(r3 + 1)

r1+s[ρΦs(P ||Q)− Φs(P ||Q)] (97)

6 Ψ∗(P ||Q) 6 2(R3 + 1)

R1+s[ρΦs(P ||Q)− Φs(P ||Q)]

BOUNDS ON TRIANGULAR DISCRIMINATION... 105

and

2(r3 + 1)

r1+s[βΦs(r, R)− Φs(P ||Q)] (98)

6 βΨ(r,R)−Ψ(P ||Q)

6 2(R3 + 1)

R1+s[βΦs(r,R)− Φs(P ||Q)] .

(b) For s > 2, we have the following inequalities:

2(R3 + 1)

R1+sΦs(P ||Q) 6 Ψ(P ||Q) 6 2(r3 + 1)

r1+sΦs(P ||Q), (99)

2(r3 + 1)

r1+s[ρΦs(P ||Q)− Φs(P ||Q)] (100)

6 Ψ∗(P ||Q) 6 2(r3 + 1)

r1+s[ρΦs(P ||Q)− Φs(P ||Q)]

and

2(R3 + 1)

R1+s[βΦs(r, R)− Φs(P ||Q)] (101)

6 βΨ(r, R)−Ψ(P ||Q)

6 2(r3 + 1)

r1+s[βΦs(r,R)− Φs(P ||Q)] ,

where

Ψ∗(P ||Q) = ρΨ(P ||Q)−Ψ(P ||Q) =n∑

i=1

(pi − qi)2(p2

i + q2i )

p2i qi

. (102)

Proof. Let us consider

gΨ(x) = x2−sf ′′Ψ(x) = 2x−1−s(x3 + 1), x ∈ (0,∞), (103)

where f ′′Ψ(x) is as given by (17).We have

g′Ψ(x) = −2x−2−s[(s− 2)x3 + (s + 1)

]

> 0, s 6 −1

6 0, s > 2. (104)

¿From (104), we conclude the followings:

m = infx∈[r,R]

gΨ(x) = minx∈[r,R]

gΨ(x) =

2(r3+1)

r1+s , s 6 −12(R3+1)

R1+s , s > 2(105)

and

M = supx∈[r,R]

gΨ(x) = maxx∈[r,R]

gΨ(x) =

2(R3+1)

R1+s , s 6 −12(r3+1)

r1+s , s > 2(106)

In view of (105) and (106) and Theorem 3.2, we have the required proof.

I.TANEJA106

Proposition 5.1. We have the following bounds in terms of χ2−divergence:

(r3 + 1)χ2(Q||P ) 6 Ψ(P ||Q) 6 (R3 + 1)χ2(Q||P ), (107)

2(r3 + 1)[3 Φ3(Q||P )− χ2(Q||P )

](108)

6 Ψ∗(P ||Q) 6 2(R3 + 1)[3, Φ3(P ||Q)− χ2(Q||P )

]

and

(r3 + 1)

[(R− 1)(1− r)

rR− χ2(Q||P )

](109)

6 (R− 1)(1− r)(R + r)−Ψ(P ||Q)

6 (R3 + 1)

[(R− 1)(1− r)

rR− χ2(Q||P )

].

Proof. Take s = −1 in (96), (97) and (98) we get respectively (107), (108) and (109).

Proposition 5.2. The following bounds on in terms of χ2−divergence hold:

R3 + 1

R3χ2(P ||Q) 6 Ψ(P ||Q) 6 r3 + 1

r3χ2(P ||Q), (110)

R3 + 1

R3χ2(P ||Q) 6 Ψ∗(P ||Q) 6 r3 + 1

r3χ2(P ||Q) (111)

and

R3 + 1

R3

[(R− 1)(1− r)− χ2(P ||Q)

](112)

6 βΨ(r,R)−Ψ(P ||Q)

6 r3 + 1

r3

[(R− 1)(1− r)− χ2(P ||Q)

].

Proof. Take s = 2 in (99), (100) and (101) we get respectively (110), (111) and (112).

We observe that the above two propositions follows from Theorem 5.2 immediatelyby taking s = −1 and s = 2 respectively. But still there are another values of s such ass = 0, s = 1 and s = 1

2for which we can obtain bounds. These values are studied below.

Proposition 5.3. We have following bounds in terms of relative information:

0 6 33√

2 K(Q||P ) 6 Ψ(P ||Q), (113)

0 6 33√

2[χ2(Q||P )−K(Q||P )

]6 Ψ∗(P ||Q) (114)

and

0 6 Ψ(P ||Q)− 33√

2 K(Q||P ) (115)

6 (R− 1)(1− r)(R + r)− 33√

2(R− 1) ln 1

r+ (1− r) ln 1

R

R− r.

BOUNDS ON TRIANGULAR DISCRIMINATION... 107

Proof. For s = 0 in (103), we have

gΨ(x) =2(x3 + 1)

x, (116)

This gives

g′Ψ(x) =2(2x3 − 1)

x2(117)

=2( 3√

2 x− 1)( 3√

4 x2 + 3√

2 x + 1)

x2

> 0, x > 1

3√2

6 0, x 6 13√2

.

Thus we conclude that the function gΨ(x) given by (116) is decreasing in x ∈ (0, 13√2

)

and increasing in x ∈ ( 13√2

,∞), and hence

m = infx∈(0,∞)

gΨ(x) = minx∈(0,∞)

gΨ(x) = gΨ(13√

2) = 3

3√

2. (118)

Now (118) together with (37), (38) and (39) give respectively (113), (114) and (115).

Proposition 5.4. We have following bounds in terms of Hellinger’s discrimination:

0 6 16 h(P ||Q) 6 Ψ(P ||Q), (119)

0 6 16

[1

2

n∑i=1

(qi − pi)

√qi

pi

− h(Q||P )

]6 Ψ∗(P ||Q) (120)

and

0 6 Ψ(P ||Q)− 16 h(P ||Q) (121)

6 (R− 1)(1− r)(R + r)− 16(√

R− 1)(1−√r)√R +

√r

.

Proof. For s = 12

in (103), we have

gΨ(x) =2(x3 + 1)

x3/2. (122)

This gives

g′Ψ(x) =3(x3 − 1)

x5/2=

3(x− 1)(x2 + x + 1)

x5/2

> 0, x > 1

6 0, x 6 1. (123)

Thus we conclude that the function gΨ(x) given by (122) is decreasing in x ∈ (0, 1)and increasing in x ∈ (1,∞), and hence

m = infx∈(0,∞)

gΨ(x) = minx∈(0,∞)

gΨ(x) = gΨ(1) = 4. (124)

Now (124) together with (37), (38) and (39) give respectively (119), (120) and (121).

I.TANEJA108

Proposition 5.5. We have the following bounds in terms of relative information:

0 6 33√

2 K(P ||Q) 6 Ψ(P ||Q), (125)

0 6 33√

2 K(Q||P ) 6 Ψ∗(P ||Q) (126)

and

0 6 Ψ(P ||Q)− 33√

2 K(P ||Q) (127)

6 (R− 1)(1− r)(R + r)− 33√

2(R− 1)r ln r + (1− r)R ln R

R− r.

Proof. For s = 1 in (103), we have

gΨ(x) =2(x3 + 1)

x2. (128)

This gives

g′Ψ(x) =2(x3 − 2)

x3(129)

=2(x− 3

√2)(x2 + 3

√2 x + 3

√4)

x3

> 0, x > 3

√2

6 0, x 6 3√

2.

Thus we conclude that the function gΨ(x) given by (128) is decreasing in x ∈ (0, 3√

2)and increasing in x ∈ ( 3

√2,∞), and hence

m = infx∈(0,∞)

gΨ(x) = minx∈(0,∞)

gΨ(x) = gΨ(3√

2) = 33√

2. (130)

Now (130) together with (37), (38) and (39) give respectively (125), (126) and (127).

References

[1] N.S. BARNETT, P. CERENE and S. S. DRAGOMIR, Some New Inequalitiesfor Hermite-Hadamard Divergence in Information Theory, http://rgmia.vu.edu.au,RGMIA Research Report Collection, (5)(4)(2002), Article 8.

[2] A. BHATTACHARYYA, Some Analogues to the Amount of Information and Theiruses in Statistical Estimation, Sankhya, 8(1946), 1-14.

[3] BULLEN, P.S., D.S. MITRINOVIC and P.M. VASIC , Means and Their Inequalities,Kluwer Academic Publishers, 1988.

[4] P. CERONE, S.S. DRAGOMIR and F. OSTERREICHER, Bounds on Extended f-Divergence for a Variety of Classes, http://rgmia.vu.edu.au, RGMIA Research ReportCollection, 6(1)(2003), Article 7.

BOUNDS ON TRIANGULAR DISCRIMINATION... 109

[5] I. CSISZAR, Information Type Measures of Differences of Probability Distributionand Indirect Observations, Studia Math. Hungarica, 2(1967), 299-318.

[6] S. S. DRAGOMIR, Some Inequalities for the Csiszar Φ-Divergence - Inequalities forCsiszar f-Divergence in Information Theory - Monograph – Chapter I – Article 1 –http://rgmia.vu.edu.au/monographs/csiszar.htm.

[7] S. S. DRAGOMIR, A Converse Inequality for the Csiszar Φ-Divergence- Inequalitiesfor Csiszar f-Divergence in Information Theory – Monograph – Chapter I – Article2 – http://rgmia.vu.edu.au/monographs/csiszar.htm.

[8] S. S. DRAGOMIR, Some Inequalities for (m, M)-Convex Mappings and Ap-plications for the Csiszar Φ-Divergence in Information Theory- Inequalities forCsiszar f-Divergence in Information Theory - Monograph – Chapter I – Article 3– http://rgmia.vu.edu.au/monographs/csiszar.htm.

[9] S. S. DRAGOMIR, Other Inequalities for Csiszar Divergence and Applications -Inequalities for Csiszar f-Divergence in Information Theory - Monograph – ChapterI – Article 4 – http://rgmia.vu.edu.au/monographs/csiszar.htm.

[10] S. S. DRAGOMIR, Upper and Lower Bounds for Csiszar’s f-divergence in termsof the Kullback-Leibler Distance and Applications - Inequalities for Csiszar f-Divergence in Information Theory - Monograph – Chapter II – Article 1 –http://rgmia.vu.edu.au/monographs/csiszar.htm.

[11] S. S. DRAGOMIR, Upper and Lower Bounds for Csiszar’s f-Divergence interms of Hellinger Discrimination and Applications - Inequalities for Csiszar f-Divergence in Information Theory - Monograph – Chapter II – Article 2 –http://rgmia.vu.edu.au/monographs/csiszar.htm.

[12] S. S. DRAGOMIR, J. SUNDE and C. BUSE, New Inequalities for Jeffreys DivergenceMeasure, Tamsui Oxford Journal of Mathematical Sciences, 16(2)(2000), 295-309.

[13] E. HELLINGER, Neue Begrundung der Theorie der quadratischen Formen von un-endlichen vielen Veranderlichen, J. Reine Aug. Math., 136(1909), 210-271.

[14] H. JEFFREYS, An Invariant Form for the Prior Probability in Estimation Problems,Proc. Roy. Soc. Lon., Ser. A, 186(1946), 453-461.

[15] S. KULLBACK and R.A. LEIBLER, On Information and Sufficiency, Ann. Math.Statist., 22(1951), 79-86.

[16] F. LIESE and I. VAJDA, Convex Statistical Decision Rule, Teubner-Texte zur Math-ematick, Band 95, Leipzig, 1987.

[17] J. LIN, Divergence Measures Based on the Shannon Entropy, IEEE Trans. on Inform.Theory, IT-37(1991), 145-151.

I.TANEJA110

[18] K. PEARSON, On the Criterion that a given system of deviations from the probablein the case of correlated system of variables is such that it can be reasonable supposedto have arisen from random sampling, Phil. Mag., 50(1900), 157-172.

[19] I.J. TANEJA, Some Contributions to Information Theory – I (A Survey: On Mea-sures of Information), Journal of Combinatorics, Information and System Sciences,4(1979), 253-274.

[20] I.J. TANEJA, On Generalized Information Measures and Their Applications, Chap-ter in: Advances in Electronics and Electron Physics, Ed. P.W. Hawkes, AcademicPress, 76(1989), 327-413.

[21] I.J. TANEJA, New Developments in Generalized Information Measures, Chapter in:Advances in Imaging and Electron Physics, Ed. P.W. Hawkes, 91(1995), 37-135.

[22] I.J. TANEJA, Generalized Information Measures and their Applications, on linebook: http://www.mtm.ufsc.br/∼taneja/book/book.html, 2001.

[23] I.J. TANEJA, Generalized Relative Information and Information Inequalities, Jour-nal of Inequalities in Pure and Applied Mathematics. 5(1)(2004), Article 21, 1-19.Also in: RGMIA Research Report Collection, http://rgmia.vu.edu.au, 6(3)(2003),Article 10.

[24] I.J. TANEJA, Bounds on Non-Symmetric Divergence Measures in terms of Symmet-ric Divergence Measures - To appear in: Journal of Combinatorics, Information andSystem Sciences.

[25] I.J. TANEJA and P. KUMAR, Relative Information of Type s, Csiszarf−Divergence, and Information Inequalities, Information Sciences, 166(1-4)(2004),105-125. Also in: http://rgmia.vu.edu.au, RGMIA Research Report Collection,6(3)(2003), Article 12.

[26] F. TOPSØE, Some Inequalities for Information Divergence and Related Mea-sures of Discrimination, IEEE Trans. on Inform. Theory, IT-46(2000), 1602-1609. Also in RGMIA Research Report Collection, 2(1)(1999), Article 9 –http://sci.vu.edu.au/∼rgmia.

[27] I. VAJDA, Theory of Statistical Inference and Information, Kluwer Academic Press,London, 1989.

BOUNDS ON TRIANGULAR DISCRIMINATION... 111

112

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TABLE OF CONTENTS,JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.1,2006 FROM GENERALIZED DE GIORGI ESTIMATES TO UPPER GAUSSIAN BOUNDS FOR THE HEAT KERNEL ASSOCIATED TO HIGHER ORDER ELLIPTIC OPERATORS,M.QAFSAOUI,……………………………………………………….9 EXPONENTIAL OPERATORS AND SOLUTION OF PSEUDO-CLASSICAL EVOLUTION PROBLEMS,C.CASSISA,P.RICCI,………………………………….33 NEWTON SUM RULES OF THE ZEROS OF SOME ASSOCIATED ORTHOGONAL q-POLYNOMIALS,C.KOKOLOGIANNAKI,P.SIAFARICAS,I.STABOLAS,……..47 KONDURAR THEOREM AND ITO FORMULA IN RIESZ SPACES,A.BOCCUTO, D.CANDELORO,E.KUBINSKA,……………………………………………………..67 BOUNDS ON TRIANGULAR DISCRIMINATION,HARMONIC MEAN AND SYMMETRIC CHI-SQUARE DIVERGENCES,I.TANEJA,…………………………91

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tel.352-392-9011 pardalos@ufl.edu Optimization,Operations Research 24) Svetlozar T.Rachev Dept.of Statistics and Applied Probability Program University of California,Santa Barbara CA 93106-3110,USA tel.805-893-4869 rachev@pstat.ucsb.edu AND Chair of Econometrics and Statistics School of Economics and Business Engineering University of Karlsruhe Kollegium am Schloss,Bau II,20.12,R210 Postfach 6980,D-76128,Karlsruhe,Germany tel.011-49-721-608-7535 rachev@lsoe.uni-karlsruhe.de Mathematical and Empirical Finance, Applied Probability, Statistics and Econometrics 25) Paolo Emilio Ricci Universita' degli Studi di Roma "La Sapienza" Dipartimento di Matematica-Istituto "G.Castelnuovo" P.le A.Moro,2-00185 Roma,ITALY tel.++39 0649913201,fax ++39 0644701007 riccip@uniroma1.it,Paoloemilio.Ricci@uniroma1.itOrthogonal Polynomials and Special functions, Numerical Analysis, Transforms,Operational Calculus, Differential and Difference equations 26) Cecil C.Rousseau Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA tel.901-678-2490,fax 901-678-2480 ccrousse@memphis.edu Combinatorics,Graph Th., Asymptotic Approximations, Applications to Physics 27) Tomasz Rychlik Institute of Mathematics Polish Academy of Sciences Chopina 12,87100 Torun, Poland T.Rychlik@impan.gov.pl Mathematical Statistics,Probabilistic Inequalities 28) Bl. Sendov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria

tel.515-294-8150 fink@math.iastate.edu Inequalities,Ordinary Differential Equations 10) Sorin Gal Department of Mathematics University of Oradea Str.Armatei Romane 5 3700 Oradea,Romania galso@uoradea.ro Approximation Th.,Fuzzyness,Complex Analysis 11) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis, Memphis,TN 38152,USA tel.901-678-2484 jgoldste@memphis.edu Partial Differential Equations, Semigroups of Operators 12) Heiner H.Gonska Department of Mathematics University of Duisburg Duisburg,D-47048 Germany tel.0049-203-379-3542 office gonska@informatik.uni-duisburg.de Approximation Th.,Computer Aided Geometric Design 13) Dmitry Khavinson Department of Mathematical Sciences University of Arkansas Fayetteville,AR 72701,USA tel.(479)575-6331,fax(479)575-8630 dmitry@uark.edu Potential Th.,Complex Analysis,Holomorphic PDE,Approximation Th.,Function Th. 14) Virginia S.Kiryakova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria virginia@diogenes.bg Special Functions,Integral Transforms, Fractional Calculus 15) Hans-Bernd Knoop Institute of Mathematics Gerhard Mercator University D-47048 Duisburg Germany tel.0049-203-379-2676

bsendov@bas.bg Approximation Th.,Geometry of Polynomials, Image Compression 29) Igor Shevchuk Faculty of Mathematics and Mechanics National Taras Shevchenko University of Kyiv 252017 Kyiv UKRAINE shevchuk@univ.kiev.ua Approximation Theory 30) H.M.Srivastava Department of Mathematics and Statistics University of Victoria Victoria,British Columbia V8W 3P4 Canada tel.250-721-7455 office,250-477-6960 home, fax 250-721-8962 harimsri@math.uvic.ca Real and Complex Analysis,Fractional Calculus and Appl., Integral Equations and Transforms,Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th. 31) Ferenc Szidarovszky Dept.Systems and Industrial Engineering The University of Arizona Engineering Building,111 PO.Box 210020 Tucson,AZ 85721-0020,USA szidar@sie.arizona.edu Numerical Methods,Game Th.,Dynamic Systems, Multicriteria Decision making, Conflict Resolution,Applications in Economics and Natural Resources Management 32) Gancho Tachev Dept.of Mathematics Univ.of Architecture,Civil Eng. and Geodesy 1 Hr.Smirnenski blvd BG-1421 Sofia,Bulgaria Approximation Theory 33) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock Germany manfred.tasche@mathematik.uni-rostock.de Approximation Th.,Wavelet,Fourier Analysis, Numerical Methods,Signal Processing,

knoop@math.uni-duisburg.de Approximation Theory,Interpolation 16) Jerry Koliha Dept. of Mathematics & Statistics University of Melbourne VIC 3010,Melbourne Australia koliha@unimelb.edu.au Inequalities,Operator Theory, Matrix Analysis,Generalized Inverses 17) Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston,RI 02881,USA kulenm@math.uri.edu Differential and Difference Equations 18) Gerassimos Ladas Department of Mathematics University of Rhode Island Kingston,RI 02881,USA gladas@math.uri.edu Differential and Difference Equations

Image Processing,Harmonic Analysis 34) Chris P.Tsokos Department of Mathematics University of South Florida 4202 E.Fowler Ave.,PHY 114 Tampa,FL 33620-5700,USA profcpt@math.usf.edu,profcpt@chuma1.cas.usf.edu Stochastic Systems,Biomathematics, Environmental Systems,Reliability Th. 35) Lutz Volkmann Lehrstuhl II fuer Mathematik RWTH-Aachen Templergraben 55 D-52062 Aachen Germany volkm@math2.rwth-aachen.de Complex Analysis,Combinatorics,Graph Theory

Power series of fuzzy numbers with crossproduct and applications to fuzzy differential

equations

Adrian Ban* and Barnabas Bede***Department of Mathematics,

University of Oradea,Str. Armatei Romane 5,410087 Oradea, Romania

E-mail: aiban@uoradea.ro,**Department of Mechanical and System Engineering,

Budapest Tech, Nepszinhaz u. 8,H-1081 Budapest, Hungary

E-mail: bede.barna@bgk.bmf.hu

Abstract

In a recent paper ([2]) the authors introduced and studied the mainproperties of the cross product of fuzzy numbers. The aim of thispaper is to study power series with the cross product. The propertiesof convergence and the expected value of fuzzy power series are mainlyapproached. As an important application, a solution of the fuzzydifferential equation y′(x) = a¯y(x)⊕b(x), where a is a strict positivefuzzy number and b is a continuous fuzzy number valued mapping, isgiven.

2000 AMS Subject Classification: 26E50, 03E72, 34G10Keywords and phrases: fuzzy number, cross product, power

series, fuzzy differential equation

125JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,125-152,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

1 Introduction

Fuzzy numbers allow us to model the linguistic expressions appeared in dif-ferent scientific areas mainly because of dependence on human judgement,finite resolution of measuring instruments, finite representation of numbersin computers. This explains the increasing interest on theoretical aspectsof fuzzy arithmetic in the last years, especially directed to: operations overfuzzy numbers and properties ([4], [16], [19], [20], [22], [23], [24], [26], [27],[30], [31]), ranking of fuzzy numbers ([6], [11], [13], [28], [36]), canonicalrepresentation of fuzzy numbers ([8], [9], [12], [14], [17]).

Generally, the addition (or triangular norm-based addition) of fuzzy num-bers and the scalar multiplication are considered. The operations of producttype of fuzzy numbers are few studied because the known definitions lead todifficult handling.

The main disadvantage is that the shape of L−R type fuzzy numbers isnot preserved. In many papers (see e.g. [5], [33], [35]) approximative methodsare given to estimate the result of the product of two fuzzy numbers, but thesemay lead to large computational errors after successive applications.

That is why in the paper [2] we introduced and studied a new product,called cross product, over fuzzy numbers and explained the advantages of theuse of this one.

In this paper we prove that good results can be obtained on power seriesof fuzzy numbers with cross product.

The paper is divided as follows. In Section 2 we present some necessarybasic concepts including the definition of a fuzzy number, addition and scalarmultiplication of fuzzy numbers, distance between fuzzy numbers, definitionand properties of the cross product of fuzzy numbers (proved in [2]). Thethird section contains the definition of a fuzzy power series with cross productas a particular fuzzy-number-valued function series, important examples andresults on convergence and expected value of a fuzzy power series. As anapplication, in Section 4, the fuzzy differential equation y′(x) = a ¯ y(x) ⊕b(x), where a is a strict positive fuzzy number and b is a continuous fuzzy-number valued mapping with all the values strict positive or all strict negativeis studied. Finally, a consequence of the main result of this section and anexample are given.

B.BAN,B.BEDE126

2 Basic concepts

Firstly, let us recall the following parametric definition of a fuzzy number(see [10], [29]).

Definition 2.1. A fuzzy number is a pair (u, u) of functions ur, ur, 0 ≤ r ≤ 1which satisfy the following requirements:

(i) ur is a bounded monotonic increasing left continuous function;(ii) ur is a bounded monotonic decreasing left continuous function;(iii) ur ≤ ur, for every r ∈ [0, 1].

We denote by RF the set of all fuzzy numbers. Obviously R ⊂ RF (hereR is understood as R =

x = χx; x is usual real number

). For arbitrary

u = (u, u) ∈ RF , v = (v, v) ∈ RF and k ∈ R we define the addition of u andv, u⊕ v, and the scalar multiplication of u by k, k · u or u · k, as

(u⊕ v)r = ur + vr,

(u⊕ v)r

= ur + vr

(k · u)r =

kur, if k ≥ 0,kur, if k < 0,

(k · u)r

=

kur, if k ≥ 0,kur, if k < 0.

We denote by ªu = (−1) · u the symmetric of u ∈ RF and 1µ· u by u

µif

µ 6= 0.A fuzzy number u ∈ RF is said to be positive if u1 ≥ 0, strict positive if

u1 > 0, negative if u1 ≤ 0 and strict negative if u1 < 0. We say that u and vhave the same sign if they are both strict positive or both strict negative. Ifu is positive (negative) then ªu is negative (positive).

The following definitions and results are useful in the paper.

Definition 2.2. For arbitrary fuzzy numbers u = (u, u) and v = (v, v) thequantity

D(u, v) = sup0≤r≤1

max|ur − vr| , |ur − vr|

is called the distance between u and v.

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 127

It is well-known (see e.g. [7]) that (RF , D) is a complete metric space andD has the following properties:

(i) D (u⊕ w, v ⊕ w) = D (u, v) , for all u, v, w ∈ RF ;(ii) D (k · u, k · v) = |k|D (u, v) , for all u, v ∈ RF and k ∈ R;(iii) D (u⊕ v, w ⊕ t) ≤ D (u,w) + D (v, t) , for all u, v, w, t ∈ RF .If we denote ||u||F = D(u, 0),∀u ∈ RF , where 0 = χ0 is the neutral

element with respect to ⊕, then || · ||F has the properties of an usual normon RF , i.e., ||u||F = 0 if and only if u = 0, ||k · u||F = |k| ||u||F , ||u⊕ v||F ≤||u||F + ||v||F , |||u||F − ||v||F | ≤ D(u, v).

Definition 2.3. (see [1]) A fuzzy-number-valued function f : [a, b] →RF iscalled Riemann integrable to I ∈RF , if for any ε > 0, there exists δ > 0 suchthat for any division P = [u, v], ξ of [a, b] with the norm ∆(P ) < δ, wehave

D

(∑P

⊕(v − u)¯ f(ξ), I

)< ε,

where∑⊕ is the addition with respect to ⊕ in RF .

We write I = (FR)∫ b

af(x)dx.

Let x, y ∈ RF . If there exists z ∈ RF such that x = y ⊕ z, then we call zthe H-difference of x and y, denoted by x− y.

Definition 2.4. ([18], [32]) A function f : A → RF is said to be differentiable

at x ∈ A if there exists f ′(x) ∈ RF such that the limits limh0f(x+h)−f(x)

hand

limh0f(x)−f(x−h)

hexist and are equal to f ′(x). We call f

′(x) the derivative

of f at x. If f is differentiable at any x ∈ A we call f differentiable.

Lemma 2.5. (i) Let f : [a, b] → RF be differentiable at c ∈ [a, b]. Then f iscontinuous at c (see [1, Lemma 5, Section 1]).

(ii) Let f, g : (a, b) → RF be continuous functions. If the H-differencefunction f − g exists on (a, b) then f − g is continuous on (a, b) (see [1,Lemma 1, Section 2]).

(iii) Let I be an open interval of R, and let f, g : I → RF be differentiablefunctions with the derivatives f ′, g′. Then (f⊕g)′ exists and (f⊕g)′ = f ′⊕g′

(see [1, Proposition 5, Section 2]).

The concept of expected value of a fuzzy number was introduced andstudied in [12]-[14] as a good representation of a fuzzy number as a crisp

B.BAN,B.BEDE128

number. In our notations, if u is a fuzzy number then the expected valueEV (u) is given by (see Definition 2 and Lemma 3 in [12])

EV (u) =1

2

1∫

0

(ur + ur) dr.

It is obvious that EV (ªu) = −EV (u), for every fuzzy number u.Let us denote R∗F = u ∈ RF ; u is strict positive or strict negative and

let R¯F = R∗F ∪ 0.Definition 2.6. ([2])The binary operation ¯ on R¯F defined by w = u ¯v, [w]r = [wr, wr], for every r ∈ [0, 1], where

wr =

urv1 + u1vr − u1v1, if u1 > 0 and v1 > 0urv1 + u1vr − u1v1, if u1 < 0 and v1 < 0urv1 + u1vr − u1v1, if u1 < 0 and v1 > 0urv1 + u1vr − u1v1, if u1 < 0 and v1 < 0

and

wr =

urv1 + u1vr − u1v1, if u1 > 0 and v1 > 0urv1 + u1vr − u1v1, if u1 > 0 and v1 < 0urv1 + u1vr − u1v1, if u1 < 0 and v1 > 0urv1 + u1vr − u1v1, if u1 < 0 and v1 < 0

for any u, v ∈ R∗F and u ¯ 0 = 0 ¯ u = 0 for any u ∈ R¯F is called the crossproduct of fuzzy numbers.

Remark 2.7. 1) Starting from the formulas of calculus in the case u and vstrict positive we can deduce: u ¯ v = ª (u¯ (ªv)) if u is strict positiveand v is strict negative, u¯ v = ª ((ªu)¯ v) if u is strict negative and v isstrict positive and u ¯ v = (ªu) ¯ (ªv) if u and v are strict negative. Thecross product u¯ v is strict positive if u and v have the same sign and strictnegative contrariwise.

2) The cross product extends the scalar multiplication of fuzzy numbersbecause R ⊂ R¯F .

3) Similar operation can be defined on the set R#F = u ∈ RF ; there exist

a unique x0 ∈ R such that u(x0) = 1 as follows w = u¯ v, [w]r = [wr, wr],for every r ∈ [0, 1] , where wr = urv1+u1vr−u1v1 and wr = urv1+u1vr−u1v1

for every u, v positive fuzzy numbers. Let also, u ¯ v = ª (u¯ (ªv)) if u

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 129

is positive and v is negative, u ¯ v = ª ((ªu)¯ v) if u is negative and v ispositive and u¯ v = (ªu) ¯ (ªv) if u and v are negative. We observe thatthe results of this paper hold with respect to this operation too. The mainalgebraic properties of the cross product are given by the following.

Theorem 2.8. ([2])If u, v, w ∈ R¯F then(i) (ªu)¯ v = u¯ (ªv) = ª (u¯ v) ;(ii) u¯ v = v ¯ u;(iii) (u¯ v)¯ w = u¯ (v ¯ w) ;(iv) If u and v are strict positive or u and v are strict negative then

(u⊕ v)¯ w = (u¯ w)⊕ (v ¯ w) .

We present other important properties of the cross product.

Theorem 2.9. ([2]) (i)If u, v have the same sign and w ∈ R¯F then

D (w ¯ u,w ¯ v) ≤ KwD (u, v) ,

where Kw = max|w1| , |w1|+ w0 − w0.(ii) If u, v ∈ R#

F and we denote u1 = u1 = u1, v1 = v1 = v1then

EV (u¯ v) = u1EV (v) + v1EV (u)− u1v1.

(iii) If u ∈ R#F then

EV(u¯n

)= n

(u1

)n−1EV (u)− (n− 1)

(u1

)n

for every n ∈ N∗, where u¯n = u¯ ...¯ u︸ ︷︷ ︸n times

.

(iv) If (un) is a sequence of strict positive or strict negative fuzzy numberssuch that un → u ∈ R¯F then c¯un → c¯u, for every c ∈ R¯F (the convergencein the metric D is considered).

(v) If f, g : [a, b] → RF are continuous at x0 ∈ [a, b] and f (x0) , g (x0) arestrict positive or strict negative then f ¯ g is continuous at x0.

(vi) If f, g : [a, b] → RF are differentiable at x0 ∈ [a, b] , f (x0) , g (x0)are strict positive or strict negative and for sufficiently small h > 0 the H-differences f(x0 + h)− f(x0) and f(x0)− f(x0 − h) exist and have the samesign as f(x0), the H-differences g(x0 +h)− g(x0) and g(x0)− g(x0−h) existand have the same sign as g(x0), then f ¯ g is differentiable at x0 and

(f ¯ g)′(x0) = f (x0)¯ g′(x0)⊕ f ′(x0)¯ g(x0).

B.BAN,B.BEDE130

(vii) If f : [a, b] → RF is differentiable at x0 ∈ [a, b] , f (x0) is strictpositive or strict negative and for sufficiently small h > 0 the H-differencesf(x0 +h)−f(x0) and f(x0)−f(x0−h) exist and have the same sign as f(x0)then c¯ f is differentiable at x0 for any c ∈ R¯F and

(c¯ f)′(x0) = c¯ f ′(x0).

3 Power series with cross product

A series of fuzzy-number-valued functions is a function series∑⊕

n fn, wherefn : A(⊆ R or RF) → RF , and

∑⊕ means the countable sum with respectto the addition of fuzzy numbers. The concepts of convergence and uniformconvergence are considered in the metric space (RF , D). We say that theseries

∑⊕n fn is absolutely (uniformly) convergent if

∑n ‖fn‖F is (uniformly)

convergent. Because (RF ,⊕, ·) is not a linear space over R, (RF , ‖·‖F) isnot a normed space. We need to prove some lemmas which are analogous tosome classical results in normed spaces.

Lemma 3.1. (i) If the series∑⊕

n fn is absolutely convergent on a set A thenit is convergent on A.

(ii) If the series∑⊕

n fn is absolutely uniformly convergent on a set A thenit is uniformly convergent on A.

Proof. (i) Let x ∈ A and∑

n ‖fn(x)‖F be convergent. By [34, Theorem 1]we have

[∑n

⊕fn(x)

]r

=∑

n

[fn(x)]r =

[∑n

fn(x)r,∑

n

fn(x)r

].

Then

∑n

‖fn(x)‖F =∑

n

D(fn(x), 0) =∑

n

supr∈[0,1]

max|fn(x)r|, |fn(x)r|.

For any r ∈ [0, 1] we obtain

∣∣∣∣∣∑

n

fn(x)r

∣∣∣∣∣ ≤∑

n

|fn(x)r| ≤∑

n

‖fn(x)‖F .

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 131

Since the last series converges, it follows that∑

n fn(x)r converges. The same

reasoning implies that∑

n fn(x)r

converges.(ii) For any ε > 0 there exists N(ε) ∈ N such that

‖fn+1(x)⊕ ...⊕ fn+p(x)‖F ≤ ‖fn+1(x)‖F + ... + ‖fn+p(x)‖F < ε,

∀n ≥ N(ε), p ≥ 1, ∀x ∈ A. The rest is obvious by Cauchy criterion and theinvariance to translation of the Hausdorff metric D.

Lemma 3.2. If fn : A(⊂ R or RF) → RF are continuous functions and ifthe series of fuzzy-number-valued functions

∑⊕n fn is uniformly convergent to

f : A → RF then f is continuous.

Proof. By the properties of the distance D we have for n ∈ N

D(f(x), f(y)) ≤ D

(f(x),

n∑

k=0

⊕fk(x)

)

+ D

(n∑

k=0

⊕fk(x),

n∑

k=0

⊕fk(y)

)+ D

(n∑

k=0

⊕fk(y), f(y)

)

≤ D

(f(x),

n∑

k=0

⊕fk(x)

)+

n∑

k=0

D(fk(x), fk(y))

+ D

(n∑

k=0

⊕fk(y), f(y)

)

for every x, y ∈ A and n ∈ N.The property of uniform convergence applied to the first and the last term

and the continuity of each function fk, k = 0, ..., n complete the proof.

In what follows we study fuzzy power series as particular series of fuzzy-number-valued functions by using the cross product of fuzzy numbers.

Definition 3.3. The function series∑∞

n=0⊕

an ¯ x¯n, where an, x ∈ R¯F ,n ∈ N and x¯n denotes x¯ ...¯ x︸ ︷︷ ︸

n times

if n ≥ 1 and by convention x¯0 = 1 = χ1,

is called ¯-power series.

The following theorem is an analogous of Abel’s first theorem.

B.BAN,B.BEDE132

Theorem 3.4. Let∑∞

n=0⊕

an ¯ x¯n be a ¯-power series. Then there existR1, R2 ∈ R, R1 ≤ R2 such that

(i)∑∞

n=0⊕

an¯x¯n converges on B(0, R1)∩(R¯F), where B(0, R1) = x ∈RF ; ‖x‖F < R1;

(ii)∑∞

n=0⊕

an ¯ x¯n converges uniformly on B(0, R) ∩ (R¯F), ∀R < R1

(A means the closure of the set A);(iii) If an and x have the same sign then

∑∞n=0

⊕an ¯ x¯n diverges for

every x with |x1| > R2 or |x1| > R2.

Proof. (i) For x ∈ R∗F , by Theorem 2.9, (i) we have:

∥∥an ¯ x¯n∥∥F = D(an ¯ x¯n, 0¯ x¯n) ≤ Kn

x D(an, 0) = Knx ‖an‖F (1)

where Kx = max|x1|, |x1|+ x0 − x0. The case x = 0 is obvious. Let us de-note by r1 the ray of convergence of the classical power series

∑∞n=0 ‖an‖F un.

Then the series∑∞

n=0 ‖an‖F un converges for u ∈ (−r1, r1) and converges uni-formly on [−r, r] for all r < r1. Let R1 = 1

3r1 and x ∈ B(0, R1) ⊆ RF i.e.

‖x‖F < R1. Thensup

r∈[0,1]

max|xr|, |xr| < R1

thereforeKx ≤ max

|x1|, |x1| + |x0|+ |x0| < 3R1 = r1.

By (1) and Lemma 3.1, (i) it follows that, for x ∈ B(0, R1), the series∑∞n=0

⊕an ¯ x¯n converges.

(ii) It follows from Lemma 3.1, (ii) and the above proof.(iii) Let R2 be the ray of convergence of the series

∑n a1

nun and R2 be theray of convergence of the series

∑n a1

nun . We denote R2 = maxR2, R2.Let |x1| > R2 or |x1| > R2 and let us suppose that the ¯-power series∑∞

n=0⊕

an ¯ x¯n converges. If an and x are positive, then by [34, Theorem1]

we have

[ ∞∑n=0

⊕an ¯ x¯n

]1

=

[ ∞∑n=0

a1n(x1)n,

∞∑n=0

a1n(x1)n

].

But at least one of the series∑∞

n=0 a1n(x1)n and

∑∞n=0 a1

n(x1)n diverges, whichis a contradiction. As a conclusion

∑∞n=0

⊕an ¯ x¯n is divergent.

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 133

Remark 3.5. On R#F a similar result can be obtained.

Corollary 3.6. If an ∈ R¯F , n ∈ N and x ∈ R then there exists R1 suchthat the series

∑∞n=0

⊕an · xn converges on (−R1, R1), converges uniformly

on [−R, R], ∀R < R1 and diverges on R \ [−R1, R1].

Proof. Since ‖an · xn‖F = |x|n · ‖an‖F , choosing R1 the ray of convergenceof the crisp power series

∑∞n=0 ‖an‖F un, we obtain the statement of the

corollary.

Remark 3.7. If∑∞

n=0⊕

an ¯ x¯n converges uniformly on a set A ⊂ R¯F thenit is continuous on A. Indeed, by Theorem 2.9, (i) and (v) we obtain thatan ¯ x¯n is continuous as a function of x and by Lemma 3.2 it follows that∑∞

n=0⊕

an ¯ x¯n is continuous.

Remark 3.8. If∑∞

n=0 ‖an‖F un converges for any u ∈ R then∑∞

n=0⊕

an¯x¯n

converges for all x ∈ R¯F and the convergence is uniform on any closed ballin this set.

Theorem 3.9. If∑∞

n=0⊕

an¯x¯n is a ¯-power series with an 6= 0, for every

n ∈ N, then the number R1 in Theorem 3.4 is R1 = 13limn→∞(‖an‖F)−

1n or

R1 = 13limn→∞

‖an‖F‖an+1‖F .

Proof. Directly from Theorem 3.4and Abel’s second Theorem for crisp series.

The following examples of ¯-power series define some fuzzy analogous ofexponential, logarithmic, sine and cosine functions.

Example 3.10. (i) The series∑∞

n=0⊕ 1

n!· x¯n converges for any x ∈ R¯F

because the number R1 in Theorem 3.4 is R1 = 13limn→∞

‖an‖F‖an+1‖F = +∞. We

denote its sum by ex¯.

(ii) The series∑∞

n=1⊕ (−1)n+1

n· x¯n converges for any strict positive or

strict negative fuzzy number x ∈ B(0, 13) because limn→∞

‖an‖F‖an+1‖F = 1. We

denote its sum by ln¯(1 + x).

(iii) The series∑∞

n=0⊕ (−1)n

(2n+1)!· x¯(2n+1) converges for any x ∈ R¯F and

we denote its sum by sin¯ x.(iv) The series

∑∞n=0

⊕ (−1)n

(2n)!· x¯(2n) converges for any x ∈ R¯F and we

denote its sum by cos¯ x.

B.BAN,B.BEDE134

Similar results can be obtained for the operation ¯ defined on R#F .

In the following theorem we compute the expected value of a ¯-powerseries.

Theorem 3.11. (i) If the power series∑∞

n=0⊕

an ¯ x¯n converges, where

an, x ∈ R#F , then

EV

( ∞∑n=0

⊕an ¯ x¯n

)= EV (a0) +

∞∑n=1

(a1nn(x1)n−1(EV (x)− x1)

+ EV (an)(x1)n);

(ii) If, in addition, an are symmetric fuzzy numbers (i.e. a1n−an

r = anr−a1

n

for all r ∈ [0, 1]) then

EV

( ∞∑n=0

⊕an ¯ x¯n

)= EV (a0)+

∞∑n=1

EV (an)[n(x1)n−1EV (x)−(n−1)(x1)n];

(iii) If, in addition, x is a symmetric fuzzy number then

EV

( ∞∑n=0

⊕an ¯ x¯n

)=

∞∑n=0

EV (an)(EV (x))n.

Proof. (i) We observe that if (un) is a sequence of fuzzy numbers convergingto u then EV (un) → EV (u). We get

EV

( ∞∑n=0

⊕an ¯ x¯n

)= lim

n→∞EV

(n∑

k=0

⊕ak ¯ x¯k

)= lim

n→∞

n∑

k=0

EV (ak¯x¯k).

Then by Theorem 2.9, (ii) we have

EV

( ∞∑n=0

⊕an ¯ x¯n

)= lim

n→∞

n∑

k=0

(a1kEV (x¯k) + (EV (ak)− a1

k)(x1)k).

By Theorem 2.9, (iii) we obtain

EV

( ∞∑n=0

⊕an ¯ x¯n

)

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 135

= EV (a0)+ limn→∞

n∑

k=1

(a1k[k(x1)k−1EV (x)−(k−1)(x1)k]+(EV (ak)−a1

k)(x1)k)

= EV (a0) + limn→∞

n∑

k=1

(a1kk(x1)k−1

(EV (x)− x1

)+ EV (ak)(x

1)k),

which proves (i). For (ii) we observe that EV (an) = a1n, ∀n ∈ N, and for (iii)

we have EV (x) = x1.

Example 3.12. We compute the expected values of the ¯-power series inExample 3.10. We have

EV (ex¯) = EV

( ∞∑n=0

⊕1

n!· x¯n

)= 1 +

∞∑n=1

1

n![n(x1)n−1EV (x)− (n− 1)(x1)n]

= 1 + EV (x)∞∑

n=1

(x1)n−1

(n− 1)!−

∞∑n=1

(n− 1)(x1)n

n!

= EV (x)ex1 − x1ex1

+ ex1

= ex1

(1 + EV (x)− x1),

for any x ∈ R#F .

We observe that if x is symmetric then EV (ex¯) = eEV (x).

Also,

EV (ln¯(1 + x)) =∞∑

n=1

(−1)n+1

n[n(x1)n−1EV (x)− (n− 1)(x1)n]

=∞∑

n=1

(−1)n+1(EV (x)− x1)(x1)n−1 + ln(1 + x1)

=EV (x)− x1

1− x1+ ln(1 + x1),

for any x ∈ R#F ∩ B

(0, 1

3

). In addition, if x is symmetric then EV (ln¯(1 +

x)) = ln(1 + EV (x)).It is easy to check that

EV (sin¯ x) = (EV (x)− x1) cos x1 + sin x1

andEV (cos¯ x) = (x1 − EV (x)) sin x1 + cos x1,

B.BAN,B.BEDE136

for every x ∈ R#F .

For almost symmetric fuzzy numbers, or fuzzy numbers with small sup-port, the expected values of the power series presented in Example 3.10 arenear to eEV (x), ln(1 + EV (x)), sin(EV (x)), cos(EV (x)), respectively. Thissuggest us the possible use of these series as definitions for fuzzy exponential,logarithmic, sine and cosine functions.

4 Application to fuzzy differential equations

In this section we study the fuzzy differential equation

y′(x) = a¯ y(x)⊕ b(x), (2)

where a is a strict positive fuzzy number and b : R→ RF is continuous andall its values are strict positive or all are strict negative.

First we prove the following lemma on differentials of function sequences.

Lemma 4.1. Let (fn)n∈N, fn : [a, b] → RF be a sequence of differentiablefunctions converging uniformly to f : [a, b] → RF . If there exists β > 0such that the H-differences fn(x + h) − fn(x), fn(x) − fn(x − h) exist forall 0 < h < β and n ∈ N and if f ′n is uniformly convergent to g then f isdifferentiable and f ′ = g.

Proof. Let x ∈ [a, b]. Since fn is differentiable, the H-differences fn(x + h)−fn(x), n ∈ N exist for sufficiently small h, i.e., there exists βn ∈ R suchthat for 0 < h < βn the H-differences fn(x + h) − fn(x), n ∈ N exist. Wecan choose βn such that infn∈N βn ≥ β. Then for 0 < h < β there existsun(h) such that un(h) ⊕ fn(x) = fn(x + h), for every n ∈ N. We prove thatthe function sequence (un) is uniformly convergent on (0, β). Indeed, by theinvariance to translation of the Hausdorff distance we obtain

D(un(h), um(h)) = D(un(h)⊕ fn(x), um(h)⊕ fn(x))

= D(fn(x + h), um(h)⊕ fn(x))

= D(fn(x + h)⊕ fm(x), um(h)⊕ fm(x)⊕ fn(x))

= D(fn(x + h)⊕ fm(x), fm(x + h)⊕ fn(x)),

for every m,n ∈ N and h ∈ (0, β). Then by the property (iii) of D (seeSection 2) we obtain

D(un(h), um(h)) ≤ D(fm(x), fn(x)) + D(fn(x + h), fm(x + h)).

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 137

The uniform convergence of the sequence (fn) implies (un) uniformly fun-damental. Since (RF , D) is complete it follows that (un) is uniformly con-vergent. Let us denote u the limit of the function sequence (un), that islimn→∞ un(h) = u(h) for every h ∈ (0, β). Since fn(x + h) = fn(x) ⊕ un(h)for every n ∈ N and h ∈ (0, β), we have

D(f(x + h), f(x)⊕ u(h)) ≤

D(f(x + h), fn(x + h)) + D(fn(x)⊕ un(h), f(x)⊕ u(h))

≤ D(f(x + h), fn(x + h)) + D(fn(x), f(x)) + D(un(h), u(h)),

for every n ∈ N and h ∈ (0, β). Passing to limit with n →∞ we obtain thatf(x)⊕ u(h) = f(x + h) for 0 < h < β, i.e., the H-difference f(x + h)− f(x)exists for 0 < h < β. Also, we get

D

(g(x),

f(x + h)− f(x)

h

)≤ D (g(x), f ′n(x)) (3)

+ D

(f ′n(x),

fn(x + h)− fn(x)

h

)

+ D

(fn(x + h)− fn(x)

h,f(x + h)− f(x)

h

),

for every n ∈ N and h ∈ (0, β). The properties of the Hausdorff distanceD in Section 2 imply

D

(fn(x + h)− fn(x)

h,f(x + h)− f(x)

h

)

=1

hD (fn(x + h)− fn(x), f(x + h)− f(x)) =

=1

hD (fn(x + h)⊕ f(x), f(x + h)⊕ fn(x)) ≤

≤ 1

h[D(fn(x + h), f(x + h)) + D(fn(x), f(x))] , (4)

for every n ∈ N and h ∈ (0, β). Passing to limit with n → ∞ in (3) andtaking into account (4) we obtain

D

(g(x),

f(x + h)− f(x)

h

)≤ lim

n→∞D

(f ′n(x),

fn(x + h)− fn(x)

h

)

B.BAN,B.BEDE138

for every h ∈ (0, β), and finally passing to limit with h 0 we obtain

D

(g(x), lim

h0

f(x + h)− f(x)

h

)≤ lim

n→∞D

(f ′n(x), lim

h0

fn(x + h)− fn(x)

h

),

i.e., the limit limh0f(x+h)−f(x)

hexists and it is equal to g(x). Analogously

can be proved that limh0f(x)−f(x−h)

h= g(x) and the proof is complete.

Lemma 4.2. If a is a strict positive fuzzy number and ea·(x−x0)¯ is defined by

the power series in Example 3.10 then ea·(x−x0)¯ is differentiable and

(ea·(x−x0))′ = a¯ ea·(x−x0)¯ ,

for every x, x0 ∈ R, x > x0.

Proof. Because (λ + µ) · u = λ · u⊕ µ · u for every λ, µ ∈ R, λ, µ > 0 and ua fuzzy number, we get

1

k!· a¯k · (x− x0 + h)k =

1

k!· a¯k · (x− x0)

k (5)

⊕ 1

k!· a¯k · [(x− x0 + h)k − (x− x0)

k],

for all h > 0, k ∈ N and x, x0 ∈ R, x > x0. Let n ∈ N∗. In the above relationwe obtain

n∑

k=0

⊕1

k!· a¯k · (x− x0 + h)k =

n∑

k=0

⊕1

k!· a¯k · (x− x0)

k (6)

⊕n∑

k=0

⊕1

k!· a¯k · [(x− x0 + h)k − (x− x0)

k],

for all h > 0, x, x0 ∈ R, x > x0. Let us denote Sn(x) =∑n

k=0⊕ 1

k!· a¯k · (x−

x0)k. Then the H-differences

Sn(x + h)− Sn(x) =n∑

k=0

⊕1

k!· a¯k · [(x− x0 + h)k − (x− x0)

k]

exist for all h > 0 (that is the first condition in Lemma 4.1

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 139

is verified), x, x0 ∈ R, x > x0. Because

(x− x0)k = (x− x0 − h + h)k = (x− x0 − h)k + v,

with v > 0 for all h > 0, h < x− x0, we similarly obtain the existence of theH-differences

1

k!· a¯k · (x− x0)

k − 1

k!· a¯k · (x− x0 − h)k = (7)

=1

k!· a¯k · [(x− x0)

k − (x− x0 − h)k],

for all 0 < h < x − x0, x, x0 ∈ R, x > x0. The same reasoning as aboveimplies that

Sn(x)− Sn(x− h) =n∑

k=0

⊕1

k!· a¯k · [(x− x0)

k − (x− x0 − h)k],

i.e., the H-differences Sn(x) − Sn(x − h) exist for all 0 < h < x − x0. Thecondition in Lemma 4.1

is satisfied with β = x − x0 > 0. Also, by (5) and the properties of themetric D we get

limh0

1

h·[

1

k!· a¯k · (x− x0 + h)k − 1

k!· a¯k · (x− x0)

k

]

= limh0

1

h

1

k!· a¯k · [(x− x0 + h)k − (x− x0)

k] =1

k!· a¯k · k(x− x0)

k−1

and by (7),

limh0

1

h·[

1

k!· a¯k · (x− x0)

k − 1

k!· a¯k · (x− x0 − h)k

]

= limh0

1

h

1

k!· a¯k · [(x− x0)

k − (x− x0 − h)k] =1

k!· a¯k · k(x− x0)

k−1,

for every x, x0 ∈ R, x > x0 and k ∈ N∗. We obtain

(1

k!· a¯k · (x− x0)

k

)′=

1

(k − 1)!· a¯k · (x− x0)

k−1, ∀k ∈ N∗.

B.BAN,B.BEDE140

Because the terms a¯(k−1) · (x−x0)k−1

(k−1)!are strict positive for every k ∈ N∗,

Lemma 2.5, (iii) implies

S ′n(x) =n∑

k=1

⊕1

(k − 1)!· a¯k · (x− x0)

k−1 = a¯ Sn−1(x).

Because (Sn)n∈N converges uniformly to ea·(x−x0)¯ and Sn is a strict positive

fuzzy number for every n ∈ N, it follows, by Theorem 2.9, (iv), that (S ′n)n∈Nconverges uniformly to a¯ e

a·(x−x0)¯ . Finally, Lemma 4.1 implies e

a·(x−x0)¯ dif-

ferentiable and(e

a·(x−x0)¯

)′= a¯ e

a·(x−x0)¯ , for every x, x0 ∈ R, x > x0.

Theorem 4.3. The fuzzy-number-valued function y : R→ RF given by

y(x) = c¯ ea·(x−x0)¯ ⊕ (FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt, (8)

where a is a strict positive fuzzy number, is a solution of the equation (2) forall x > x0 and c ∈ R¯F having the same sign as b(t), t > x0 (if c 6= 0).

Proof. First we observe that, by Remark 3.7, ea·(x−t)¯ is continuous for t ∈

[x0, x]. Then in Theorem 2.9, (v) it follows that b(t)¯ ea·(x−t)¯ is continuous

as a fuzzy-number-valued function of t, and by [7, Theorem 3.7]it is integrable. So, we have proved that the function y is well defined. If

the last term in (8) is differentiable, then by Theorem 2.9, (vii), Lemma 2.5,(iii) and Lemma 4.2

we get

y′(x) = c¯ a¯ ea·(x−x0)¯ ⊕

((FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt

)′(9)

Let us prove that (FR)∫ x

x0b(t)¯e

a·(x−t)¯ dt is differentiable and to compute

its derivative. We have

(FR)

∫ x+h

x0

b(t)¯ ea·(x+h−t)¯ dt = (FR)

∫ x

x0

b(t)¯ ea·(x+h−t)¯ dt

⊕ (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt.

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 141

Since the H-difference u(t) = ea·(x+h−t)¯ −e

a·(x−t)¯ exists (by Lemma 4.2) and it

is continuous (by Lemma 2.5, (ii)) for any h > 0 and t ∈ [x0, x], it follows that

u(t)⊕ea·(x−t)¯ = e

a·(x+h−t)¯ . We observe that u(t) and e

a·(x−t)¯ are strict positive

and by Theorem 2.8, (iv) we have b(t)¯u(t)⊕b(t)¯ea·(x−t)¯ = b(t)¯e

a·(x+h−t)¯

with all the terms continuous and so integrable (see [7, Theorem 3.7]). Thenwe get [7, Theorem 2.5 and Theorem 2.6]

(FR)

∫ x

x0

b(t)¯ ea·(x+h−t)¯ dt = (FR)

∫ x

x0

b(t)¯ u(t)dt

⊕(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt

and it follows

(FR)

∫ x+h

x0

b(t)¯ ea·(x+h−t)¯ dt = (FR)

∫ x

x0

b(t)¯ u(t)dt

⊕ (FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt⊕ (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt.

Then the H-difference (FR)∫ x+h

x0b(t)¯e

a·(x+h−t)¯ dt−(FR)

∫ x

x0b(t)¯e

a·(x−t)¯ dt

exists and we have

limh0

1

h·(

(FR)

∫ x+h

x0

b(t)¯ ea·(x+h−t)¯ dt− (FR)

∫ x

x0

b(t)¯ ea·(x+h−t)¯ dt

)

= limh0

1

h·(

(FR)

∫ x

x0

b(t)¯ u(t)dt⊕ (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)

= limh0

1

h·(

(FR)

∫ x

x0

b(t)¯(e

a·(x+h−t)¯ − e

a·(x−t)¯

)dt (10)

⊕(FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)

By [7, Theorem 2.5 and Theorem 2.6]we obtain

limh0

1

h·(

(FR)

∫ x

x0

b(t)¯(e

a·(x+h−t)¯ − e

a·(x−t)¯

)dt

)=

B.BAN,B.BEDE142

= limh0

(FR)

∫ x

x0

b(t)¯ ea·(x+h−t)¯ − e

a·(x−t)¯

hdt

Also, [7, Remark 3.1 and Remark 3.2]and Theorem 2.9, (i) imply

D

((FR)

∫ x

x0

b(t)¯ ea·(x+h−t)¯ − e

a·(x−t)¯

hdt, (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt

)≤

∫ x

x0

D

(b(t)¯ e

a·(x+h−t)¯ − e

a·(x−t)¯

h, b(t)¯ a¯ e

a·(x−t)¯

)dt ≤

∫ x

x0

Kb(t)D

(e

a·(x+h−t)¯ − e

a·(x−t)¯

h, a¯ e

a·(x−t)¯

)dt,

where Kb(t) = supt∈[x0,x]max|b(t)1|, |b(t)1|+ b(t)0− b(t)0. Lemma 4.2 im-

plies

limh0

D

(e

a·(x+h−t)¯ − e

a·(x−t)¯

h, a¯ e

a·(x−t)¯

)= 0,

therefore

limh0

1

h·(

(FR)

∫ x

x0

b(t)¯(e

a·(x+h−t)¯ − e

a·(x−t)¯

)dt

)=

= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt.

In relation (10) we get

limh0

1

h·(

(FR)

∫ x+h

x0

b(t)¯ ea·(x+h−t)¯ dt− (FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt

)=

= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ lim

h0

1

h· (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt.

Theorem 2.6, Remark 3.2 in [7]and the properties of the distance D imply

D

(b(x),

1

h· (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)=

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 143

= D

((FR)

∫ x+h

x

1

h· b(x)dt,

1

h· (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)≤

≤∫ x+h

x

D

(1

h· b(x), b(t)¯ 1

h· ea·(x+h−t)¯

)dt

=1

h

∫ x+h

x

D(b(x), b(t)¯ e

a·(x+h−t)¯

)dt.

In addition,

D(b(x), b(t)¯ e

a·(x+h−t)¯

)≤ D (b(x), b(t)) + D

(b(t), b(t)¯ e

a·(x+h−t)¯

)≤

(Theorem 2.9, (i)) ≤ ω(b, h) + Kb(t)D(1, ea·(x+h−t)¯ ) (11)

for every t ∈ [x, x + h], with ω and Kb(t) as above. We get

D

(b(x),

1

h· (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)

≤ 1

h

∫ x+h

x

ω(b, h)dt + Kb(t)1

h

∫ x+h

x

D(1, ea·(x+h−t)¯ )dt.

We also observe that

D(1, ea·(x+h−t)¯ ) = lim

n→∞D

(1, 1⊕ 1

1!a · (x + h− t)⊕ ...⊕ 1

n!a¯n · (x + h− t)n

)

(12)

= limn→∞

D

(0,

1

1!a · (x + h− t)⊕ ...⊕ 1

n!a¯n · (x + h− t)n

)

≤ limn→∞

(∥∥∥∥1

1!a · (x + h− t)

∥∥∥∥F

+ ... +

∥∥∥∥1

n!a¯n · (x + h− t)n

∥∥∥∥F

)

≤ limn→∞

(|x + h− t|

∥∥∥∥1

1!· a

∥∥∥∥F

+ ... + |x + h− t|n∥∥∥∥

1

n!· a¯n

∥∥∥∥F

),

therefore1

h

∫ x+h

x

D(1, ea·(x+h−t)¯ )dt

B.BAN,B.BEDE144

≤ 1

hlim

n→∞

(∥∥∥∥1

1!· a

∥∥∥∥F

∫ x+h

x

|x + h− t|dt + ...

+

∥∥∥∥1

n!· a¯n

∥∥∥∥F

∫ x+h

x

|x + h− t|ndt

)

=1

hlim

n→∞

(h2

2

∥∥∥∥1

1!· a

∥∥∥∥F

+ ... +hn+1

n + 1

∥∥∥∥1

n!· a¯n

∥∥∥∥F

).

Passing to limit we obtain

limh0

1

h

∫ x+h

x

D(1, ea·(x+h−t)¯ )dt ≤

≤ limn→∞

limh0

1

h

(h2

2

∥∥∥∥1

1!· a

∥∥∥∥F

+ ... +hn+1

n + 1

∥∥∥∥1

n!· a¯n

∥∥∥∥F

)= 0.

Since b is continuous, ω(b, h) → 0 for h 0 (see e.g. [21]). We obtain

D

(b(x), lim

h0

1

h· (FR)

∫ x+h

x

b(t)¯ ea·(x+h−t)¯ dt

)= 0 (13)

for every x > x0, which means limh01h· (FR)

∫ x+h

xb(t)¯e

a·(x+h−t)¯ dt = b(x),

for every x > x0. Then in (10) we have

limh0

1

h·(

(FR)

∫ x+h

x0

b(t)¯ ea·(x+h−t)¯ dt− (FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt

)

= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ b(x), (14)

for every x > x0.For the symmetric case we observe that

(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt = (FR)

∫ x−h

x0

b(t)¯ ea·(x−t)¯ dt

⊕ (FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt,

the H-difference v(t) = ea·(x−t)¯ − e

a·(x−h−t)¯ exists and it is continuous for

any t ∈ [x0, x−h] and 0 < h < x−x0 (see Lemma 2.5, (ii)). Also, by Lemma

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 145

4.2, v(t) and ea·(x−h−t)¯ are strict positive and we get

(FR)

∫ x−h

x0

b(t)¯ ea·(x−t)¯ dt = (FR)

∫ x−h

x0

b(t)¯ v(t)dt

⊕ (FR)

∫ x−h

x0

b(t)¯ ea·(x−h−t)¯ dt,

for every 0 < h < x− x0, therefore

(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt = (FR)

∫ x−h

x0

b(t)¯ v(t)dt

⊕ (FR)

∫ x−h

x0

b(t)¯ ea·(x−h−t)¯ dt⊕ (FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt.

Then the H-difference (FR)∫ x

x0b(t)¯e

a·(x−t)¯ dt− (FR)

∫ x−h

x0b(t)¯e

a·(x−h−t)¯ dt

exists and similar to (10) we have

limh0

1

h·(

(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt− (FR)

∫ x−h

x0

b(t)¯ ea·(x−h−t)¯ dt

)

= limh0

1

h

((FR)

∫ x−h

x0

b(t)¯(e

a·(x−t)¯ − e

a·(x−h−t)¯

)dt

)(15)

⊕ limh0

1

h

((FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt

)

= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ lim

h0

1

h· (FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt.

Similar to the symmetric case we get

D

(b(x),

1

h· (FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt

)

≤ 1

h

∫ x

x−h

D(b(x), b(t)¯ ea·(x−t)¯ )dt,

for every 0 < h < x− x0. As in (11) we have

D(b(x), b(t)¯ e

a·(x−t)¯

)≤ ω(b, h) + Kb(t)D(1, e

a·(x−t)¯ )

B.BAN,B.BEDE146

for every t ∈ [x− h, x], 0 < h < x− x0. Similar to (12) we obtain

D(1, ea·(x−t)¯ )

≤ limn→∞

(|x− t|

∥∥∥∥1

1!· a

∥∥∥∥F

+ ... + |x− t|n∥∥∥∥

1

n!· a¯n

∥∥∥∥F

),

for every t ∈ [x− h, x], 0 < h < x− x0. Similar to (13) we get

limh0

1

h· (FR)

∫ x

x−h

b(t)¯ ea·(x−t)¯ dt = b(x),

for every x > x0. Then in (15) we have

limh0

1

h·(

(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt− (FR)

∫ x−h

x0

b(t)¯ ea·(x−h−t)¯ dt

)

= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ b(x), (16)

for every x > x0, and by (14) and (16) we obtain that(

(FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt

)′= (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ b(x),

for every x > x0. In (9) we get

y′(x) = c¯ a¯ ea·(x−x0)¯ ⊕ (FR)

∫ x

x0

b(t)¯ a¯ ea·(x−t)¯ dt⊕ b(x)

Because c and b(t) have the same sign it is easy to prove that the fuzzy-number-valued function given by (9) is a solution of the fuzzy differentialequation (2) and the proof is complete.

It is immediate the following result.

Corollary 4.4. The fuzzy-number-valued function y : R→ RF given by

y(x) = y0 ¯ ea·(x−x0) ⊕ (FR)

∫ x

x0

b(t)¯ ea·(x−t)¯ dt,

where a is a strict positive fuzzy number, is a solution of the fuzzy initialvalue problem

y′(x) = a¯ y(x)⊕ b(x)y(x0) = y0

, (17)

for all x > x0, where y0 ∈ R¯F , b : R→ RF is continuous, y0 and b(x) havethe same sign for all x > x0 (if y0 6= 0).

POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 147

The following example gives the solution of a fuzzy initial value problemwith triangular fuzzy numbers as data. We recall that for a < b < c, a, b, c ∈R the triangular fuzzy number u = (a, b, c) determined by a, b, c is givensuch that ur = a + (b − a)r and ur = c − (c − b)r are the endpoints of ther−level sets, for every r ∈ [0, 1]. If ui = (ai, bi, ci), i ∈ 1, 2 are triangularfuzzy numbers then

u1 ¯ u2 = (a1b2 + a2b1 − b1b2, b1b2, c1b2 + c2b1 − b1b2) (see [3])

andu1 ⊕ u2 = (a1 + a2, b1 + b2, c1 + c2).

If u = (a, b, c) is a triangular fuzzy number and k ∈ R then k ·u = (ka, kb, kc)if k is positive and k · u = (kc, kb, ka) if k is negative.

Example 4.5. Let a = (α(1− ε), α, α(1 + ε)) , α, ε > 0 and b = (β(1 −ε′), β, β(1 + ε′)), β, ε′ > 0, strict positive triangular fuzzy numbers and let usconsider the fuzzy initial value problem

y′(x) = a¯ y(x)⊕ by(0) = 1

. (18)

Firstly, let us compute ea·x¯ for x > 0, x ∈ R. Because (see [3])

(a · x)¯n = (αnxn(1− nε), αnxn, αnxn(1 + nε)),

by direct computation we get

ea·x¯ =

∞∑n=0

⊕1

n!· (a · x)¯n

=∞∑

n=0

⊕ (αnxn

n!(1− nε),

αnxn

n!,αnxn

n!(1 + nε)

)

=

( ∞∑n=0

αnxn

n!(1− nε),

∞∑n=0

αnxn

n!,

∞∑n=0

αnxn

n!(1 + nε)

)

=

( ∞∑n=0

αnxn

n!− ε

∞∑n=1

αnxn

(n− 1)!,

∞∑n=0

αnxn

n!,

∞∑n=0

αnxn

n!+ ε

∞∑n=1

αnxn

(n− 1)!

)

B.BAN,B.BEDE148

= (eαx(1− εαx), eαx, eαx(1 + εαx)) .

It follows that

(FR)

∫ x

0

b¯ ea·(x−t)¯ dt = [2, Theorem 4.9, (iv)] = b¯ (FR)

∫ x

0

ea·(x−t)¯ dt

= b¯ (FR)

∫ x

0

(eα(x−t)(1− εα(x− t)), eα(x−t), eα(x−t)(1 + εα(x− t))

)dt

= [7, Theorem 3.2]

= b¯(∫ x

0

eα(x−t)(1− εα(x− t))dt,

∫ x

0

eα(x−t)dt,

∫ x

0

eα(x−t)(1 + εα(x− t))dt

)

= (β(1− ε′), β, β(1 + ε′))¯

¯(

eαx − 1

α(1 + ε)− εxeαx,

eαx − 1

α,eαx − 1

α(1− ε) + εxeαx

)

= β ·(

eαx − 1

α(1 + ε− ε′)− εxeαx,

eαx − 1

α,eαx − 1

α(1 + ε′ − ε) + εxeαx

)

Theny(x) = eαx · (1− εαx, 1, 1 + εαx)⊕

⊕β ·(

eαx − 1

α(1 + ε− ε′)− εxeαx,

eαx − 1

α,eαx − 1

α(1 + ε′ − ε) + εxeαx

)

is the solution of the problem (18).

Acknowledgement. We express our thanks to Professor S.G. Gal,whose comments helped us to improve the paper.

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POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 149

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POWER SERIES OF FUZZY NO...FUZZY DIFFERENTIAL EQUATIONS 151

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B.BAN,B.BEDE152

On a delay integral equation in biomathematics

Alexandru BicaDepartment of Mathematics, University of Oradea,Str. Armatei Române no 5, Oradea, Romaniasmbica@yahoo.com ; abica@uoradea.ro

Craciun IancuFaculty of Mathematics and Computer Science,

”Babes-Bolyai” University,Str. N. Kogalniceanu no 1, Cluj-Napoca, Romania

ciancu@math.ubbcluj.ro

Abstract

In this paper we obtain a numerical method for the solution of thedelay integral equation (1) which is known in some anterior given points.In this purpose we construct a cubic fitting spline function through theleast squares method on the anterior interval which realize here a subop-timal approximation of the solution. Afterwards, we use the successiveapproximations method combined with a quadrature rule to approximatethe solution on [0,T].

2000 Mathematics Subject Classification: Primary 65R20, 65D10, Secondary45D05.Keywords and Phrases: fitting spline, delay integral equation, perturbed

trapezoidal quadrature rule.

1 IntroductionThe aim of this paper is to propose a numerical method for the approximationof the smooth positive solution of the delay integral equation,

x(t) =

Z t

t−τf(s, x(s))ds. (1)

This integral equation has the origin in biomathematics where it models thespread of certain infectious diseases with a contact rate that varies seasonally.So that x(t) is the proportion of infectious in the population at the time t, τ isthe lenght of time in which an individual remains infectious and f(t, x(t)) is theproportion of new infections on unit time.

153JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,153-170,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

In [4], [14], [9] and [10] sufficient conditions for the existence of the positivecontinuous solution of (1), which can be periodic, were given by D. Guo, V.Laksmikantham, L.R. Williams, R.W. Legget and R. Precup. Using the Ba-nach’s fixed point principle, I.A. Rus obtains existence and uniqueness resultsof this solution in [11] and [12]. In [7] C. Iancu obtains a numerical methodfor the approximation of the positive bounded solution of (1) using the Picard’stechnique of successive approximations and the trapezoidal quadrature rule. In[2] we obtain some numerical methods for the approximation of the positivesmooth solution of (1) according to the properties of the function f , ( that isLipschitzian or of C1, C2 smoothness class ). Continuing our ideas, in this paperwe propose a numerical method using fitting spline functions and a pertubedtrapezoidal quadrature rule in C3 smoothness case.In this paper we consider τ > 0, and T > 0 fixed, suppose that we know the

solution of the equation (1) in some p discrete points in the interval [−τ , 0] andwe want to obtain an approximation of this solution for t ∈ [−τ , T ]. Firstly wegive sufficient conditions for the existence and uniqueness of a positive boundedsmooth solution of the equation (1). Using the above discrete values and theIchida-Yoshimoto-Kiyono’s method based on the least squares principle we con-struct a fitting spline function which approximates the solution on [−τ , 0] ([8] and [13] ). Afterwards, using the knots and the parameter values obtainedwith this method we construct an approximation of the solution for t ∈ [0, T ]based on the succesive approximations method and on a perturbed trapezoidalquadrature rule ([1] and [3]).

2 The existence and the uniqueness of the pos-itive smooth solution

Let T > 0 be a fixed real number. Considering the initial condition, x(t) = Φ(t),t ∈ [−τ , 0], we have the initial value problem :½

x(t) =R tt−τ f(s, x(s))ds, ∀t ∈ [0, T ]

x(t) = Φ(t), ∀t ∈ [−τ , 0]. (2)

We suppose that Φ ∈ C[−τ , 0] satisfies the condition

Φ(0) =

Z 0

−τf(s,Φ(s))ds. (3)

Supposing that f ∈ C ([−τ , T ]×R) and using the Leibniz’s formula ofderivation for an integral with parameters, we obtain the equivalence between(2) and the following Cauchy problem for a delay differential equation½

x0(t) = f(t, x(t))− f(t− τ , x(t− τ)), ∀t ∈ [0, T ]x(t) = Φ(t), ∀t ∈ [−τ , 0]. (4)

To obtain the existence of an unique positive, bounded solution we considerthese assumptions :

A.BICA,C.IANCU154

(i) ∃a,β > 0 such that Φ is continuous on [−τ , 0] and0 < a ≤ Φ(t) ≤ β,∀t ∈ [−τ , 0];(ii) f ∈ C ([−τ , T ]× [a,β]) , f(t, x) ≥ 0,∀t ∈ [−τ , T ],∀x ≥ 0 and ∃M > 0

such that f(t, y) ≤M,∀t ∈ [−τ , T ],∀y ∈ [a,β];(iii) Φ ∈ C1[−τ , 0] satisfies the condition (3) and

Φ0(0) = f(0, b)− f(−τ ,Φ(−τ))

where b = Φ(0) ;(iv) Mτ ≤ β and there is an integrable function g(t) such that

f(t, x) ≥ g(t), ∀t ∈ [−τ , T ], ∀x ≥ a,

with Z t

t−τg(s)ds ≥ a, ∀t ∈ [0, T ] ;

(v) ∃L > 0 such that |f(t, x)− f(t, y)| ≤ L |x− y| for all t ∈ [−τ , T ] andx, y ∈ [a,β).

Theorem 1 Suppose that assumptions (i)-(v) are satisfied. Then the equation(1) has a unique continuous on [−τ , T ] solution x(t) , with a ≤ x(t) ≤ β ,∀t ∈ [−τ , T ] such that x(t) = Φ(t) for t ∈ [−τ , 0]. Moreover,

max |xn(t)− x(t)| : t ∈ [0, T ] −→ 0

as n → ∞ where xn(t) = Φ(t) for t ∈ [−τ , 0] , n ∈ N , x0(t) = b and xn(t) =R tt−τ f(s, xn−1(s))ds for t ∈ [0, T ] , n ∈ N∗. The solution x belongs to C1[−τ , T ].

Proof. In the functional space C[−τ , T ] we consider the Bielecki’s norm

kxkB = max|x(t)| · e−θ(t+τ) : t ∈ [−τ , T ]

for θ > 0 and define the operator A : C[−τ , T ] −→ C[−τ , T ]

A(x(t)) =

½ R tt−τ f(s, x(s))ds, t ∈ [0, T ]Φ(t), t ∈ [−τ .0].

We have

|A(x1(t))−A(x2(t))| ≤Z t

t−τ|f(s, x1(s))− f(s, x2(s))| ds

≤Z t

t−τL kx1 − x2kB · eθ(s+τ)ds ≤

L

θkx1 − x2kB · eθ(t+τ)

for t ∈ [0, T ] and consequently

kA(x1)−A(x2)k ≤L

θ· kx1 − x2kB

ON A DELAY INTEGRAL EQUATION... 155

for x1, x2 ∈ C[−τ , T ]. This operator is contraction for θ > L. Using the Banach’sfixed point principle we obtain the existence of an unique continuous on [−τ , T ]solution for (1) such that x(t) = Φ(t) for t ∈ [−τ .0] and

max |xn(t)− x(t)| : t ∈ [0, T ] −→ 0

as n→∞ . Using now the Chebyshev’s norm,

kxkC = max|x(t)| : t ∈ [−τ , T ]

we obtain

|xn(t)− x(t)| ≤τmLm

1− τL· kx0 − x1kC ,∀t ∈ [0, T ].

From (iv) we see that

x(t) =

Z t

t−τf(s, x(s))ds ≥

Z t

t−τg(s)ds ≥ a

x(t) =

Z t

t−τf(s, x(s))ds ≤

Z t

t−τMds ≤ β,∀t ∈ [0, T ].

Because a ≤ Φ(t) ≤ β,∀t ∈ [−τ , 0] and x(t) = Φ(t) for t ∈ [−τ .0] we inferthat a ≤ x(t) ≤ β,∀t ∈ [−τ , T ], and the solution is bounded. Since x issolution for (1) and we have x(t) =

R tt−τ f(s, x(s))ds, for all t ∈ [0, T ], and

because f ∈ C([−τ , T ] × [a,β]) we infer that x is derivable on [0, T ] , with x0is continuous on [0, T ]. From the condition (iii) follow that x is derivable withx0 continuous on [−τ , 0] (including the continuity at the point t = 0 ). Thenx ∈ C1[−τ , T ] and the proof is complete.

Corollary 2 In the conditions of Theorem 1, if f ∈ C1 ([−τ , T ]× [a,β]) , Φ ∈C2[−τ , 0] and

Φ00(0) =∂f

∂t(0, b) +

∂f

∂x(0, b) · [f(0, b)− f(−τ ,Φ(−τ))]−

−∂f∂t(−τ ,Φ(−τ))− ∂f

∂x(−τ ,Φ(−τ)) · Φ0(−τ)

then x ∈ C2[−τ , T ].

Proof. Follows directly from the above theorem and from hypothesis.

3 The construction of the fitting spline function

The initial condition in (2) means that the proportion Φ(t) of infectious inpopulation is known for t ∈ [−τ , 0].

A.BICA,C.IANCU156

But in the real world this knowledge is incomplete, more exactly we canknow the function Φ only in some discret moments when we make measure-ments. For this reason we can consider many sufficient values of Φ, f1, f2, ..., fpexperimentally found at the moments ui, i = 1, p such that

−τ = u1 < u2 < ... < up = 0.

Because these values are affected by errors we have to find an adequatefitting spline function which approximate the function Φ on [−τ , 0], using theleast squares method combined with the algorithm of K. Ichida, F. Yoshimoto,T. Kiyono (see [8] and [13]) and the spline function of C. Iancu (see [5] and [6]).This algorithm have the knots as variables. Starting from three knots we

insert another knot successively until a certain criterion is satisfied. The es-timator of the variance of errors (as well as in [8]) is used to determine if asatisfactory fit is reached. Then we obtain the new knots u(i), i = 1, n+ 1. Lethi = u

(i+1) − u(i), i = 1, n.

3.1 The cubic spline function

The spline function, above mentioned, is the following piecewise cubic polyno-mial on [−τ , 0] (see [5] and [6]) for which if t ∈ [u(i), u(i+1)] we have :

si(t) =Mi+1 −Mi

6hi· (t− u(i))3 + Mi

2· (t− u(i))2+ (5)

+mi · (t− u(i)) + yi, ∀i = 1, n− 1.From [5] and [6] we have,(

Mi+1 + 2Mi = 6 · yi+1−yi−mi·hih2i

Mi+1 +Mi =2(mi+1−mi)

hi, i = 1, n− 1.

(6)

From relation (6) we obtain the following recurrent relations(Mi+1 = 6 · yi+1−yih2i

− 6mi

hi− 2Mi

mi+1 = 3 · yi+1−yihi− 2mi − 1

2Mihi, i = 1, n− 1. (7)

and we can also obtain(mi =

yi+1−yihi

− 16hi(Mi+1 + 2Mi)

mi+1 =yi+1−yi

hi+ 1

6hi(4Mi+1 +Mi), i = 1, n− 1. (8)

Replacing this value of mi in (5), we obtain a new expresion for si(t).If we denote,

ai(t) =(t− u(i))36hi

− hi · (t− u(i))

6

ON A DELAY INTEGRAL EQUATION... 157

bi(t) = −(t− u(i))36hi

+(t− u(i))2

2− hi · (t− u

(i))

3

ci(t) =t− u(i)6

, di(t) = 1−t− u(i)hi

then relation (5) becomes,

si(t) =Mi+1ai(t) +Mibi(t) + yi+1ci(t) + yidi(t) . (9)

Here, yi,mi,Mi are the values of si, s0i, s00i on u

(i), i = 1, n. The computingalgorithm may be summarized as follows ( [8] and [6] ):Step 1 : Calculate least squares fitting for initial two intervals. Construct

the normal equation and go to step 4.Step 2 : Determine the interval to divide and insert a new knot.Step 3 : Construct the normal equation.Step 4 : Solve the normal equation.Step 5 : If the number of parameters (yi togetherMi, i = 1, n) is greater than

that of data, then end. Else test the criterion. If the criterion is not satisfied,go to step 2. If it is satisfied, then end.The algorithm run if we choose a satisfactory ε > 0 to test the criterion.

3.2 The algorithm

First we take three knots u(1), u(2), u(3) where u(1) = u1 = −τ , u(3) = up = 0and u(2) is determined in order that there are equal numbers of data ui in eachinterval [u(1), u(2)] and [u(2), u(3)]. We have

y1 = s1

³u(1)

´, y2 = s1

³u(2)

´= s2

³u(2)

´, y3 = s2

³u(3)

´m1 = s

01

³u(1)

´, m2 = s

01

³u(2)

´= s02

³u(2)

´, m3 = s

02

³u(3)

´M1 = s

001

³u(1)

´, M2 = s

001

³u(2)

´= s002

³u(2)

´, M3 = s

002

³u(3)

´.

The values of parameters y1, y2, y3,M1,M2,M3 will be determined by leastsquares fitting. From (9) we have,

s1(t) =M2a1(t) +M1b1(t) + y2c1(t) + y1d1(t) (10)

s2(t) =M3a2(t) +M2b2(t) + y3c2(t) + y2d2(t).

Let the least and the largest values ui of data in the interval [u(1), u(2)] berespectively up1 = u1 and uq1 . Similarly we denote up2 and uq2 = up in theinterval [u(2), u(3)]. The sum of squares of residuals is,

R(y1,M1, y2,M2, y3,M3) =

q1Xk=p1

[s1(uk)− fk]2 +q2X

k=p2

[s2(uk)− fk]2 . (11)

A.BICA,C.IANCU158

We want to obtain the values of yi , Mi ,i = 1, 3 which minimize R. Differ-entiating this expresion with the 6 parameters yi , Mi and setting to zero, thatis

∂R

∂y1= 0,

∂R

∂M1= 0,

∂R

∂y2= 0

∂R

∂M2= 0,

∂R

∂y3= 0,

∂R

∂M3= 0

we get a linear system in this 6 parameters, which can be written in the followingnormal equation :

A3z3 = g3 (12)

where the subscripts denote the number of knots and,

zT3 = (y1,M1, y2,M2, y3,M3) ∈ R6 (13)

gT3 = (

q1Xk=p1

d1(uk)fk,

q1Xk=p1

b1(uk)fk,

q1Xk=p1

c1(uk)fk +

q2Xk=p2

d2(uk)fk,

q1Xk=p1

a1(uk)fk +

q2Xk=p2

b2(uk)fk,

q2Xk=p2

c2(uk)fk,

q2Xk=p2

a2(uk)fk) ∈ R6 (14)

A3 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

a(3)11 a

(3)12 a

(3)13 a

(3)14 0 0

a(3)21 a

(3)22 a

(3)23 a

(3)24 0 0

a(3)31 a

(3)32 a

(3)33 a

(3)34 a

(3)35 a

(3)36

a(3)41 a

(3)42 a

(3)43 a

(3)44 a

(3)45 a

(3)46

0 0 a(3)53 a

(3)54 a

(3)55 a

(3)56

0 0 a(3)63 a

(3)64 a

(3)65 a

(3)66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠a(3)ij = a

(3)ji , i, j = 1, 6, i 6= j

(15)

a(3)11 =

Xk

[d1(uk)]2, a

(3)22 =

Xk

[b1(uk)]2, a

(3)12 =

Xk

b1(uk)d1(uk)

a(3)13 =

Xk

c1(uk)d1(uk), a(3)14 =

Xk

a1(uk)d1(uk) (16)

a(3)23 =

Xk

b1(uk)c1(uk), a(3)24 =

Xk

a1(uk)b1(uk), ∀k = p1, q1.

a(3)33 =

q1Xk=p1

[c1(uk)]2 +

q2Xk=p2

[d2(uk)]2 , a

(3)34 =

q1Xk=p1

a1(uk)c1(uk)+

+

q2Xk=p2

b2(uk)d2(uk), a(3)44 =

q1Xk=p1

[a1(uk)]2 +

q2Xk=p2

[b2(uk)]2 . (17)

ON A DELAY INTEGRAL EQUATION... 159

a(3)35 =

Xk

c2(uk)d2(uk), a(3)36 =

Xk

a2(uk)d2(uk)

a(3)45 =

Xk

b2(uk)c2(uk), a(3)46 =

Xk

a2(uk)b2(uk), a(3)55 =

Xk

[c2(uk)]2

a(3)56 =

Xk

a2(uk)c2(uk), a(3)66 =

Xk

[a2(uk)]2 , ∀k = p2, q2. (18)

Solving the equation (12) by the Gauss’s elimination method we obtain thevector of parameters, zT3 , and the minimal value of R.We denote by R1(y1,M1, y2,M2) and R2(y2,M2, y3,M3) the first sum and

the second sum of R. We see that R1 is associated to the interval [u(1), u(2)]and R2 to the interval [u(2), u(3)]. We compute R1 and R2 and make compar-ison between R1

q1−p1+1 andR2

q2−p2+1 , where qi − pi + 1 is the number of dataui from the interval [u(i), u(i+1)]. If Rj

qj−pj+1 is greater , j ∈ 1, 2 , then wedivide the interval [u(j), u(j+1)] and insert here the new knot in order that thetwo new subintervals have equal number of data ui. In this way we have fournew knots and three intervals, three cubic polynomial as new functions, andeight parameters yi,Mi, i = 1, 4. With the least squares method we minimizeR(y1,M1, y2,M2, y3,M3, y4,M4), which is an expression with 3 sums as in (10)and (11), obtaining the values of these eight parameters, as above. We com-pute R1(y1,M1, y2,M2) , R2(y2,M2, y3,M3) , R3(y3,M3, y4,M4) and

Rjqj−pj+1 =

maxn

Riqi−pi+1 , i = 1, 2, 3

o, in order to divide the interval [u(j), u(j+1)] and to

insert a new knot. The procedure continue and comes up with n− 1 intervals.Computing,

Rjqj − pj + 1

= max

½Ri

qi − pi + 1, i = 1, n− 1

¾we divide the interval [u(j), u(j+1)] and insert a new knot. We obtain n+1 newknots, u(1), u(2), ..., u(n), u(n+1), i = 1, n. Similarly, we have n cubic polynomialfunctions si, i = 1, n and 2(n+ 1) parameters yi,Mi, i = 1, n+ 1,

y1 = s1

³u(1)

´, M1 = s

001

³u(1)

´, yn+1 = sn

³u(n+1)

´Mn+1 = s

00n

³u(n+1)

´, yi+1 = si(u

(i+1)) = si+1(u(i+1))

Mi+1 = s00i (u

(i+1)) = s00i+1(u(i+1)), i = 1, n− 1

The values of the parameters were obtained by the least squares method,minimizing the similar expression R(y1,M1, ..., yn+1,Mn+1) and solving the nor-mal equation An+1zn+1 = gn+1 , where zTn+1 and g

Tn+1 are (n + 1)-vectors

analogous with (13) and (14), and An+1 is a symmetric matrix of dimension(n + 1) × (n + 1) analogous with A3. The elements of An+1 have analogousform with (16), (17) and (18). This matrix has a band structure ( [8] and[13] ) and therefore we can apply the Householder reductions or the Gaussianelimination method to solve the normal equation.

A.BICA,C.IANCU160

In the following we present the criterion of convergence (as in [8] and [13]).Let,

δ(n+ 1) =R(y1,M1, ..., yn+1,Mn+1)

p− 2(n+ 1) ,

be the estimator of the variance of errors.This estimator decrease when we insert a new knot. When δ(n+1)−δ(n) < ε,

for ε > 0 choosen, we consider that we have reached the adequate fitting. Thenthe number n+ 1 is enough and we store the new knots u(1), ..., u(n+1) and thenew spline parameters yi,Mi, i = 1, n+ 1.

3.3 The fitting spline function

With the knots u(i) and the values yi,Mi, i = 1, n+ 1 we construct the fittingspline function s : [−τ , 0] → R , which has the restrictions si ( as in (5 ))on [u(i), u(i+1)] for each i = 1, n. With the relations (8) we obtain the valuesmi, 1, n+ 1 and then

s(u(i)) = yi, s0(u(i)) = mi, s

00(u(i)) =Mi;∀i = 1, n+ 1.

We have that s ∈ C2[−τ , 0]. Indeed, from (5) we obtain that

si(u(i+1)) = si+1(u

(i+1)), ∀i = 1, n.

The relations (8) are equivalent with

s0i(u(i+1)) = s0i+1(u

(i+1)), ∀i = 1, n

and since

s00i (t) =Mi +Mi+1 −Mi

hi· (t− u(i))

we have s00i (u(i+1)) = s00i+1(u

(i+1)), ∀i = 1, n .

Remark 1 Unlike the papers [5] and [8], here we use only the parametersyi,Mi. Of course, the values mi are directly obtained from the relations (8).In [5] the parameters are yi,mi,Mi and the expression of the spline function isthe same (5). In [8] the parameters are yi,mi, but the expression of the splinefunction is more complicated ( see expression (1) in [8] ). Because we choose atthe beginning of the algorithm u(1) = −τ and u(3) = 0 , then after the stop ofthis algorithm we have u(1) = −τ and u(n+1) = 0. The knots u(i), i = 1, n+ 1are not equidistant in general. Since the positive values f1, f2, ..., fp are largeenough we can have s(t) ≥ 0 for t ∈ [−τ , 0].

4 The approximation of the solution

From Theorem 1 it follows that the equation (1) has an unique positive,bounded and smooth solution on [−τ , T ]. Let ϕ be this solution, which can beobtained by successive approximations method on [0, T ].

ON A DELAY INTEGRAL EQUATION... 161

So, we have :⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ϕm(t) = Φ(t) , for t ∈ [−τ , 0] , ∀m ∈ Nϕ0(t) = Φ(0) = b =

R 0−τ f(s,Φ(s))ds ,

ϕ1(t) =R tt−τ f(s,ϕ0(s))ds ,

.............................for t ∈ [0, T ].ϕm(t) =

R tt−τ f(s,ϕm−1(s))ds ,

.............................

(19)

To obtain the sequence of successive approximations (19) we compute theabove integrals using a quadrature rule.

4.1 The quadrature rule

We have seen that the fitting spline function approximates the solution ϕ onthe interval [−τ , 0] if ϕ ∈ C2[−τ , 0], and the knots u(i), i = 1, n+ 1 cover thisinterval. We choose T > 0 such that ∃ l ∈ N∗ with T = lτ , and we move bytranslation the knots u(i), i = 1, n+ 1 over the each subinterval of [0, T ] havingthe length τ . In this way we have q+1 = l·n+1 knots on [0, T ]. We denote theseknots by t0, t1, ..., tq , and obtain a partition of [0, T ], 0 = t0 < t1 < ... < tq = Twith

t0 − τ = −τ = u(1), ..., tn − τ = u(n+1) = 0.

Let h(i) = u(i+1) − u(i), i = 1, n and hi = ti − ti−1, i = 0, q.We see that hi = h(i),∀i = 1, n, and

hn = h(1), ..., hn+ki = h

(i), hq = h(n),

∀i = 1, n,∀k = 1, l − 1. If we denote ti−n = u(i+1),∀i = 0, n then we havetk − τ = tk−n ∀k = 1, q.We apply the following quadrature rule from [1]:Z b

a

F (t)dt = Pn(F, In) +Rn(F, In) , (20)

where In is a division of [a, b] , In : a = x0 < x1 < x2 < ... < xn−1 < xn = b ,with hi = xi+1 − xi, i = 0, n− 1 and

Pn(F, In) =1

2

n−1Xi=0

hi [F (xi) + F (xi+1)]−1

12

n−1Xi=0

h2i [F0 (xi+1)− F 0 (xi)]

with the remainder’s estimation, in the case F ∈ C3[a, b]:

|Rn(F, In)| ≤1

160· kF 000k ·

n−1Xi=0

h4i . (21)

A.BICA,C.IANCU162

For the integrals (19) we apply the rule (20) with the function Fm(t) =f(t,ϕm−1(t)), and according to (21) we need to realize an estimation for F

000m (t) =

[f(t, x(t))]000t , where x = ϕm−1.After an elementary calculus we obtain :

F 000(t) =∂3f

∂t3(t, x(t)) + 3

∂3f

∂t2∂x(t, x(t))x0(t) + 3

∂3f

∂t∂x2(t, x(t))· (22)

· [x0(t)]2 + ∂3f

∂x3(t, x(t)) [x0(t)]

3+ 3

∂2f

∂t∂x(t, x(t))x00(t)+

+3∂2f

∂x2(t, x(t))x0(t) · x00(t) + ∂f

∂x(t, x(t))x000(t).

Remark 2 [f(t, x(t)]000t has 7 terms and [f(t, x(t)](IV )t has (after an elementary

calculus) 13 terms. For this reason the use of a remainder which contains thefourth derivative is not efficient. Then we avoid the Simpson’s or Newton’squadrature formulas.

LetM0 =M = max |f(t, x)| : t ∈ [−τ , T ] , x ∈ [a,β]°°°° ∂αf

∂tα1∂xα2

°°°° = max½¯ ∂αf(t, x)∂tα1∂xα2

¯: (t, x) ∈ [−τ , T ]× [a,β]

¾for α = 1, 3 , α1,α2 = 0, 3 , α1 + α2 = α,

M1 = max

½°°°°∂f∂t°°°° ,°°°°∂f∂x

°°°°¾

M2 = max

½°°°°∂2f∂t2

°°°° ,°°°° ∂2f

∂t∂x

°°°° ,°°°°∂2f∂x2

°°°°¾M3 = max

½°°°°∂3f∂t3

°°°° ,°°°° ∂3f

∂t2∂x

°°°° ,°°°° ∂3f

∂t∂x2

°°°° ,°°°°∂3f∂x3

°°°°¾ .Then from (22) we obtain :

|F 000m (t)| ≤M3(1 +M0)3 + 5M2M1(1 +M0)

2+

+2M21 (1 +M0) =M

000, ∀t ∈ [−τ , T ].

4.2 The algorithm

For the relations (19) and adapting the quadrature rule (20) at the knots ti, i =0, q we obtain for m ∈ N∗, and k = 1, q:

ϕm(tk) =

Z tk

tk−τf(s,ϕm−1(s))ds =

1

2

kXi=k−n

hi[f(ti,ϕm−1(ti))+ (23)

ON A DELAY INTEGRAL EQUATION... 163

+f(ti+1,ϕm−1(ti+1))]−1

12

kXi=k−n

h2i · [∂f

∂t(ti+1,ϕm−1(ti+1))+

+∂f

∂x(ti+1,ϕm−1(ti+1)) · ϕ0m−1(ti+1)−

∂f

∂t(ti,ϕm−1(ti))−

−∂f∂x(ti,ϕm−1(ti)) · ϕ0m−1(ti)] + r

(n)m,k(f) .

Also, we have ¯r(n)m,k(f)

¯≤ M

000

160·

kXi=k−n

h4i , ∀k = 1, q. (24)

if f ∈ C3 ([−τ , T ]× [a,β]) . In (23), ϕm−1(ti−τ) = s(ti−τ) and ϕ0m−1(ti−τ) =s0(ti − τ), for i = 1, n. Here, the values of the first derivative of ϕm−1 are form ∈ N,m ≥ 2,

ϕ0m−1(t) = f(t,ϕm−2(t))− f(t− τ ,ϕm−2(t− τ)),∀t ∈ [0, T ]

and ϕ00(ti) = ϕ01(ti) = s0(ti) for ti ∈ [−τ , 0], ϕ00(t) = 0 , ∀t ∈ [0, T ]. We see that

ϕ01(ti) =

½f(ti, b)− f(ti − τ , s(ti)), ∀i = 1, nf(ti, b)− f(ti − τ , b), ∀i = n+ 1, q

and ϕ0(ti) = s(ti) for ti ∈ [−τ , 0], ϕ0(t) = b, ∀t ∈ [0, T ].Using for the successive approximations (19) and the formulas (23) with

the remainder estimation (24), we can obtain an algorithm for the approximatesolution of (2).So, we have for k = 1, q:

ϕ1(tk) =1

2

kXi=k−n

hi · [f(ti,ϕ0(ti)) + f(ti+1,ϕ0(ti+1))]−

1

12

kXi=k−n

h2i [∂f

∂t(ti+1,ϕ0(ti+1)) +

∂f

∂x(ti+1,ϕ0(ti+1))ϕ

00(ti+1)

−∂f∂t(ti,ϕ0(ti))−

∂f

∂x(ti,ϕ0(ti))ϕ

00(ti)] + r

(n)1,k (f)

that is ϕ1(tk) = fϕ1(tk) + r(n)1,k (f) and

ϕ2(tk) =1

2

kXi=k−n

hi[f(ti,fϕ1(ti) + r(n)1,i (f)) + f(ti+1,fϕ1(ti+1)++r

(n)1,i+1(f))]−

1

12

kXi=k−n

h2i · [∂f

∂t(ti+1,fϕ1(ti+1) + r(n)1,i+1(f))+

A.BICA,C.IANCU164

+∂f

∂x(ti+1,fϕ1(ti+1) + r(n)1,i+1(f)) · ϕ01(ti+1)−

∂f

∂t(ti,fϕ1(ti)+

+r(n)1,i (f))−

∂f

∂x(ti,fϕ1(ti) + r(n)1,i (f)) · ϕ01(ti)] + r

(n)2,k (f)

With the Lipschitz’s property of f we have

ϕ2(tk) =1

2

kXi=k−n

hi · [f (ti,fϕ1(ti)) + f(ti+1,fϕ1(ti+1))]− (25)

− 112

kXi=k−n

h2i · [∂f

∂t(ti+1,fϕ1(ti+1)) + ∂f

∂x(ti+1,fϕ1(ti+1)) ·

· (f (ti+1,ϕ0(ti+1))− f (ti+1−n,ϕ0(ti+1−n)))−∂f

∂t(ti,fϕ1(ti))−

−∂f∂x(ti,fϕ1(ti)) · (f (ti,ϕ0(ti))− f (ti−n,ϕ0(ti−n)))]++gr(n)2,k (f) = fϕ2(tk) +gr(n)2,k (f), ∀k = 1, q.

Let h = ν(In) = max©h(i), i = 1, n

ª. Then, h = max

©hi, i = 1, q − 1

ª. With

these, from (24) and (25) we obtain for the remainders the estimation :¯r(n)1,k (f)

¯≤ M 000nh4

160 , and¯gr(n)2,k (f)

¯≤ M

000

160

µnh4 + n2h5L+

n2h6

3M2

¶,∀k = 1, q (26)

Continuing by induction for m ≥ 3 , from (23), (25) and (26) we obtain form ∈ N, m > 2, k = 1, q:

ϕm(tk) =1

2

kXi=k−n

hi[f(ti, ϕm−1(ti) + r(n)m−1,i(f)) + f(ti+1, ϕm−1(ti+1)

+^r(n)m−1,i+1(f))]−

1

12

kXi=k−n

h2i [∂f

∂t(ti+1, ϕm−1(ti+1) +

^r(n)m−1,i+1(f))+

+∂f

∂x(ti+1, ϕm−1(ti+1) +

^r(n)m−1,i+1(f)) · [f(ti+1, ϕm−2(ti+1)+

+^r(n)m−2,i+1(f))− f(ti+1−n, ϕm−2(ti+1−n) +

^r(n)m−2,i+1−n(f))]−

−∂f∂t(ti, ϕm−1(ti) + +r

(n)m−1,i(f))−

∂f

∂x(ti, ϕm−1(ti) + r

(n)m−1,i(f))·

·f(ti, ϕm−2(ti)+r(n)m−2,i(f))− f(ti−n, ϕm−2(ti−n) +

^r(n)m−2,i−n(f))]

ON A DELAY INTEGRAL EQUATION... 165

+r(n)m,k(f) =

1

2

kXi=k−n

hi£f¡ti, ϕm−1(ti)

¢+ f

¡ti+1, ϕm−1(ti+1)

¢¤−

− 112

kXi=k−n

h2i · [∂f

∂t

¡ti+1, ϕm−1(ti+1)

¢+

∂f

∂x

¡ti+1, ϕm−1(ti+1)

¢·

·f¡ti+1, ϕm−2(ti+1)

¢− f

¡ti+1−n, ϕm−2(ti+1−n)

¢−

−∂f∂t

¡ti, ϕm−1(ti)

¢− ∂f

∂x

¡ti, ϕm−1(ti)

¢· f

¡ti, ϕm−2(ti)

¢−

−f¡ti−n, ϕm−2(ti−n)

¢] + g

r(n)m,k(f) = ϕm(tk) +

gr(n)m,k(f). (27)

4.3 Main result

We can obtain a concise estimation for

¯gr(n)m,k(f)

¯. Indeed, form ≥ 2 and k = 1, q

we have: ¯gr(n)m,k(f)

¯≤¯r(n)m,k(f)

¯+ Lnh ·

¯r(n)m−1,k(f)

¯+

+1

12· 2nh2(M2(M + 1) +M1L)

¯r(n)m−1,k(f)

¯=¯r(n)m,k(f)

¯+

+[Lnh+1

6· nh2(M2(M + 1) +M1L)] ·

¯r(n)m−1,k(f)

¯.

and

¯gr(n)1,k (f)

¯=¯r(n)1,k (f)

¯≤ M 000nh4

160 , ∀k = 1, q . Then we obtain

¯gr(n)2,k (f)

¯≤¯r(n)2,k (f)

¯+ [Lnh+

nh2

6(M2(M + 1) +M1L)]·

·¯r(n)1,k (f)

¯≤ [1 + Lnh+ nh

2

6(M2(M + 1) +M1L)] ·

M 000nh4

160

and ¯gr(n)3,k (f)

¯≤¯r(n)3,k (f)

¯+ [Lnh+

nh2

6(M2(M + 1) +M1L)]·

·¯gr(n)2,k (f)

¯≤ [1 + (Lnh+ nh

2

6(M2(M + 1) +M1L))+

+(Lnh+nh2

6(M2(M + 1) +M1L))

2] · M000nh4

160,∀k = 1, q.

By induction we obtain for m ∈ N,m ≥ 2, and k = 1, q:¯gr(n)m,k(f)

¯≤ [1 + (Lnh+ nh

2

6(M2(M + 1) +M1L)) + ...+

A.BICA,C.IANCU166

+(Lnh+1

6· nh2(M2(M + 1) +M1L))

m−1] · M000nh4

160=

=1− [Lnh+ 1

6 · nh2(M2(M + 1) +M1L)]m

1− [Lnh+ 16 · nh2(M2(M + 1) +M1L)]

· M000nh4

160.

Since h = O( τn) we can consider µ =hnτ . If Lµτ <

34 and n ∈ N is such that

Lnh+1

6· nh2(M2(M + 1) +M1L) < 1,

then for m ∈ N,m ≥ 2, and k = 1, q, we have the estimation¯gr(n)m,k(f)

¯≤ M 000nh4

160(1− [Lnh+ 16 · nh2(M2(M + 1) +M1L)])

. (28)

In this way we got the sequence (fϕm(tk))m∈N∗ , which approximates thesequence of successive approximations (19) on the knots tk, k = 1, q, with the

error : |ϕm(tk)− fϕm(tk)| = ¯gr(n)m,k(f)

¯which has the estimation (28).

Proposition 3 Consider the initial value problem (2) under the conditions ofTheorem 1 and of Corollary 2. If f ∈ C3 ([−τ , T ]× [a,β]) , Φ ∈ C3[−τ , 0],Lµτ < 3

4 and the exact solution ϕ is approximated by the sequence (fϕm(tk))m∈N∗ ,, on the knots tk, k = 1, q , through the successive approximation method (19),combined with the quadrature rule (20), then the following error estimation holds

|ϕ(tk)− fϕm(tk)| ≤ M 000nh4

160(1− [Lnh+ 16 · nh2(M2(M + 1) + LM1)])

+

+τm · Lm1− τL

· kϕ0 − ϕ1kC[0,T ] , ∀m ∈ N∗,∀k = 1, q, (29)

for n ∈ N large enough.

Proof. From Theorem 1, we have the estimation

|ϕ(tk)− ϕm(tk)| ≤τmLm

1− τL· kϕ0 − ϕ1kC[0,T ] ,∀k = 1, q,

∀m ∈ N∗. Since |ϕm(tk)− fϕm(tk)| = ¯gr(n)m,k(f)

¯, from this estimation and from

the inequality (28), we get the estimation (29).

Remark 3 In the equidistant case, hi = h = τn ,∀i = 1, n, after analogous

calculus we obtain for m ∈ N, m ≥ 2, and k = 1, q the estimation¯gr(n)m,k(f)

¯≤1− [τL+ 1

6n2 (τ2(M2(M + 1)) + LM1)]

m

1− [τL+ 16n2 (τ

2(M2(M + 1) + LM1))]· τ

4M 000

160n3.

ON A DELAY INTEGRAL EQUATION... 167

Finally, we obtain the main result of our paper :

Theorem 4 In the conditions of Proposition 3 , if hi = τn ,∀i = 1, n, τL <

34

and n∈ N is such that n > [23 (M2(M +1)+LM1)]12 then for any m ∈ N,m ≥ 2

we have

|ϕ(tk)− fϕm(tk)| ≤ τmLm

1− τL· kϕ0 − ϕ1kC[0,T ]+

+τ4M 000

160n3¡1− [τL+ 1

6n2 (τ2(M2(M + 1) + LM1))]

¢ ,∀k = 1, q.

Proof. Follows from Proposition 3 and previous remark.The above presented numerical method is also useful for first order ODE.

Remark 4 If we use the trapezoidal quadrature rule, then we obtain the valuesfϕm(tk) from the first sum in (23). The remainder estimation is¯gr(n)m,k(f)

¯≤ 1− (Lnh)

m

1− Lnh · M00nh3

12

where, M 00 =M2(M0 + 1)2 +M1(M0 + 1).

4.4 An example

To illustrate the algorithm we give an example in the conditions of Theorem 4.Here, τ = 1, T = 3, n = 10, q = 30 and

Φ(t) =1

4[cos2 πt+ 0.25 · sin2 2πt] + 0.1

f(t, x) =1

20sin2 πt+

4x+ 3

10(4x+ 5)+ 0.25832.

We have β = 2, a = 0.1, m = 0.2, M = 0.5, ϕ(0) = 720 , L =

4125 , M1 =

120π,

M2 =110π

2 and M 000 ≤ 24.For given ε > 0 we find the smallest m ∈ N such that

| fϕm(tk)− ϕm−1(tk) |< ε, ∀k = 1, q.In the following table there are the values fϕm(tk), for k = 1, 30. For ε = 10−12we get m = 8.

tk fϕm(tk) tk fϕm(tk) tk fϕm(tk)t1 0.34795627524 t11 0.35207611617 t21 0.35211158318t2 0.34816606071 t12 0.35208360617 t22 0.35211127612t3 0.34849749458 t13 0.35209051377 t23 0.36211086721t4 0.34906707741 t14 0.35209665275 t24 0.35211053170t5 0.34986045867 t15 0.35210167720 t25 0.35211041254t6 0.35065567756 t16 0.35210539510 t26 0.35211056646t7 0.35123022656 t17 0.35210798938 t27 0.35211094266t8 0.35156856971 t18 0.35210979766 t28 0.35211140258t9 0.35178615824 t19 0.35211100019 t29 0.35211177352t10 0.35197711605 t20 0.35211158784 t30 0.35211191551

A.BICA,C.IANCU168

Acknowledgement

The authors are indebted to Professor Horea Oros from University of Oradeafor his help in computer implementation of the algorithm.

References[1] Barnett N.S., Dragomir S.S.,Perturbed trapezoidal inequality in terms of

the third derivative and applications, RGMIA Preprint vol.4, no.2, (2001),221-233.

[2] Bica A.,Numerical methods for approximating the solution of a delayinte-gral equation from biomathematicsolution of a delay , (submitted ).

[3] Bica A., Mangrau I., Mezei R.,Numerical method for Volterra integral equa-tions, Proceedings of the 11-th Conference on Applied and Industrial Math-ematics (2003), 29-33, Oradea, Romania.

[4] Guo D., Laksmikantham V.,Positive solutions of nonlinear integral equa-tions arising in infectious diseases, J. Math. Anal. Appl. 134 (1988), 1-8.

[5] Iancu C.,The analysis and data processing with spline functions, PhDThesis, ”Babes-Bolyai” University, Cluj-Napoca, Romania, 1983 (in Ro-manian).

[6] Iancu C.,On the cubic spline of interpolation, Seminar of Functional Analy-sis and Numerical Methods, Preprint 4, Cluj-Napoca, (1981), 52-71.

[7] Iancu C.,A numerical method for approximating the solution of an integralequation from biomathematics, Studia Univ. ”Babes-Bolyai”, Mathematica,vol.XLIII, No.4, (1988), 37-45.

[8] Ichida K., Yoshimoto F., Kiyono T.,Curve Fitting by a Piecewise CubicPolynomial, Computing 16, (1976), 329-338.

[9] Precup R.,Positive solutions of the initial value problem for an integralequation modelling infectious diseases, Seminar of Fixed Point Theory,Preprint 3, Cluj-Napoca, (1991), 25-30.

[10] Precup R.,Periodic solutions for an integral equation from biomathematicsvia Leray-Schauder principle, Studia Univ. ”Babes-Bolyai”, Mathematica,vol.XXXIX, 1, (1994), 47-58.

[11] Rus I.A.,Principles and applications of the fixed point theory, Ed. Dacia,Cluj-Napoca, 1979 (in Romanian).

[12] Rus I.A.,A delay integral equation from biomathematics, Seminar of Dif-ferential Equations, Preprint3, Cluj-Napoca, (1989), 87-90.

ON A DELAY INTEGRAL EQUATION... 169

[13] Yoshimoto F., Studies on data fitting with spline functions, PhD Thesis,Kyoto University, Japan, 1977.

[14] L.R. Williams, R.W. Legget,Nonzero solutions of nonlinear integral equa-tions modeling infectious disease, SIAM J. Math. Anal., 13 (1982), 112-121.

A.BICA,C.IANCU170

Set-valued Variational Inclusions with Fuzzy Mappings inBanach Spaces

A. H. SiddiqiDepartment of Mathematical Sciences

King Fahd University of Petroleum and MineralsP.O. Box 1745, Dhahran 31261, Saudi Arabia

E-mail: ahasan@kfupm.edu.sa

Rais AhmadDepartment of MathematicsAligarh Muslim University

Aligarh 202 002, IndiaE-mail: raisain@lycos.com

and

S. S. IrfanDepartment of MatematicasAligarh Muslim University

Aligarh 202 002, IndiaE-mail: shakaib11@rediffmail.com

171JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,171-181,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

Abstract. In this paper, we study set-valued variational inclusions with fuzzy map-pings in the setting of real Banach spaces. By using Nadler’s Theorem and theresolvent operator technique for m-accretive mappings, we propose an iterative algo-rithm for computing the approximate solutions of this class of set-valued variationalinclusions. We also discuss the existence of a solution of our problem without com-pactness assumption and prove the convergence of the iterative sequences generatedby the proposed algorithm.

Keywords: Variational inclusion, Algorithm, Convergence, Resolvent operators,m-accretive mappings.

2000 AMS subject Classification: 49J40, 47S40, 47H06

A.SIDDIQI ET AL172

1. INTRODUCTION

The theory of variational inequalities have had a great impact and influence inthe development of almost all branches of pure and applied sciences. Variationalinequalities have been generalized and extended in different directions using noveland innovative techniques. An important and useful generalization of variationalinequalities is called the variational inclusion, which is mainly due to Hassouni andMoudafi [10]. Ansari [1] and Chang and Zhu [8], separately, introduced and studieda class of variational inequalities for fuzzy mappings. Several classes of variationaland quasi-variational inequalities with fuzzy mappings are considered and studied byChang [3], Chang and Huang [7], Huang [11], Irfan [13] and Lee et al [14]. Recently,Chang et al [5, 6] introduced and studied the following class of set-valued variationalinclusion problems in the setting of Banach space E.

For a given m-accretive mapping A : E → 2E, a nonlinear mapping N : E×E →E, set-valued mappings T, F : E → CB(E) and a single-valued mapping g : E → E,find x ∈ E, w ∈ T (x) and q ∈ F (x) such that

0 ∈ N(w, q) + A(g(x)), (1)

where CB(E) denotes the family of all nonempty closed and bounded subsets of E.The purpose of this paper is to study the set-valued variational inclusion (1)

with fuzzy mappings without compactness condition in real Banach spaces. By us-ing the resolvent technique for m-accretive mappings, we establish the equivalencebetween the set-valued variational inclusions with fuzzy mappings and the resolventequation in the setting of real Banach spaces. We use this equivalence and Nadler’sTheorem [15] to propose an iterative algorithm for solving our class of set-valuedvariational inclusions with fuzzy mappings in real Banach spaces. The inequalityof Petryshyn [16] is used to establish the existence of a solution of our inclusionwithout compactness assumption and prove the convergence of iterative sequencesgenerated by the algorithm.

2. PRELIMINARIES

Throughout the paper, we assume that E is a real Banach space, E∗ is thetopological dual space of E, CB(E) is the family of nonempty closed and boundedsubsets of E,D(., .) is the Hausdorff metric on CB(E) defined by

D(A,B) = max

supx∈A

d(x,B), supy∈B

d(A, y)

, A,B ∈ CB(E),

We denote by 〈., .〉 the dual pairing between E and E∗ and by D(T ) the domain ofT . We also assume that J : E → 2E

∗is the normalized duality mapping defined by

J(x) = f ∈ E∗ : 〈x, f〉 =‖ x ‖ ‖ f ‖ and ‖ f ‖=‖ x ‖ , x ∈ E.Definition 1. [2] A set-valued mapping A : D(A) ⊂ E → 2E is said to be

SET-VALUED VARIATIONAL INCLUSIONS... 173

(i) accretive if for any x, y ∈ D(A) there exists j(x− y) ∈ J(x− y) such that forall u ∈ Ax and v ∈ Ay,

〈u− v, j(x− y)〉 ≥ 0;

(ii) k-strongly accretive, k ∈ (0, 1), if for any x, y ∈ D(A), there exists j(x− y) ∈J(x− y) such that for all u ∈ Ax and v ∈ Ay,

〈u− v, j(x− y)〉 ≥ k ‖ x− y ‖2;

(iii) m-accretive if A is accretive and (I + ρA)(D(A)) = E for every (equivalently,for some) ρ > 0, where I is the identity mapping, (equivalently, if A is accretiveand (I + A)(D(A)) = E).

Remark 1. If E = E∗ = H is a Hilbert space, then A : D(A) ⊂ H → 2H is anm-accretive mapping if and only if A : D(A) ⊂ H → 2H is a maximal monotonemapping, see for example [9].

Let E be a real Banach space. Let F(E) be a collection of fuzzy sets over E. Amapping F : E → F(E) is said to be fuzzy mapping. For each x ∈ E, F (x) (in thesequel, it will be denoted by Fx) is a fuzzy set on E and Fx(y) is the membershipfunction of y in F (x).

A fuzzy mapping F : E → F(E) is said to be closed if for each x ∈ E, thefunction y 7−→ Fx(y) is upper semicontinuous, that is, for any given net yα ⊂ Esatisfying yα → y0 ∈ E we have lim supα Fx(yα) ≤ Fx(y0).

For B ∈ F(E) and λ ∈ [0, 1], the set (B)λ = x ∈ E : B(x) ≥ λ is called aλ-cut set of B.

A closed fuzzy mapping A : E → F(E) is said to satisfy condition (∗):if there exists a function a : E → [0, 1] such that for each x ∈ E, (Ax)a(x)

is a nonempty bounded subset of E.

It is clear that if A is a closed fuzzy mapping satisfying condition (∗), then foreach x ∈ E, the set (Ax)a(x) ∈ CB(E).

In fact, let yαα∈Γ ⊂ (Ax)a(x) be a net and yα → y0 ∈ E. Then (Ax)(yα) ≥ a(x)for each α ∈ Γ. Since A is closed, we have

Ax(y0) ≥ lim supα∈Γ

Ax(yα) ≥ a(x).

This implies that y0 ∈ (Ax)ax and so (Ax)a(x) ∈ CB(E).

Let T, F : E → F(E) be two closed fuzzy mappings satisfying condition (∗).Then there exists two functions a, b : E → [0, 1] such that for each x ∈ E, wehave (Tx)a(x), (Fx)b(x) ∈ CB(E). Therefore, we can define two set-valued mappings

T , F : E → CB(E) by

T (x) = (Tx)a(x) , F (x) = (Fx)b(x), for each x ∈ E.

A.SIDDIQI ET AL174

In the sequel, T and F are called the set-valued mappings induced by fuzzy mappingsT and F , respectively.

Let N : E × E → E and g : E → E be the single-valued mappings and letT, F : E → F(E) be fuzzy mappings. Let a, b : E → [0, 1] be given functions.Suppose that A : E → 2E is an m-accretive mapping. We consider the followingset-valued variational inclusion problem with fuzzy mappings in Banach spaces.

(SVIPFM)

Find x,w, q ∈ E such thatTx(w) ≥ a(x), Fx(q) ≥ b(x) and0 ∈ N(w, q) + A(g(x)).

If T, F : E → CB(E) are classical set-valued mappings, we can define fuzzymappings T, F : E → F(E) by

x→ XT (x), x→ XF (x),

where XT (x) and XF (x) are characteristic functions of T (x) and F (x), respectively.Taking a(x) = b(x) = 1 for all x ∈ E, then (SVIPFM) is equivalent to problem

(1). For a suitable choice of mappings T, F,A, g,N and the space E, a number ofknown and new classes of variational inequalities and variational inclusions studiedin [4, 5, 6, 13] can be obtained from (SVIPFM).

3. ITERATIVE ALGORITHM

Definition 2. Let A : D(A) ⊂ E → 2E be an m-accretive mapping. For any ρ > 0,the mapping RA

ρ : E → D(A) associated with A defined by

RAρ (x) = (I + ρA)−1(x), x ∈ E,

is called the resolvent operator.

Remark 2. We note that RAρ is a single-valued and nonexpansive mapping, see for

example [2].

We first transfer (SVIPFM) into a fixed point problem.

Theorem 1. (x,w, q) is a solution of (SVIPFM) if and only if (x,w, q) satisfies thefollowing relation

g(x) = RAρ [g(x)− ρN(w, q)], (2)

where w ∈ T (x), q ∈ F (x), and ρ > 0 is a constant.

Proof. By the definition of resolvent operator RAρ associated with A, we have that

(2) holds if and only if w ∈ T (x) and q ∈ F (x) such that

g(x)− ρN(w, q) ∈ g(x) + ρA(g(x))

SET-VALUED VARIATIONAL INCLUSIONS... 175

if and only if0 ∈ N(w, q) + A(g(x)).

Hence (x,w, q) is a solution of (SVIPFM) if and only if w ∈ T (x) and q ∈ F (x) aresuch that (2) holds.

We also need the following lemma.

Lemma 1. [12] Let g : E → E be a continuous and k-strongly accretive mapping.Then g maps E onto E.

We now invoke Theorem 1, Lemma 1 and Nadler’s Theorem [15] to propose thefollowing iterative algorithm.

Algorithm 1. Let T, F : E → F(E) be two closed fuzzy mappings satisfyingcondition (∗) and T , F : E → CB(E) be the set-valued mappings induced by thefuzzy mappings T and F , respectively. Let N : E × E → E be a single-valuedbifunction and g : E → E a continuous and k-strongly accretive mapping. Forgiven x0 ∈ E, w0 ∈ T (x0) and q0 ∈ F (x0), let

g(x1) = RAρ [g(x0)− ρN(w0, q0)].

Since w0 ∈ T (x0) ∈ CB(E) and q0 ∈ F (x0) ∈ CB(E), by Nadler [15], there existw1 ∈ T (x1) and q1 ∈ F (x1) such that

‖ w0 − w1 ‖≤ (1 + 1)D(T (x0), T (x1)),

‖ q0 − q1 ‖≤ (1 + 1)D(F (x0), F (x1)).

Letg(x2) = RA

ρ [g(x1)− ρN(w1, q1)].

Continuing the above process inductively, we can obtain sequences xn, wn andqn satisfying

g(xn+1) = RAρ [g(xn)− ρN(wn, qn)],

wn ∈ T (xn), ‖ wn − wn+1 ‖≤(

1 +1

n+ 1

)D(T (xn), T (xn+1)),

qn ∈ F (xn), ‖ qn − qn+1 ‖≤(

1 +1

n+ 1

)D(F (xn), F (xn+1)),

for all n = 0, 1, 2 . . .

4. EXISTENCE AND CONVERGENCE RESULTS

In this section, we establish the existence of a solution of (SVIPFM) and provethe convergence of iterative sequences generated by Algorithm 1.

A.SIDDIQI ET AL176

Definition 3. A set-valued mapping T : E → CB(E) is said to be ξ-Lipschitzcontinuous if for any x, y ∈ E

D(Tx, Ty) ≤ ξ ‖ x− y ‖,

where ξ > 0 is a constant.

Definition 4. A mapping N : E ×E → E is said to be Lipschitz continuous in thefirst argument if there exists a constant α > 0 such that

‖ N(x1, .)−N(x2, .) ‖≤ α ‖ x1 − x2 ‖, for all x1, x2 ∈ E.

In a similar way, we can define the Lipschitz continuity of N in the second argument.

The following lemma plays an important role to prove our main result.

Lemma 2. [16] Let E be a real Banach space and J : E → 2E∗

be the normalizedduality mapping. Then for any x, y ∈ E,

‖ x+ y ‖2≤‖ x ‖2 +2 〈y, j(x+ y)〉 , for all j(x+ y) ∈ J(x+ y).

Theorem 2. Let E be a real Banach space. Let T, F : E → F(E) be two closedfuzzy mappings satisfying condition (?) and T , F : E → CB(E) the set-valuedmappings induced by the fuzzy mappings T, F , respectively. Let A : E → 2E

be an m-accretive mapping and g : E → E σ-Lipschitz continuous and k-stronglyaccretive. Assume that T is µ-Lipschitz continuous, F is γ-Lipschitz continuous andN : E ×E → E is Lipschitz continuous in both the arguments with constants α, β,respectively. If

ρ <2k − 1− σ2

2(βγ + αµ)√

2k − 1, k >

1

2(3)

then there exist x ∈ E, w ∈ T (x), q ∈ F (x) such that (x,w, q) is a solution of(SVIPFM) and xn → x, wn → w, qn → q as n→∞.

Proof. Since T is µ-Lipschitz continuous and F is γ-Lipschitz continuous, it followsfrom Algorithm 1 that

‖ wn−wn+1 ‖ ≤(

1 +1

n+ 1

)D(T (xn), T (xn+1) ≤

(1 +

1

n+ 1

)µ ‖ xn−xn+1 ‖,

‖ qn − qn+1 ‖ ≤(

1 +1

n+ 1

)D(F (xn), F (xn+1) ≤

(1 +

1

n+ 1

)γ ‖ xn − xn+1 ‖,

(4)for all n = 0, 1, 2 · · ·. Let zn+1 = g(xn) − ρ(N(wn, qn)) for all n = 0, 1, 2, · · ·. Sinceg is σ-Lipschitz continuous, N is Lipschitz continuous in both the arguments withconstants α and β, respectively, and by using Lemma 2 and (4), we have

‖ zn+1−zn ‖2=‖ g(xn)−g(xn−1)−ρ [N(wn, qn)−N(wn−1, qn−1)] ‖2

SET-VALUED VARIATIONAL INCLUSIONS... 177

≤ ‖ g(xn)− g(xn−1) ‖2 −2ρ 〈(N(wn, qn)−N(wn−1, qn−1)), j(zn+1, zn)〉≤ σ2 ‖ xn − xn−1 ‖2 +2ρ ‖ N(wn, qn)−N(wn−1, qn−1) ‖‖ zn+1 − zn ‖

= σ2 ‖ xn − xn−1 ‖2 +2ρ ‖ N(wn, qn)−N(wn, qn−1) +N(wn, qn−1)

−N(wn−1, qn−1) ‖‖ zn+1 − zn ‖≤ σ2 ‖ xn − xn−1 ‖2 +2ρ ‖ N(wn, qn)−N(wn, qn−1) ‖+ ‖ N(wn, qn−1)−N(wn−1, qn−1) ‖ ‖ zn+1 − zn ‖≤ σ2 ‖ xn−xn−1 ‖2 +2ρ β ‖ qn − qn−1 ‖ +α ‖ wn − wn−1 ‖ ‖ zn+1−zn ‖

≤ σ2 ‖ xn−xn−1 ‖2 +2ρβγ

(1 +

1

n

)+ αµ

(1 +

1

n

)‖ xn−xn−1 ‖‖ zn+1−zn ‖

= σ2 ‖ xn−xn−1 ‖2 +2ρ(

1 +1

n

)(βγ+αµ) ‖ xn−xn−1 ‖‖ zn+1− zn ‖ .

(5)Since g is k-strongly accretive, RA

ρ is non-expansive and from Lemma 2, it followsthat for any j(xn − xn−1) ∈ J(xn − xn−1),

‖ xn − xn−1 ‖2=‖ [RAρ (zn)−RA

ρ (zn−1)]− [g(xn)− xn − (g(xn−1)− xn−1)] ‖2

≤‖ RAρ (zn)−RA

ρ (zn−1) ‖2 −2 〈g(xn)− xn − (g(xn−1)− xn−1), j(xn − xn−1)〉≤‖ zn − zn−1 ‖2 −2k ‖ xn − xn−1 ‖2 +2 ‖ xn − xn−1 ‖2

which implies that

‖ xn − xn−1 ‖≤ 1√2k − 1

‖ zn − zn−1 ‖ . (6)

¿From (5) and (6), we have

‖ zn+1−zn ‖2≤ σ2

2k − 1‖ zn−zn−1 ‖2 +

2ρ(1 + 1/n)(βγ + αµ)√2k − 1

‖ zn−zn−1 ‖‖ zn+1−zn ‖

≤ σ2

2k − 1‖ zn−zn−1 ‖2 +

ρ(1 + 1/n)(βγ + αµ)√2k − 1

[‖ zn − zn−1 ‖2 + ‖ zn+1 − zn ‖2

]

Finally, we have‖ zn+1 − zn ‖≤ θn ‖ zn − zn−1 ‖ (7)

where

θn =[ σ2 + ρ(1 + 1/n)(βγ + αµ)

√2k − 1

(2k − 1)[1− ρ(1 + 1/n)(βγ + αµ)/√

2k − 1]

]1/2

Let

θ =[ σ2 + ρ(βγ + αµ)

√2k − 1

(2k − 1)[1− ρ(βγ + αµ)/√

2k − 1]

]1/2.

A.SIDDIQI ET AL178

Then we know that θn → θ as n → ∞. ¿From condition (3) it follows that θ < 1.Hence θn < 1 for n sufficiently large. Therefore (7) implies that zn is a Cauchysequence in E. Since E is a Banach space, there exists z ∈ E such that zn → zas n → ∞. From (4) and (6), we know that xn, wn and qn are also Cauchysequences in E. Therefore, there exist x ∈ E, w ∈ E and q ∈ E such that xn → x,wn → w and qn → q as n→∞. Note that wn ∈ T (xn), we have

d(w, T (x)) = inf‖ w − p ‖ : p ∈ T (x)

≤ ‖ w − wn ‖ +d(wn, T (x))

≤ ‖ w − wn ‖ +D(T (xn), T (x))

≤ ‖ w − wn ‖ +µ ‖ xn − x ‖→ 0, n→∞,which implies that d(w, T (x)) = 0. Since T (x) ∈ CB(E), it follows that w ∈ T (x).Similarly we can show that q ∈ F (x). Since g, RA

ρ , N(., .), T and F are continuous,it follows from Algorithm 1 that

g(x) = RAρ [g(x)− ρN(w, q)]

By Theorem 1, we know that (x,w, q) is a solution of set-valued variational in-clusions with fuzzy mappings (1) in real Banach spaces. This completes the proof.

References

[1] Q. H. Ansari, Certain Problems Concerning Variational Inequalities, Ph. D.Thesis, Aligarh Muslim University, Aligarh, India, 1988.

[2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,Noordhoff International Publishing, Leyden, The Netherlands, 1976.

[3] S. S. Chang, Coincidence theorem and variational inequalities for fuzzy map-pings, Fuzzy Sets and Systems 61, 359-368 (1994).

[4] S. S. Chang, On the Mann and Ishikawa iterative approximation of solutionsto variational inclusions with accretive type mappings, Comput. Math. Appl.37 (9), 17-24 (1999).

[5] S. S. Chang, Set-valued variational inclusions in Banach Spaces, J. Math. Anal.Appl. 248, 438-454 (2000).

[6] S. S. Chang, Y. G. Cho, B. B. Lee and I. H. Jung, Generalized set-valuedvariational inclusions in Banach Spaces, J. Math. Anal. Appl. 246, 409-422(2000).

[7] S. S. Chang and N. J. Huang, Generalized complementarity problem for fuzzymappings, Fuzzy Sets and systems 55 (2), 227-234 (1993).

SET-VALUED VARIATIONAL INCLUSIONS... 179

[8] S. S. Chang and Y. G. Zhu, On Variational inequalities for fuzzy mappings,Fuzzy Sets and Systems 32, 359-367 (1989).

[9] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

[10] A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions,J. Math. Anal. Appl. 185, 706-712 (1994).

[11] N. J. Huang, A new method for a class of nonlinear variational inequalities withfuzzy mappings, Appl. Math. Lett. 10 (6), 129-133 (1997).

[12] N. J. Huang, A new class of generalized set-valued implicit variational inclusionsin Banach spaces with an application, Comput. Math. Appl. 41, 937-943 (2001).

[13] S. S. Irfan, Existence Results and Solution Methods for Certain VariationalInequalities and Complementarity Problems, Ph. D. Thesis, Aligarh MuslimUniversity, Aligarh, India, (2002).

[14] G. M. Lee, D. S. Kim, B. S. Lee and S. J. Cho, Generalized vector variationalinequality and fuzzy extension, Appl. Math. Lett. 66, 47-51 (1993).

[15] S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30, 475-488(1969).

[16] W. V. Petryshyn, A characterization of strictly convexity of Banach spaces andother uses of duality mappings, J. Func. Anal. 6, 282-291 (1970).

A.SIDDIQI ET AL180

Acknowledgments. The first author would like to express his thanks to the De-partment of Mathematical Sciences, King Fahd University of Petroleum & Minerals,Dhahran, Saudi Arabia for providing excellent facilities for carrying out this work.In this research, second author was supported by the Chilean Government AgencyCONICYT under FONDAP program.

SET-VALUED VARIATIONAL INCLUSIONS... 181

182

On some properties of semilinear functional

differential inclusions in abstract spaces

Candida Goria - Valeri Obukhovskiib ∗- Marcello Ragnia - Paola Rubbionic

a Dipartimento di Matematica e Informatica, Universita di Perugia

via L. Vanvitelli 1, 06123 Perugia, ITALY

b Department of Mathematics, Voronezh University

Universitetskaya pl. 1, 394006 Voronezh, RUSSIA

c INFM and Dipartimento di Matematica e Informatica, Universita di Perugia

via L. Vanvitelli 1, 06123 Perugia, ITALY

Telephone: +39-075-5855042/5040; Fax: +39-075-5855024

e-mails: cgori@unipg.it - ingar@dipmat.unipg.it - valerio@math.vsu.ru - rubbioni@dipmat.unipg.it

Abstract

We consider local and global existence results for differential inclusions with infinite delay

in a Banach space of the form y′(t) ∈ Ay(t) + F (t, yt) , t ≥ σ , yσ = ϕ ∈ B , properties of

the translation multioperator along trajectories and existence of periodic solutions.

∗The work of the author is supported by RFBR Grants 05-01-00100, 04-01-00081 and NATO Grant

ICS.NR.CLG 981757.

1

183JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,183-214,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

AMS Subject Classification : 34K30, 34K13, 34G25

Key words and Phrases: Functional differential inclusion, infinite delay, phase space, measure of

noncompactness, topological degree, fixed point, periodic solutions.

1 Introduction

Semilinear functional differential inclusions in a Banach space have been the object of in-

tensive study by many researchers in recent years (see, e.g. [7], [12], [13], [16], [17], [18],

[19]).

Several works related to this subject are concerned with the Cauchy problem

y′(t) ∈ Ay(t) + F (t, yt) , t ≥ σ yσ = ϕ ∈ B

where yt represents the ”history” of the system, B is the abstract Hale-Kato phase space

(see, e.g. [8], [10]) and A is the densely defined linear operator infinitesimal generator of a

strongly continuous semigroup of linear operators eAt , t ≥ 0.

Notice that inclusions of that type appear in a very natural way in the description of processes

of controlled heat transfer (see, e.g. [2], [15]), in obstacle problems, in the study of hybrid

systems with dry friction, in the control of a transmission line process and other problems

(see [12] and references therein).

In this paper we prove the existence of local, global and periodic solutions for semilinear

differential inclusions with infinite delay in a Banach space.

Here we continue the study we began in [7], but in a slightly modified setting, more suitable

C.GORI ET AL184

to deal with the periodic problem. In particular, the regularity conditions posed on the

right-hand side as well as the condensivity of integral and translation multioperators are

expressed in terms of the Kuratowski measure of noncompactness.

An analogous problem for differential equations was studied by H.Henriquez in [9] under

a stronger assumption of compactness on the semigroup generated from the linear part of

the equation and by using the quotient space of the phase space.

About the periodic problem for differential inclusions without delay, there exists a de-

scription in the recent book of M.Kamenskii, V.Obukhovskii, P.Zecca ([12]), where topolog-

ical methods in multivalued analysis are developed and applied.

The paper is organized as follows.

In Section 2, we give some definitions and preliminary results. Afterwards, in Section 3,

we provide local and global existence results for our abstract Cauchy problem. In Section

4, under some additional hypotheses, we study the upper semicontinuous dependence of

the solution set on the initial data as well as its topological properties. Further we apply

the properties of the translation multioperator along the trajectories of the solutions to the

study of the periodic problem. In particular, in order to use the corresponding topological

degree theory, we give a sufficient condition under which the translation multioperator is

condensing with respect to the Kuratowski measure of noncompactness of the phase space.

2 Preliminaries

Let X and Y be topological spaces and let us denote by P (Y ) the collection of all

nonempty subsets of Y . A multivalued map (multimap) F : X → P (Y ) is said to be:

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 185

(i) upper semicontinuous (u.s.c) if F−1(V ) = x ∈ X : F(x) ⊂ V is an open subset of

X for every open V ⊂ Y ;

(ii) closed if its graph ΓF = (x, y) : y ∈ F (x) is a closed subset of the space X × Y .

Recall also the following notions (see, e.g. [1], [12]).

Let E be a Banach space and (A,≥) a (partially) ordered set. A function β : P (E) →

A is called a measure of noncompactness (MNC) in E if

β(coΩ) = β(Ω)

for every Ω ∈ P (E).

A MNC is called:

i) monotone if Ω0,Ω1 ∈ P (E), Ω0 ⊂ Ω1 implies β(Ω0) ≤ β(Ω1) ;

ii) nonsingular if β(a ∪ Ω) = β(Ω) for every a ∈ E , Ω ∈ P (E);

iii) real if A = [0,+∞] with the natural ordering, and β(Ω) < +∞ for every bounded Ω.

If A is a cone in a Banach space we will say that the MNC β is regular if β(Ω) = 0 is

equivalent to the relative compactness of Ω.

As the example of the MNC possessing all these properties, we may consider the Kura-

towskii MNC

α(Ω) = infδ > 0 : Ω has a partition into a finite number of sets with diameter less than δ

and the Hausdorff MNC

χ(Ω) = infε > 0 : Ω has a finite ε− net .

C.GORI ET AL186

¿From the definitions it follows that the Kuratowski and Hausdorff MNCs are connected by

the following relations:

χ(Ω) ≤ α(Ω) ≤ 2χ(Ω) . (1)

By means of the Kuratowski MNC we can define some important characteristics of a

linear operator in a Banach space.

Definition 2.1. Let L : E → E be a bounded linear operator and B the unit ball of E .

The number

‖L‖(α) ≡ α(LB)

is said to be the (α)-norm of the operator L.

It is easy to see that the following properties hold:

‖L‖(α) ≤ ‖L‖

α(LΩ) ≤ ‖L‖(α)α(Ω) (2)

Let X be a closed subset of a Banach space E and K(E) [Kv(E)] denote the collection

of all nonempty compact [compact convex] subsets of E .

Definition 2.2. Let β be a real MNC in E and 0 ≤ k < 1. A multimap F : X → K(E)

is said to be (k, β)-condensing or simply β-condensing if

β(F(Ω)) ≤ kβ(Ω)

for every Ω ⊆ X .

The following fixed point theorem (see [12], Corollary 3.2, Proposition 3.5.1) will be

useful in the forthcoming local existence result.

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 187

Theorem 2.1. If M is a bounded closed convex subset of E , and F : M→ Kv(M) is

a closed β-condensing multimap, where β is a real nonsingular and regular MNC in E , then

the fixed points set FixF = x : x ∈ F(x) is nonempty and compact.

Everywhere in the following E will denote a separable Banach space with the norm ‖·‖ .

Let us recall some notions (see, e.g. [5], [11], [12]).

A multifunction G : [c, d] → K(E) is said to be strongly measurable if there exists a

sequence Gn∞n=1 of step multifunctions such that

h(Gn(t),G(t)) → 0

as n → ∞ for µ-a.e. t ∈ [c, d] where µ denotes a Lebesgue measure on [c, d] and h is the

Hausdorff metric on K(E). Every strongly measurable multifunction G admits a strongly

measurable selection g : [c, d] → E , i.e., g(t) ∈ G(t) for a.e. t ∈ [c, d] .

By the symbol L1([c, d];E) we will denote the space of all Bochner summable functions.

A multifunction G : [c, d] → K(E) is said to be:

i) integrable provided it has a summable selection g ∈ L1([c, d];E) ;

ii) integrably bounded if there exists a summable function ω(·) ∈ L1+([c, d]) such that

‖G(t)‖ := sup‖g‖ : g ∈ G(t) ≤ ω(t)

for a.e. t ∈ [c, d].

It is clear that strongly measurable and integrably bounded multifunction is integrable.

The set of all summable selections of the multifunction G will be denoted by S1G .

C.GORI ET AL188

Put αE the Kuratowski measure of noncompactness in the space E, the following result

about αE-estimates for a multivalued integral is the consequence of Theorem 4.2.3 of [12]

and the estimates (1):

Lemma 2.1. Let a multifunction G : [c, d] → P (E) be integrable, integrably bounded and

αE(G(t)) ≤ q(t)

for a.e. t ∈ [c, d], where q ∈ L1+[c, d].

Then

αE

(∫ t

cG(s)ds

)≤ 2

∫ t

cq(s)ds

for all t ∈ [c, d].

Consider an abstract operator S : L1([c, d];E) → C([c, d];E) satisfying the following

conditions (cf. [12]):

(S1) there exists ζ ≥ 0 such that

‖Sf(t)− Sg(t)‖ ≤ ζ

∫ t

c‖f(s)− g(s)‖ds

for every f, g ∈ L1([c, d];E) , c ≤ t ≤ d;

(S2) for any compact K ⊂ E and sequence fn∞n=1 ⊂ L1([c, d];E) such that fn(t)∞n=1 ⊂

K for a.e. t ∈ [c, d] the weak convergence fn f0 implies Sfn → Sf0.

Of course, condition (S1) implies that the operator S satisfies the Lipschitz condition

(S1′) ‖Sf − Sg‖C ≤ ζ‖f − g‖L1

where the first norm is the usual sup-norm in the space C([c, d];E) .

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 189

In the following we will suppose that

(A) the linear operator A : D(A) ⊆ E → E is the densely defined infinitesimal generator

of a strongly continuous semigroup of linear operators eAt , t ≥ 0 which is immediately

norm continuous, i.e. the function t→ eAt is norm continuous from (0,∞) into L(E),

the space of bounded linear operators in E (see [6]).

Remark 2.1. Note that the Cauchy operator G : L1([c, d];E) → C([c, d];E) defined as

Gf(t) =∫ t

ceA(t−s)f(s)ds

satisfies properties (S1) and (S2) (see [12], Lemma 4.2.1).

The sequence fn∞n=1 ⊂ L1([c, d];E) is said to be semicompact if:

(i) it is integrably bounded: ‖fn(t)‖ ≤ ω(t) for a.e. t ∈ [c, d] and every n ≥ 1 where

ω(·) ∈ L1+[c, d]

(ii) the set fn(t)∞n=1 is relatively compact for a.e. t ∈ [c, d].

Let us mention the following important property of semicompact sequences (see, e.g.

[12], Proposition 4.2.1).

Lemma 2.2. Every semicompact sequence is weakly compact in the space L1([c, d];E).

In the sequel we will need also the following properties of the operator S satisfying

assumptions mentioned above (see [12], Theorem 5.1.1).

Lemma 2.3. Let S : L1([c, d];E) → C([c, d];E) be an operator satisfying the conditions

(S1′) and (S2). Then for every semicompact sequence fn∞n=1 ⊂ L1([c, d];E) the sequence

Sfn∞n=1 is relatively compact in C([c, d];E) and, moreover, if fn f0 then Sfn → Sf0.

C.GORI ET AL190

Now let σ be a real number and a > 0 . For any function y : (−∞, σ + a] → E and

for every t ∈ (−∞, σ + a] , yt represents the function from (−∞, 0] into E defined by

yt(θ) = y(t+ θ) , −∞ < θ ≤ 0 .

Throughout this paper we will employ the axiomatic definition of the phase space B

introduced by Hale and Kato [8]. The space B will be considered as a linear topological

space of functions mapping (−∞, 0] into E endowed with a seminorm ‖ · ‖B. We will assume

that B satisfies the following set of axioms.

If y : (−∞, σ+a] → E is continuous on [σ, σ+a] and yσ ∈ B, then for every t ∈ [σ, σ+a]

we have

(B1) yt ∈ B;

(B2) the function t 7→ yt is continuous;

(B3) ‖yt‖B ≤ K(t− σ) supσ≤τ≤t ‖y(τ)‖+M(t− σ)‖yσ‖B

where K,M : [0,+∞) → [0,+∞) are independent of y, K is strictly positive and

continuous, and M is locally bounded.

3 Existence results

We consider the following problem for systems governed by a semilinear functional differential

inclusion

y′(t) ∈ Ay(t) + F (t, yt) , t ≥ σ (3)

yσ = ϕ ∈ B (4)

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 191

where the multivalued nonlinearity F : [σ, σ + a] × B → Kv(E) will satisfy the following

hypotheses:

(F1) for every fixed ψ ∈ B the multifunction F (·, ψ) : [σ, σ + a] → Kv(E) admits a

strongly measurable selector;

(F2) for a.e. t ∈ [σ, σ + a] the multimap F (t, ·) : B → Kv(E) is u.s.c.;

(F3) for every nonempty bounded set Ω ⊂ B there exists a number µΩ ≥ 0 such that for

every ψ ∈ Ω

‖F (t, ψ)‖ ≤ µΩ

for a.e. t ∈ [σ, σ + a];

(F4) there exists a function k ∈ L1+([σ, σ + a]) such that for every bounded D ⊂ B

αE(F (t,D)) ≤ k(t)αB(D) ,

for a.e. t ∈ [σ, σ + a] where αB is Kuratowski measure of noncompactness in the

space B generated by the seminorm ‖ · ‖B.

For 0 < b ≤ a , let us denote by the symbol C((−∞, σ + b];E) the linear topological

space of functions y : (−∞, σ + b] → E such that yσ ∈ B and the restriction y| [σ,σ+b] is

continuous, endowed with a seminorm

‖y‖C = ‖yσ‖B + ‖y| [σ,σ+b]‖C .

Definition 3.1. A function y ∈ C((−∞, σ+ b];E) (0 < b ≤ a) is a mild solution of (3),

(4) if

C.GORI ET AL192

(i) yσ = ϕ

(ii) y(t) = eA(t−σ)ϕ(0) +∫ tσ e

A(t−s)f(s)ds, t ∈ [σ, σ + b], where f ∈ S1F (·,y(·))

.

Note that conditions under which a mild solution satisfies some regularity properties are

described in Section 5.2.1 of [12].

Let X be any set of functions x : (−∞, σ + a] → E such that xσ ∈ B and x| [σ,σ+a] is

continuous for all x ∈ X. We denote, for t ∈ [σ, σ + a],

Xt = xt ∈ B : x ∈ X

X[σ, t] = x| [σ,t] : x ∈ X .

The main result of this section is an existence result for problem (3), (4). For the proof

we will need the following result of J.S.Shin ([20], Theorem 2.1):

Lemma 3.1. For any t ∈ [σ, σ + a] the following relation holds:

αB(Xt) ≤ K(t− σ)αC(X[σ, t]) +M(t− σ)αB(Xσ) .

Let us also mention the following useful property (see, e.g. [3]).

Lemma 3.2. If ∆ ⊂ C([c, d];E) is a bounded equicontinuous set, then

αC(∆) = supc≤t≤d

αE(∆(t)) .

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 193

Theorem 3.1. Under assumptions (A), (B1)-(B3) and (F1)-(F4) there exist h ∈

(0, a] and a mild solution y∗ of (3), (4) on (−∞, σ + h].

Proof. The proof follows the lines of Theorem 2 in [7].

Let us take r > 0 .

Since the semigroup eAt is strongly continuous, there exists h1 ∈ (0, a] such that

‖(eAτ − I)ϕ(0)‖ ≤ r

2, 0 ≤ τ ≤ h1

and there exists C ∈ IR+ such that

‖eAθ‖ ≤ C , 0 ≤ θ ≤ a . (5)

Let us take the continuous function s : [0, a] → B defined as

s(t)(θ) =

ϕ(t+ θ), −∞ < θ ≤ −t

ϕ(0), −t < θ ≤ 0 ,

and consider the set

Q = s([0, a]) ⊂ B ,

which is compact.

Let K∗ > 0 be the value

K∗ = max0≤t≤a

K(t) , (6)

where K(·) is the function introduced in the axiom (B3).

Then, taking the closure W of the rK∗-neighbourhood of Q,

W = QrK∗

C.GORI ET AL194

and the corresponding number µW (see condition (F3)) we may choose h2 ∈ (0, a] so that

CµWh2 ≤r

2.

Moreover, there exists h3 ∈ (0, a] such that

2K∗C

∫ σ+h3

σk(τ) dτ < 1 (7)

where K∗ and C are as above, and k(·) is from the condition (F4).

We put

h = minh1, h2, h3 . (8)

For ϕ ∈ B given in (4), we consider

D(ϕ, h) = x ∈ C([σ, σ + h];E) : x(σ) = ϕ(0)

which is a closed convex set.

Further, for any x ∈ D(ϕ, h), we consider the function x[ϕ] ∈ C((−∞, σ + h];E)

x[ϕ](t) =

ϕ(t− σ) −∞ < t < σ

x(t) σ ≤ t ≤ σ + h .

(9)

Then, for any t ∈ [σ, σ + h],

x[ϕ]t(θ) =

ϕ(t− σ + θ) −∞ < θ < σ − t

x(t+ θ) σ − t ≤ θ ≤ 0 .

(10)

We consider the map j : [σ, σ + h]×D(ϕ, h) → B defined by

j(t, x) = x[ϕ]t

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 195

and recall ([7], Theorem 2) that j(·, x) is continuous and j(t, ·) is Lipschitz continuous in

the seminorm ‖ · ‖B uniformly with respect to t ∈ [σ, σ + h] .

Now, we consider the multivalued superposition operator

PF : D(ϕ, h) → P (L1([σ, σ + h];E))

defined as

PF (x) = S1F (·,j(·,x)) =

= f ∈ L1([σ, σ + h];E) : f(s) ∈ F (s, j(s, x)) = F (s, x[ϕ]s) for a.e. s ∈ [σ, σ + h].

PF is correctly defined and, using Lemma 4 in [7], one may verify that it is weakly closed.

We consider the integral multioperator

Γ : D(ϕ, h) → P (D(ϕ, h))

defined by

Γ(x) =z : z(t) = eA(t−σ)ϕ(0) +

∫ t

σeA(t−s)f(s)ds, f ∈ PF (x)

.

Of course, every mild solution y ∈ C((−∞, σ + h];E) of (3), (4) is determined by a fixed

point x of Γ by means of

y(t) = x[ϕ](t).

The fact that Γ is a closed multioperator with compact convex values and that it trans-

forms the ball Br(ϕ) into itself, where ϕ(t) ≡ ϕ(0) , t ∈ [σ, σ + h] , can be proved by the

same method as in [7].

To prove the α−condensivity of Γ on bounded sets, we use the following important

property of the integral multioperator.

C.GORI ET AL196

Lemma 3.3. For any bounded Ω ⊂ D(ϕ, h) the image Γ(Ω) is bounded equicontinuous.

Proof. Let Ω be a bounded subset of D(ϕ, h) and let N be a positive number such that

‖x‖C ≤ N for any x ∈ Ω.

First of all, we prove that Γ(Ω) is bounded.

Let y ∈ Γ(Ω). By means of the definition of Γ(Ω), there exists x ∈ Ω such that

y(t) = eA(t−σ)ϕ(0) +∫ t

σeA(t−s)f(s)ds, t ∈ [σ, σ + h]

where f ∈ PF (x), so, in particular, f(s) ∈ F (s, x[ϕ]s), a.e. s ∈ [σ, t].

Let Ω be the set defined as

Ω = x[ϕ]t ∈ B : x ∈ Ω, t ∈ [σ, σ + h] .

It is a bounded set. In fact, for any x ∈ Ω and t ∈ [σ, σ + h], from (B3) we get

‖x[ϕ]t‖B ≤ K∗ supσ≤τ≤t

‖x(τ)‖+M∗‖ϕ‖B ≤

≤ K∗‖x‖C +M∗‖ϕ‖B ≤ K∗N +M∗‖ϕ‖B

where K∗ is from (6) and

M∗ = sup0≤t≤a

M(t). (11)

Therefore, by using (5) and by applying condition (F3) with D = Ω, for every t ∈ [σ, σ+ h]

we have the following estimation :

‖y(t)‖ ≤ C‖ϕ(0)‖+ C

∫ t

σ‖f(s)‖ds ≤ C‖ϕ(0)‖+ CµΩh ,

where C is from (5).

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 197

To prove the equicontinuity of Γ(Ω), for any t1, t2 ∈ [σ, σ + h] with t1 < t2, we consider

y(t2)− y(t1) = J1 + J2 + J3,

where

J1 =(eA(t2−σ) − eA(t1−σ)

)ϕ(0) ,

J2 =∫ t1

σ

(eA(t2−s) − eA(t1−s)

)f(s) ds

and

J3 =∫ t2

t1eA(t2−s)f(s) ds .

Given arbitrary ε > 0, let us estimate every term Ji, i = 1, 2, 3. At first,

‖J1‖ = ‖eA(t1−σ)‖ ‖(eA(t2−t1) − I

)ϕ(0)‖ ≤ C‖

(eA(t2−t1) − I

)ϕ(0)‖ < ε

provided t2 − t1 < δ1.

To estimate J2 we may assume w.l.o.g. that t1 > σ. Let us take κ > 0 such that

2CµΩκ <ε2 and κ < t1 − σ. By virtue of condition (A), there exists δ2 > 0 such that

‖eA(t2−s) − eA(t1−s)‖ < ε

2µΩh

for t2 − t1 < δ2; s ≤ t1 − κ, where h is from (8). Therefore

‖J2‖ ≤∫ t1−κ

σ‖eA(t2−s) − eA(t1−s)‖ ‖f(s)‖ ds+

∫ t1

t1−κ‖eA(t2−s) − eA(t1−s)‖ ‖f(s)‖ ds

≤ ε

2µΩhµΩ(t1 − κ− σ) + 2CµΩκ < ε.

At last, notice that ‖J3‖ is obviously estimated by CµΩ(t2 − t1). 2

Lemma 3.4. Γ is αC-condensing on bounded subsets of D(ϕ, h).

C.GORI ET AL198

Proof. Let Ω ⊂ D(ϕ, h) be a bounded set and let t ∈ [σ, σ + h] be fixed.

Of course,

Γ(Ω)(θ) ⊂ eA(θ−σ)ϕ(0) +∫ θ

σeA(θ−τ)F (τ,Ω[ϕ]τ ) dτ . (12)

First of all, we consider the MNC αE of the integrand in the right-hand side of the above

relation. From (5) and assumption (F4), we have

αE(eA(θ−τ)F (τ,Ω[ϕ]τ )) ≤ CαE(F (τ,Ω[ϕ]τ )) ≤ Ck(τ)αB(Ω[ϕ]τ ) .

It is clear that Ω[ϕ][σ, τ ] = Ω[σ, τ ] and Ω[ϕ]σ = ϕ, therefore, by applying Lemma 3.1, we

have the following estimate:

αE(eA(θ−τ)F (τ,Ω[ϕ]τ )) ≤ Ck(τ) [K(τ − σ)αC(Ω[σ, τ ]) +M(τ − σ)αB(ϕ)] ≤

≤ CK∗k(τ)αC(Ω)

where K∗ is from (6).

Then, by means of Lemma 2.1 and by applying the above inequality, from (12) we have

αE(Γ(Ω)(θ)) ≤ αE

(∫ θ

σeA(θ−τ)F (τ,Ω[ϕ]τ ) dτ

)≤ 2CK∗αC(Ω)

∫ θ

σk(τ) dτ .

Since the right-hand part does not depend on t, by using Lemma 3.2 we obtain

αC(Γ(Ω)) = supσ≤t≤σ+h

αE(Γ(Ω)(t)) ≤ 2K∗C

∫ σ+h

σk(τ) dτ αC(Ω) .

Afterwards, since h is small enough to provide 2K∗C∫ σ+hσ k(τ) dτ < 1 (see (7)), we

conclude that Γ is αC-condensing. 2

As a direct consequence of Theorem 2.1, Γ has a fixed point. 2

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 199

Under a condition stronger than (F3), we can obtain a global existence result.

Theorem 3.2. Assume that conditions (A), (B1)-(B3), (F1), (F2), (F4) are satisfied

and that

(F3′) there exists a number µ ≥ 0 such that for every ψ ∈ B we have the estimate

‖F (t, ψ)‖ ≤ µ(1 + ‖ψ‖B)

for a.e. t ∈ [σ, σ + a].

Assume also that

(H1) 2K∗C∫ σ+aσ k(τ) dτ < 1 where K∗, C, k(·) are from (6), (5), (F4) respectively.

Then the set Σϕ of all mild solutions of problem (3), (4) on (−∞, σ + a] is nonempty and

compact.

Proof. The proof is analogous to that of Theorem 3 in [7] by using the Kuratowski

MNC instead of the MNC adopted in the cited theorem. We just note that, by applying the

same arguments of Lemma 3.4 under the further assumption (H1), the multifunction Γ is

αC-condensing on the whole interval [σ, σ + a] . 2

4 The translation multioperator and the periodic problem

In this section we study the periodic problem associated to (3), (4) by developing the method

of the translation multioperator along the trajectories of solutions.

To this end, we need to investigate on the α-condensivity of the translation multioperator.

C.GORI ET AL200

In fact, from the structure of the integral funnel and the theory of topological degree for con-

densing nonconvex-valued multimaps ([12], Section 3.4), the existence of periodic solutions

follows.

We need some additional hypotheses on the phase space B.

The first is the following:

(B4) there exists l > 0 such that

‖ϕ(0)‖ ≤ l‖ϕ‖B

for all ϕ ∈ B.

Then, we need two hypotheses concerning continuous representatives in B.

It is easy to see ([10], Proposition 1.2.1) that the space C00 of all continuous functions

from (−∞, 0] into E with compact support is a subset of any space B.

Our first assumption is nothing but hypothesis (C2) in [10]:

(D1) if a uniformly bounded sequence ϕn+∞n=1 ⊂ C00 converges to a function ϕ uniformly

on every compact subset of (−∞, 0], then ϕ ∈ B and limn→+∞ ‖ϕn − ϕ‖B = 0.

The hypothesis (D1) yields that the Banach space BC((−∞, 0];E) of bounded continuous

functions is continuously imbedded into B (see [10], Proposition 7.1.1).

Our next assumption means that the seminorm ‖ · ‖B is sensitive enough to distinguish

continuous functions with compact support:

(D2) if x ∈ C00 and ‖x‖BC 6= 0, then ‖x‖B 6= 0.

This hypothesis implies that the space C00 endowed by ‖ · ‖B is a normed space. We will

denote it as BC00.

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 201

We recall that, under assumptions of existence of solutions, the translation multioperator

maps every ϕ ∈ B into the set Σ(ϕ)σ+h ⊂ B (see (10)), where Σ(ϕ) is the set of all mild

solutions of (3), (4).

By the definition, if y : (−∞, σ + h] → E is a mild solution of the problem (3),(4) with

the initial function ϕ in BC00, then yσ+h ∈ BC00; therefore, we may consider the translation

multioperator along the trajectories of the inclusion (3) as acting in the space BC00:

Pσ+h : BC00 → P (BC00) .

Now, we assume that the multimap F is T -periodic in the first argument:

(FT ) F (t+ T, ψ) = F (t, ψ) for every (t, ψ) ∈ IR+ × B .

We consider on the space BC00 × C([σ, σ + T ];E) the subset

V = (ϕ, x) : ϕ(0) = x(σ)

which is closed by (B4), and we define an integral multioperator G : V → P (C([σ, σ +

T ];E)) by

G(ϕ, x) = z : z(t) = eA(t−σ)ϕ(0) +∫ t

σeA(t−s)f(s)ds, f ∈ L1([σ, σ + T ];E),

f(s) ∈ F (s, x[ϕ]s) for a.e. s

where, as in Theorem 3.1,

x[ϕ]s(θ) =

ϕ(s− σ + θ), −∞ < θ ≤ σ − s

x(s+ θ), σ − s < θ ≤ 0 .

It is clear that if x ∈ G(ϕ, x) , then the element x[ϕ] ∈ C((−∞, σ + T ];E) (see (9)) is a

mild solution of our Cauchy problem.

C.GORI ET AL202

In the forthcoming Lemmas, we assume that the hypotheses of Theorem 3.2 on the

interval [σ, σ + T ] and the further assumptions (B4), (D1), (D2) are fullfilled.

Lemma 4.1. The multimap Σσ : BC00 → K(C([σ, σ + T ];E)) defined as

Σσ(ϕ) = x ∈ D(ϕ, T ) : x ∈ G(ϕ, x)

is upper semicontinuous.

Proof. Let us assume, to the contrary, that there exist ε0 > 0 and sequences ϕn∞n=1 ⊂

BC00, ‖ϕn − ϕ0‖B → 0, xn∞n=1 ⊂ C([σ, σ + T ];E) , xn ∈ Σσ(ϕn), such that

xn /∈Wε0(Σσ(ϕ0)) , n ≥ 1 (13)

where Wε0 denotes the open ε0-neighborhood of Σσ(ϕ0).

It is clear by definition that

xn ∈ G(ϕn, xn) , n ≥ 1 ,

i.e.

xn(t) = eA(t−σ)ϕn(0) +∫ t

σeA(t−s)fn(s)ds

where fn ∈ L1([σ, σ + T ];E) , fn(s) ∈ F (s, xn[ϕn]s) , for a.e. s ∈ [σ, t].

Further, the sequence xn∞n=1 is a priori bounded. In fact, for every n ≥ 1, by applying

condition (F3′) and axiom (B3), we have the following estimations:

‖xn(t)‖ ≤ C‖ϕn(0)‖+ Cµ

∫ t

σ(1 + ‖xn[ϕn]s‖B)ds ≤

≤ C‖ϕn(0)‖+ CµT + Cµ

∫ t

σ

(K∗ sup

σ≤τ≤s‖xn(τ)‖+M∗‖ϕn‖B

)ds ≤

≤ C‖ϕn(0)‖+ CµT (1 +M∗‖ϕn‖B) + CK∗µ

∫ t

σsup

σ≤τ≤s‖xn(τ)‖ds

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 203

where C, K∗ and M∗ are defined by (5), (6) and (11) respectively.

Since the last expression does not decrease, we have

supσ≤τ≤t

‖xn(τ)‖ ≤ C‖ϕn(0)‖+ CµT (1 +M∗‖ϕn‖B) + CK∗µ

∫ t

σsup

σ≤τ≤s‖xn(τ)‖ds.

Applying to the function wn(t) = supσ≤τ≤t

‖xn(τ)‖ the Gronwall-Bellmann inequality we

obtain that

wn(t) ≤ Nn exp (CK∗µ(t− σ))

where Nn = C‖ϕn(0)‖+ CµT (1 +M∗‖ϕn‖B) .

From the convergence of the sequence ϕn∞n=1 in the space BC00, which is normed, it follows

that ϕn∞n=1 is bounded in BC00; therefore, by using (B4), the sequence ϕn(0)∞n=1 is also

bounded in E, implying the desired boundedness of sequence xn∞n=1.

Besides, the sequence xn∞n=1 is equicontinuous. In fact, for any t1, t2 ∈ [σ, σ + T ] with

t1 < t2 and for any n ≥ 1, we have

xn(t2)− xn(t1) = In1 + In2 + In3,

where

In1 =(eA(t2−σ) − eA(t1−σ)

)ϕn(0);

In2 =∫ t1

σ

(eA(t2−s) − eA(t1−s)

)fn(s)ds;

and

In3 =∫ t2

t1eA(t2−s)fn(s)ds.

Given arbitrary ε > 0 and taking into account, by (B4), that the sequence ϕn(0)∞n=1 is

convergent and hence relatively compact, we conclude that the term In1 may be estimated

C.GORI ET AL204

uniformly with respect to n, i.e. there exists δ1 > 0 such that ‖In1‖ < ε, n = 1, 2, ...

provided t2 − t1 < δ1.

Consider the sequence xn[ϕn]s∞n=1. It is bounded in BC00, in fact, by applying axiom

(B3), for every n ≥ 1 the following estimation holds:

‖xn[ϕn]s‖B ≤ K∗ supσ≤τ≤s

‖xn(τ)‖+M∗‖ϕn‖B ≤

≤ K∗‖xn‖C +M∗‖ϕn‖B

where K∗,M∗ are as above. Bearing in mind that sequences xn∞n=1 and ϕn∞n=1 are

bounded in C([σ, σ + T ];E) and BC00 respectively, the conclusion follows.

Now, from (F3′) we have that

‖fn(s)‖ ≤ µ(1 + ‖xn[ϕn]s‖B) for a.e. s ∈ [0, t]

and hence the sequence fn∞n=1 is bounded and we may apply the same arguments as while

the proof of Lemma 3.3 to obtain the estimates for In2, In3 uniform with respect to n yielding

the equicontinuity of the sequence xn∞n=1.

Now, for any t ∈ [σ, σ + T ], it is easy to see that

xn(t)∞n=1 ⊂ eA(t−σ)ϕn(0)∞n=1 +∫ t

σeA(t−s)F (s, xn[ϕn]s∞n=1)ds

hence, by means of the properties of the MNC,

αE(xn(t)∞n=1) ≤ αE(eA(t−σ)ϕn(0)∞n=1) + αE

(∫ t

σeA(t−s)F (s, xn[ϕn]s∞n=1)ds

).

¿From the relative compactness of the sequence ϕn(0)∞n=1 it follows that the first term of

the right hand side vanishes. Estimating, by means of (2), (F4) and Lemma 3.1,

αE

(eA(t−s)F (s, xn[ϕn]s∞n=1)

)≤ K∗Ck(s)αC(xn∞n=1)

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 205

and using Lemma 2.1 we get

αC(xn∞n=1) = supσ≤t≤σ+T

αE(xn(t)∞n=1) ≤ 2K∗C

∫ σ+T

σk(τ) dτ αC(xn∞n=1) .

Now from hypothesis (H1) it follows that αC(xn∞n=1) = 0 and the sequence xn∞n=1 is

relatively compact. We may assume, w.l.o.g., that xn → x0 ∈ C([σ, σ + T ];E).

Our aim is to show that x0 ∈ G(ϕ0, x0), i.e. x0 ∈ Σσ(ϕ0), contrary to (13).

To this end, we consider the identity

xn(t) = eA(t−σ)ϕn(0) +∫ t

σeA(t−s)fn(s)ds , fn(s) ∈ F (s, xn[ϕn]s) a.e. s ∈ [0, t]. (14)

From the convergence of ϕn(0)∞n=1 to ϕ0(0), we have

eA(t−σ)ϕn(0) → eA(t−σ)ϕ0(0).

Now, let us estimate the MNC of the sequence fn(t)∞n=1 for a.e. t ∈ [σ, σ + T ].

By using assumption (F4) and Lemma 3.1, we have

αE(fn(t)∞n=1) ≤ αE(F (t, xn[ϕn]t∞n=1)) ≤ k(t)αB(xn[ϕn]t∞n=1) ≤

≤ k(t)[K∗αC(xn| [σ,t]∞n=1) +M∗αBC00(ϕn∞n=1)

].

Since the sequences xn∞n=1 and ϕn∞n=1 converge, the last expression is equal to zero,

therefore

αE(fn(t)∞n=1) = 0 for a.e. t ∈ [σ, σ + T ]. (15)

Afterwards, by means of (F3′) and (15), the sequence fn∞n=1 is semicompact. By applying

Lemma 2.2, we deduce that, w.l.o.g., fn∞n=1 weakly converges to a function f0 ∈ L1([σ, σ+

T ];E).

C.GORI ET AL206

By using axiom (B3) and the assumptions on F , with the same arguments as in Lemma

5.1.1 in [12], we have

f0(t) ∈ F (t, x0[ϕ0]t) , a.e. t .

Hence, by applying Remark 2.1 and Lemma 2.3, the integral term in (14) converges to

∫ tσ e

A(t−s)f0(s)ds, which concludes the proof. 2

To formulate the next statement, let us recall (see e.g. [12]) that a nonempty space is

said to be an Rδ-set if it can be represented as the intersection of a decreasing sequence of

compact, contractible sets. It is clear that every Rδ-set is acyclic.

Lemma 4.2. The set Σσ(ϕ) is compact and, moreover, it is Rδ.

Proof. For any ϕ ∈ B the set Σσ(ϕ) is a priori bounded and it can be seen as the set

FixG(ϕ, ·).

As proved in Lemma 3.4, the integral multioperator G(ϕ, ·) is α-condensing, then the com-

pactness of Σσ(ϕ) follows from Proposition 3.5.1 of [12].

The fact that Σσ(ϕ) is an Rδ-set can be proved following the lines of Theorem 5.3.1 in

[12]. 2

Lemma 4.3. The translation multioperator PT is quasi Rδ-multimap, i.e. it can be

represented as a composition of an u.s.c. multimap with Rδ values and a continuous map.

Proof. We consider the multimap Π : BC00 → P (V ) defined as

Π(ϕ) = ϕ × Σσ(ϕ) .

A multimap Σσ(ϕ) can be regarded as a fixed point set of a condensing multioperator G(ϕ, ·)

depending on parameter and so it is u.s.c. by virtue of Proposition 3.5.2 of [12]. The set

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 207

Π(ϕ) is an Rδ since so is Σσ(ϕ). Further, the map X : V → BC00 defined as

X (ϕ, x) = x[ϕ]σ+T ,

is continuous from (B3).

The translation multioperator PT can be regarded as the composition of Π and X , i.e.

PT = X Π, hence it is quasi Rδ. 2

We will find conditions under which PT will be α-condensing.

First of all, we suppose the following additional hypothesis holds

(A1) the semigroup eAt is uniformly continuous and α-decreasing:

‖eAt‖(α) ≤ C1e−γt

where γ and C1 are positive constants (see Definition 2.1).

By αBC00 we will denote the Kuratowski MNC generated in the space BC00 by the norm

‖ · ‖B.

Let Ω ⊂ BC00 be a bounded set and t ∈ [σ, σ + T ].

We can apply Lemma 3.1 to the set X = Σ(Ω) and, bearing in mind that Σ(Ω)t = Pt(Ω)

and Σ(Ω)σ = Ω, we obtain the following estimate:

αBC00(Pt(Ω)) ≤ K(t− σ)αC(Σ(Ω)[σ, t]) +M(t− σ)αBC00(Ω) . (16)

At first, let us estimate αC(Σ(Ω)[σ, t]).

To this aim we need the following statement which may be verified using hypothesis (A1).

Lemma 4.4. For any bounded Ω ⊂ BC00 the set Σ(Ω)[σ, t] is bounded equicontinuous.

C.GORI ET AL208

¿From the Lemma above and Lemma 3.2, we get

αC(Σ(Ω)[σ, t]) = supσ≤θ≤t

αE(Σ(Ω)(θ)) .

Moreover, from the definition of the integral multioperator, for θ ∈ [σ, t] we have that

Σ(Ω)(θ) ⊂ eA(θ−σ)Ω(0) +∫ θ

σeA(θ−s)F (s, Ps(Ω)) ds . (17)

The MNC of the first term of the previous sum may be estimated, by using (2) and (A1),

in the following way:

αE(eA(θ−σ)Ω(0)) ≤ ‖eA(θ−σ)‖(α)αE(Ω(0)) ≤

≤ C1e−γ(θ−σ)αE(Ω(0)) .

Moreover, from (B4) we obtain

αE(Ω(0)) ≤ lαBC00(Ω).

Therefore,

αE(eA(θ−σ)Ω(0)) ≤ C1e−γ(θ−σ)lαBC00(Ω) . (18)

On the other hand, by applying (2), (A1) and (F4), for the integrand we have the following

estimate:

αE(eA(θ−s)F (s, Ps(Ω))) ≤ ‖eA(θ−s)‖(α)αE(F (s, Ps(Ω))) ≤

(19)

≤ C1e−γ(θ−s)k(s)αBC00(Ps(Ω)).

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 209

Taking into account the fact that the space BC00 is separable, hypothesis (B2), and following

the lines of Theorem 4.2.4 in [12] we may demonstrate that the function s 7→ αBC00(Ps(Ω))

is summable. Therefore, by means of (17), (18), Lemma 2.1 and (19), we obtain

αE(Σ(Ω)(θ)) ≤ C1e−γ(θ−σ)lαBC00(Ω) + C1

∫ θ

σe−γ(θ−s)k(s)αBC00(Ps(Ω)) ds .

Hence, taking the supremum for θ ∈ [σ, t],

αC(Σ(Ω)[σ, t]) ≤ C1lαBC00(Ω) + C1e−γσ

∫ t

σeγsk(s)αBC00(Ps(Ω)) ds .

At last, from (16) we have

αBC00(Pt(Ω)) ≤ K(t− σ)C1lαBC00(Ω) +K(t− σ)C1e−γσ

∫ t

σeγsk(s)αBC00(Ps(Ω)) ds+

+M(t− σ)αBC00(Ω) =

= [K(t− σ)C1l +M(t− σ)]αBC00(Ω) +

+K(t− σ)C1e−γσ

∫ t

σeγsk(s)αBC00(Ps(Ω)) ds .

So, denoting u(t) = αBC00(Pt(Ω)) , we have the following integral inequality:

u(t) ≤ R(t)u(σ) +Q(t)∫ t

σeγsk(s)u(s) ds

where

R(t) = C1lK(t− σ) +M∗, with M∗ as in (11);

Q(t) = C1e−γσK(t− σ) .

C.GORI ET AL210

Finally, by applying Theorem 17.1 of [4], we get the following Gronwall-type inequality

for u(t):

u(t) ≤ R(t)u(σ) +Q(t)u(σ)∫ t

σR(s)eγsk(s)e

∫ t

sQ(τ)eγτ k(τ) dτ ds =

=[R(t) +Q(t)

∫ t

σR(s)eγsk(s)e

∫ t

sQ(τ)eγτ k(τ) dτ ds

]u(σ).

Therefore, we obtain the following condition for the condensivity of PT :

(H2) L = R(T ) +Q(T )∫ Tσ R(s)eγsk(s)e

∫ T

sQ(τ)eγτ k(τ) dτ ds < 1 .

Remark 4.1. In the particular case when F (t, ψ) is completely u.s.c. in the second

argument for a.e. t, i.e. k(t) ≡ 0, then condition (H2) reduces to

R(T ) < 1 .

Now we may summarize our reasonings in the form of the following statement.

Theorem 4.1. Let us suppose that assumptions (A1), (B1)-(B4), (D1)-(D2), (F1)-(F2),

(F3′), (F4), (H1)-(H2) hold. Then the translation multioperator Pσ+T : BC00 → P (BC00) is

α-condensing on bounded sets of BC00.

Since the translation multioperator PT is quasi Rδ (Lemma 4.3) the corresponding topo-

logical degree theory (see [14], [12]) may be applied to obtain fixed points of PT which

obviously are initial values of T -periodic mild solutions of the inclusion (3). In fact, we have

the following general principle.

Theorem 4.2. Suppose that conditions of Theorem 4.1 hold. Let U ⊂ BC00 be an open

bounded set such that FixPT ∩ ∂U = ∅ and deg(i − PT , U) 6= 0 . Then the differential

inclusion (3) has a T -periodic mild solution.

...SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS... 211

Corollary 4.1 Suppose that conditions of Theorem 4.1 hold. Let N ⊂ BC00 be a convex

bounded set such that PT (N) ⊆ N . Then the differential inclusion (3) has a T -periodic

mild solution.

Notice that sufficient conditions for the existence of such invariant set N may be found,

e.g. in [9].

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Math. 35, 407-415 (1987).

[20] J.S.Shin, An existence theorem for functional differential equations with infinite delay in a

Banach space, Funkcial. Ekvac. 30, 19-29 (1987).

C.GORI ET AL214

COMPLETELY GENERALIZED NONLINEAR VARIATIONALINCLUSIONS WITH FUZZY SETVALUED MAPPINGS

Ravi. P. Agarwal1, M. Firdosh Khan2, Donal O’Regan3 and Salahuddin2

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida32901-6975, USA2Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India3Department of Mathematics, National University of Ireland, Galway, Ireland.

Abstract : In this paper, we study a class of completely generalized nonlinear variationalinclusions for fuzzy setvalued mappings and construct some general iterative algorithms.We prove the existence of solutions for completely generalized nonlinear variational inclu-sions for fuzzy setvalued mappings and the convergence of iterative sequences generatedby algorithms.

Key words : Variational inclusions, fuzzy mappings, proximal operators, existence andconvergence theorems, Hilbert spaces and Hausdorff metric.AMS Subject Classification : 49J40, 47H10

1. INTRODUCTION

In recent years, fuzzy set theory introduced by Zadeh [14] in 1965 has emerged asan interesting and fascinating branch of pure and applied sciences. The applications offuzzy set theory can be found in many branches of regional, physical, mathematical andengineering sciences including artificial intelligence, computer sciences, control engineer-ing, management science, economics, transportation problems and operation research, see[5,11,14,15] and references therein. The concept of variational inequalities and comple-mentarity problem for fuzzy mappings were introduced and studied by many authors (see[2,3,6,8,9,10,13,15]). Motivated and inspired by recent research in this field, in this paperwe consider a class of completely generalized nonlinear variational inclusions for fuzzyset-valued mappings and its equivalence with a class of proximal operator equations forfuzzy mappings will be shown. Using this equivalence, the iterative algorithms for theclass of inclusions will be developed and we will prove that the approximate solutionsobtained by this iterative algorithms, converge to the exact solutions of the variationalinclusion.

2. PRELIMINARIES

Let H be a real Hilbert space with inner product and norm define by 〈u, u〉 = ‖u‖2,we denote the collection of all fuzzy sets on H by F(H) = µ : H → [0, 1]. A mapping

215JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,215-228,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

F : H → F(H) is said to be a fuzzy mapping. For each x ∈ H, F (x) (denote it by Fx, inthe sequel) is a fuzzy set on H and Fx(y) is the membership function of y in Fx.

A fuzzy mapping F : H → F(H) is said to be closed, if for any x ∈ H, the functionFx(y) is upper semicontinuous with respect to y, (i.e., for any given point y0 ∈ H and anynet yα ⊂ H, when yα → y0, we have Fx(y0) ≥ lim sup

αFx(yα)).

Let E ∈ F(H), q ∈ [0, 1]. Then the set

(E)q = x ∈ H : E(x) ≥ q

is called a q-cut set of E.

Let M, S, T : H → F(H) be closed fuzzy mappings satisfying the following condition:

Condition (I) : There exist functions a, b, c : H → [0, 1] such that for all x ∈ H, we have(Mx)a(x), (Sx)b(x), (Tx)c(x) ∈ CB(H), where CB(H) denote the family of all nonemptyclosed and bounded subsets of H. Therefore, we can define three multivalued mappings,M, S, T : H → CB(H) by

M(x) = (Mx)a(x), S(x) = (Sx)b(x), T (x) = (Tx)c(x),

for each x ∈ H. In the sequel M, S, T are called the multivalued mappings induced by thefuzzy mappings M, S and T , respectively. More precisely, let ∂φ denote the subdifferentialof a proper convex and lower semicontinuous function φ : H × H → R ∪ +∞. Leta, b, c : H → [0, 1] be given functions and M, S, T : H → F(H) fuzzy mappings. Letg, F, G, P : H → H be single-valued mappings with Img∩ dom∂φ(., v) 6= ∅, and weconsider the following completely generalized nonlinear variational inclusion with fuzzysetvalued mappings for finding u, x, y, z ∈ H such that g(u)∩dom∂φ 6= ∅ and Mu(x) ≥a(u), Su(y) ≥ b(u), Tu(z) ≥ c(u),

〈P (x) − (Fy − Gz), v − g(u)〉 ≥ φ(g(u), u)− φ(v, u), ∀ v ∈ H. (2.1)

Inequality (2.1) is called the completely generalized nonlinear variational inclusion forfuzzy setvalued mappings.

It is clear that completely generalized nonlinear variational inclusions for fuzzy set-valued mappings (2.1) includes many kinds of variational inclusions and inequalities asspecial cases, such as those in [1,2,3,7,8,9,10,13].

Definition 2.1[5]. If G : H → 2H (2H denotes the power set of H) is a maximalmonotone multivalued mappings, then for any η > 0, the mapping JG

η : H → H definedby

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R.P.AGARWAL ET AL216

JGη (u) = (I + ηG)−1(u), ∀ u ∈ H,

is said to be the proximal operator of index η of G, where I is the identity mapping onH. Furthermore, the proximal operator JG

η is single-valued and nonexpansive i.e.,

‖JGη (u) − JG

η (v)‖ ≤ ‖u − v‖, ∀ u, v ∈ H.

Since the subdifferential ∂φ of a proper convex and lower semicontinuous function φ : H →R ∪ +∞ is a maximal monotone multivalued mappings, it follows that the proximaloperator J∂φ

η of index η of ∂φ is given by

J∂φη (u) = (I + η∂φ)−1(u), ∀ u ∈ H.

Lemma 2.1[5]. A given u, w ∈ H satisfies the inequality

〈u − w, v − u〉 + ηφ(v) − ηφ(u) ≥ 0, ∀ v ∈ H

if and only ifu = J∂φ

η (w),

where J∂φη = (I + η∂φ)−1 is the proximal operator and η > 0 is a constant.

We now consider the problem of finding u, w, x, y, z ∈ H, Mu(x) ≥ a(u),Su(y) ≥ b(u),Tu(z) ≥ c(u) and

P (x) + η−1R∂φ(.,u)η (w) = Fy − Gz, (2.2)

where R∂φη = I − J∂φ

η , I is the identity mapping and η > 0. Equation (2.2) is called theproximal operator equation for fuzzy setvalued mappings.

3. EQUIVALENCE RELATION AND ITERATIVE ALGORITHMS

In this section, we establish the equivalence between (2.1) and (2.2), and we developthe iterative algorithms.

Lemma 3.1. The set (u, x, y, z) is the solution for (2.1) if and only if there exist x ∈M(u), y ∈ S(u), z ∈ T (u) such that

g(u) = J∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))], (3.1)

where η > 0 is a constant and J∂φ(.,u)η = (I + η∂φ)−1(u) is the so-called proximal mapping

on H.

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...GENERALIZED NONLINEAR VARIATIONAL.. 217

Let u ∈ H, x ∈ M(u), y ∈ S(u) z ∈ T (u) and from the definition of the proximaloperator J∂φ(.,u)

η of index η of ∂φ(., u) and relation (3.1), we have

g(u) = J∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))]

= (I + η∂φ(., u))−1[g(u) − η(P (x) − (Fy − Gz))]

and

g(u) − η(P (x) − (Fy − Gz)) ∈ g(u) + η∂φ(., u)(g(u)),

which gives

(Fy − Gz) − P (x) ∈ ∂φ(., u)(g(u)).

From the definition of ∂φ(., u), we have

φ(v, u) ≥ φ(g(u), u) + 〈(Fy − Gz) − P (x), v − g(u)〉, ∀ v ∈ H.

Thus u, x, y and z are solutions of problem (2.1).

From Lemma 3.1, we conclude that (2.1) is equivalent to the fuzzy fixed point problem

u ∈ N(u) (3.2)

where

N(u) = u − g(u) + J∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))].

Based on (3.1) and (3.2), we have the following iterative algorithm.

Algorithm 3.1. Suppose P, F, G, g : H → H are single-valued mappings. Let M, S, T :H → F(H) be fuzzy mappings satisfying condition (I) and M, S, T : H → CB(H) thefuzzy mappings induced by M, S, T respectively. For a given u0 ∈ H, we take x0 ∈ M(u0),y0 ∈ S(u0) and z0 ∈ T (u0) and let

u1 = u0 − g(u0) + J∂φ(.,u0)η [g(u0) − η(P (x0) − (Fy0 − Gz0))] .

where η > 0 is a constant. Since x0 ∈ M(u0) ∈ CB(H), y0 ∈ S(u0) ∈ CB(H) andz0 ∈ T (u0) ∈ CB(H), by Nadler [12 pp.480], there exist x1 ∈ M(u1), y1 ∈ S(u1) andz1 ∈ T (u1) such that

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R.P.AGARWAL ET AL218

‖x0 − x1‖ ≤ (1 + 1)H(M(u0), M(u1)),

‖y0 − y1‖ ≤ (1 + 1)H(S(u0), S(u1)),

‖z0 − z1‖ ≤ (1 + 1)H(T (u0), T (u1)),

where H(., .) is the Hausdorff metric on CB(H). Let

u2 = u1 − g(u1) + J∂φ(.,u1)η [g(u1) − η(P (x1) − (Fy1 − Gz1))].

By induction we can obtain sequences un, xn, yn and zn such that

xn ∈ M(un), ‖xn − xn+1‖ ≤ (1 + (n + 1)−1)H(M(un), M(un+1)),

yn ∈ S(un), ‖yn − yn+1‖ ≤ (1 + (n + 1)−1)H(S(un), S(un+1)),

zn ∈ T (un), ‖zn − zn+1‖ ≤ (1 + (n + 1)−1)H(T (un), T (un+1)),

un+1 = un − g(un) + J∂φ(.,un)η [g(un) − η(P (xn) − (Fyn − Gzn))], (3.3)

where η > 0 is a constant.

Now, we show that (2.1) is equivalent to (2.2).

Theorem 3.1. The problem (2.1) has a solution u ∈ H such that x ∈ M(u), y ∈S(u), z ∈ T (u) if and only if (2.2) has a solution set (u, x, y, z), where

g(u) = J∂φ(.,u)η (w) (3.4)

andw = g(u)− η(P (x) − (Fy − Gz)), (3.5)

here η > 0 is a constant.

Proof. Let u ∈ H, x ∈ M(u), y ∈ S(u), z ∈ T (u) be a solution of (2.1), then by Lemma3.1,

g(u) = J∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))]. (3.6)

Using R∂φ(.,u)η = I − J∂φ(.,u)

η and (3.6), we have

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...GENERALIZED NONLINEAR VARIATIONAL.. 219

R∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))]

= [g(u) − η(P (x) − (Fy − Gz))] − J∂φ(.,u)η [g(u) − η(P (x) − (Fy − Gz))]

= [g(u) − η(P (x) − (Fy − Gz))] − g(u) = −η[P (x) − (Fy − Gz)],

which implies that

P (x) + η−1R∂φ(.,u)η (w) = Fy − Gz,

where w = g(u) − η(P (x) − (Fy − Gz)), here η > 0 is a constant.

Conversely, let u ∈ H, x ∈ M(u), y ∈ S(u), z ∈ T (u) be a solution of (2.2), then

−η(Fy − Gz) + ηP (x) = −R∂φ(.,u)η (w)

= J∂φ(.,u)η (w) − w,

g(u) = J∂φ(.,u)η (w). (3.7)

Now from Lemma 2.1 and equation (3.7), we have

0 ≤ 〈g(u)− w, v − g(u)〉 + ηφ(v, u)− ηφ(g(u), u)

= η〈P (x) − (Fy − Gz), v − g(u)〉+ φ(v, u) − φ(g(u), u).

Note that η > 0 is a constant, so the above relation implies u ∈ H, x ∈ M(u),y ∈ S(u), z ∈ T (u) a solution set of (2.1).

The (2.2) can be written as

R∂φ(.,u)η (w) = η(Fy − Gz) − ηP (x),

from which it follows that

w = g(u)− η(P (x) − (Fy − Gz)), (3.8)

completing the proof of the Theorem 3.1.

This fixed point formulation for fuzzy mappings enables us to suggest the followingAlgorithm.

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Algorithm 3.2. Let g, F, G, P, S, T and M be as in Algorithm 3.1. For any givenu0 ∈ H, compute the sequences un, wn, xn, yn and zn for fuzzy mappings bythe iteration method

g(un) = J∂φ(.,un)η (wn),

xn ∈ M(un), ‖xn − xn+1‖ ≤ (1 + (n + 1)−1)H(M(un), M(un+1)),

yn ∈ S(un), ‖yn − yn+1‖ ≤ (1 + (n + 1)−1)H(S(un), S(un+1)),

zn ∈ T (un), ‖zn − zn+1‖ ≤ (1 + (n + 1)−1)H(T (un), T (un+1)), n ≥ 0,

(3.9)

wn+1 = g(un) − η(P (xn) − (Fyn − Gzn)), (3.10)

here η > 0 is a constant.

(ii) (2.2) may be written as

0 = − η−1R∂φ(.,un)η (wn) − (P (xn) − (Fyn − Gzn)),

which implies that

R∂φ(.,un)η (wn) = (1 − η−1)R∂φ(.,un)

η (wn) − (P (xn) − (Fyn − Gzn)),

that is

wn = J∂φ(.,un)η (wn) − (P (xn) − (Fyn − Gzn)) + (1 − η−1)R∂φ(.,un)

η (wn)

= g(un) − (P (xn) − F (yn − Gzn)) + (1 − η−1)R∂φ(.,un)η (wn). (3.11)

Using the fixed point formulation for fuzzy mappings, the following iterative algo-rithm is proposed.

Algorithm 3.3. Let g, F, G, P, M, S and T be as in Algorithm 3.1. For any givenu0 ∈ H, compute the sequences un, wn, xn, yn and zn for fuzzy mappings bythe iterative schemes

g(un) = J∂φ(.,un)η (wn), (3.12)

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...GENERALIZED NONLINEAR VARIATIONAL.. 221

xn ∈ M(un), ‖xn − xn+1‖ ≤ (1 + (n + 1)−1)H(M(un), M(un+1)),

yn ∈ S(un), ‖yn − yn+1‖ ≤ (1 + (n + 1)−1)H(S(un), S(un+1)),

zn ∈ T (un), ‖zn − zn+1‖ ≤ (1 + (n + 1)−1)H(T (un), T (un+1)),

wn+1 = g(un) − (P (xn) − (Fyn − Gzn)) + (1 − η−1)R∂φ(.,un)η (wn), (3.13)

here η > 0 is a constant.

Now, we consider for each fixed u ∈ H, the sequences φn of proper convex lowersemicontinuous φn : H × H → R ∪ +∞, approximating φ, then we have followingmore general iterative algorithms for fuzzy set-valued mappings.

Algorithm 3.4. For any given u0 ∈ H, x0 ∈ M(u0), y0 ∈ S(u0), z0 ∈ T (u0), computethe sequences un, wn, xn, yn and zn for fuzzy mappings by iterative schemes

g(un) = J∂φn(.,un)η (wn), (3.14)

xn ∈ M(un), ‖xn − xn+1‖ ≤ (1 + (n + 1)−1)H(M(un), M(un+1)),

yn ∈ S(un), ‖yn − yn+1‖ ≤ (1 + (n + 1)−1)H(S(un), S(un+1)),

zn ∈ T (un), ‖zn − zn+1‖ ≤ (1 + (n + 1)−1)H(T (un), T (un+1)),

wn+1 = g(un) − η(P (xn) − (Fyn − Gzn)), η ≥ 0, (3.15)

here η > 0 is a constant.

Algorithm 3.5. For any given u0 ∈ H, compute the sequences un, wn, xn, ynand zn for fuzzy mappings by iterative schemes

g(un) = J∂φn(.,un)η (wn), (3.16)

xn ∈ M(un), ‖xn − xn+1‖ ≤ (1 + (n + 1)−1)H(M(un), M(un+1)),

yn ∈ S(un), ‖yn − yn+1‖ ≤ (1 + (n + 1)−1)H(S(un), S(un+1)),

zn ∈ T (un), ‖zn − zn+1‖ ≤ (1 + (n + 1)−1)H(T (un), T (un+1)),

wn+1 = g(un) − (P (xn) − (Fyn − Gzn)) + (I − η−1)R∂φn(.,un)η (wn), (3.17)

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here η > 0 is a constant and

R∂φn(.,un)η (wn) = I − J∂φn(.,un)

η (wn), for each u ∈ H.

4. MAIN RESULTS

In this section, we consider those conditions under which the solutions of (2.1) existsand for which the sequences of approximate solutions obtained by our iterative algorithmsconverge to the exact solution of (2.2). For this purpose, we need the following concepts.

Definition 4.1. A mapping g : H → H is said to be

(i) strongly monotone if there exists a constant r > 0 such that

〈g(u1) − g(u2), u1 − u2〉 ≥ r‖u1 − u2‖2, ∀ u1, u2 ∈ H,

(ii) Lipschitz continuous if there exists a constant s > 0 such that

‖g(u1) − g(u2)‖ ≤ s‖u1 − u2‖, ∀ u1, u2 ∈ H.

Definition 4.2. Let S : H → CB(H) be a mapping

(i) S is said to be relaxed Lipschitz continuous with respect to a mapping F : H → H ifthere exists a constant α > 0 such that

〈Fy1 − Fy2, u1 − u2〉 ≤ − α‖u1 − u2‖2, ∀ ui ∈ H, yi ∈ S(ui), i = 1, 2;

(ii) S is said to be relaxed monotone with respect to a mapping G : H → H if thereexists a constant β > 0 such that

〈Gy1 − Gy2, u1 − u2〉 ≥ − β‖u1 − u2‖2, ∀ yi ∈ S(ui), ui ∈ H, i = 1, 2.

Definition 4.3[11]. A sequence φn of convex, proper, lower semicontinuous functionφn : H → R ∪ +∞ is said to be convergent to φ in the sense of Mosco if

(i) for every u ∈ H, we have

φ(u) ≤ limn→∞

inf φn(un),

for every sequence un in H which converges weakly to u, and

(ii) there exists a sequence un in H which converges strongly to u and satisfies

φ(u) ≥ limn→∞

sup φn(un).

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Lemma 4.1[4]. If the sequence φn of convex, proper, lower semicontinuous functionφn : H → R ∪ +∞ converges to φ in the sense of Mosco, then

Jφnη (u) → Jφ

η (u) as n → ∞, for any u ∈ H,

where η > 0 is a constant.

Assumption 4.1. For each fixed v ∈ H, let φ(., v) be a proper, convex and lowersemicontinuous function φ : H ×H → R∪+∞. Then for each u, v, w ∈ H, there existsa coefficient µ > 0 such that

‖J∂φ(.,u)η (w) − J∂φ(.,v)

η (w)‖ ≤ µ‖u − v‖,

where η > 0 is a constant.

Theorem 4.1. Let M, S, T : H → F(H) be the closed fuzzy mappings satisfying con-dition (I) and M, S, T : H → CB(H) the multivalued mappings induced by the fuzzymappings M, S, T, respectively. Let M, S, T be δ-Lipschitz continuous, γ-Lipschitz contin-uous, ε-Lipschitz continuous, respectively. Let F, G, P : H → H be Lipschitz continuouswith corresponding coefficients ξ, ρ and σ, respectively. Let g : H → H be Lipschitzcontinuous and strongly monotone with corresponding coefficients s ≥ 0 and r ≥ 0, re-spectively. Let S be a relaxed Lipschitz continuous with respect to F with constant α > 0and T be a relaxed monotone with respect to G with constant β > 0. Let the sequenceφn of proper, convex and lower semicontinuous function φn : H × H → R ∪ +∞converges to φ in the sense of Mosco and assume Assumption 4.1 holds (for φn).

Suppose that there exists a constant η > 0 such that

∣∣∣∣∣η − α − β − σδ(1 − k − µ)

(ξγ + ρε)2 − σ2δ2

∣∣∣∣∣

<

√(α − β − σδ(1 − k − µ))2 − ((ξγ + ρε)2 − σ2δ2)(k + µ)(2 − k − µ)

(ξγ + ρε)2 − σ2δ2,

α > β + σδ(1 − k − µ) +√

((ξγ + ρε)2 − σ2δ2)(k + µ)(2 − k − µ), (4.1)

σδ < ξγ + ρε,

where k = 2√

1 − 2r + s2.

Then there exists a set of elements u ∈ H, x ∈ M(u), y ∈ S(u), z ∈ T (u), whichsatisfies (2.2) and the sequences un, wn, xn, yn and zn generated by Algorithm

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3.4 converge to u, w, x, y and z strongly in H, respectively.

Proof. From Algorithm 3.4, we have

‖wn+1−wn‖ ≤ ‖g(un)−g(un−1)+η(Fyn−Fyn−1)−η(Gzn−Gzn−1)‖+η‖P (xn)−P (xn−1)‖

≤ ‖un − un−1 − (g(un) − g(un−1))‖ + η‖P (xn) − P (xn−1)‖

+‖un − un−1 + η(Fyn − Fyn−1) − η(Gzn − Gzn−1)‖. (4.2)

By Lipschitz continuity and the strong monotonicity of the operator g, we obtain

‖un − un−1 − (g(un) − g(un−1))‖2 ≤ (1 − 2r + s2)‖un − un−1‖2. (4.3)

Again from the H-Lipschitz continuity of the fuzzy operators M, S, T and the Lips-chitz continuity of the operators P, F, G, we have

‖P (xn) − P (xn−1)‖ ≤ σ‖xn − xn−1‖

≤ σ (1 + (n + 1)−1) H(M(un), M(un−1))

≤ σδ(1 + n−1)‖un − un−1‖, (4.4)

‖F (yn) − F (yn−1)‖ ≤ ξ‖yn − yn−1‖

≤ ξ (1 + (n + 1)−1) H(S(un), S(un−1))

≤ γξ(1 + n−1)‖un − un−1‖, (4.5)

‖G(zn) − G(zn−1)‖ ≤ ρ‖zn − zn−1‖

≤ ρ (1 + (n + 1)−1) H(T (un), T (un−1))

≤ ρε(1 + n−1)‖un − un−1‖. (4.6)

Further, since S is relaxed Lipschitz continuous with respect to the operator F andthe operator T is relaxed monotone with respect to the operator G, we have

‖un − un−1 + η(Fyn − Fyn−1) − η(Gzn − Gzn−1)‖2 ≤ ‖un − un−1‖2

+2η〈Fyn − Fyn−1, un − un−1〉 − 2η〈Gzn − Gzn−1, un − un−1〉

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+η2‖(Fyn − Fyn−1) − (Gzn − Gzn−1)‖2

≤ [1 − 2η(α − β) + η2(1 + n−1)2(ξγ + ρε)2]‖un − un−1‖2. (4.7)

From (4.2)-(4.7), we obtain

‖wn+1−wn‖ ≤ √

1 − 2r + s2+ησδ(1+n−1)+√

1 − 2η(α − β) + η2(1 + n−1)2(ξγ + ρε)2

‖un − un−1‖

= k

2+ ησδ(1 + n−1) + Πn(η)‖un − un−1‖, (4.8)

where k = 2√

1 − 2r + s2 and Πn(η) =√

1 − 2η(α − β) + η2(1 + n−1)2(ξγ + ρε)2.

From(3.14) and (4.3), we have

‖un − un−1‖ = ‖un − un−1 − (g(un) − g(un−1)) + J∂φn(.,un)η (wn) − J∂φn−1(.,un−1)

η (wn−1)‖

≤ ‖un − un−1 − (g(un) − g(un−1))‖ + ‖J∂φn(.,un)η (wn) − J∂φn(.,un)

η (wn−1)‖

+‖J∂φn(.,un)η (wn−1) − J∂φn(.,un−1)

η (wn−1)‖ + ‖J∂φn(.,un−1)η (wn−1) − J∂φn−1(.,un−1)

η (wn−1)‖

≤ k

2‖un − un−1‖ + ‖wn − wn−1‖ + µ‖un − un−1‖ + εn,

since J∂φ(.,u)η is nonexpansive and Assumption 4.1 holds (with φn), which implies that

‖un − un−1‖ ≤ 1

1 − k2− µ

[‖wn − wn−1‖ + εn], (4.9)

where

εn = ‖J∂φn(.,un−1)η (wn−1) − J∂φn−1(.,un−1)

η (wn−1)‖.

Combining (4.8)-(4.9), we have

‖wn+1 − wn‖ ≤

k2

+ ησδ(1 + n−1) + Πn(η)

1 − k2− µ

‖wn − wn−1‖

+

k2

+ ησδ(1 + n−1) + Πn(η)

1 − k2− µ

εn

≤ θn‖wn − wn−1‖ + θnεn, (4.10)

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R.P.AGARWAL ET AL226

where θn =k2

+ ησδ(1 + n−1) + Πn(η)

1 − k2− µ

. Let

θ =k2

+ ησδ + Π(η)

1 − k2− µ

with Π(η) =√

1 − 2η(α − β) + η2(ξγ + ρε)2.

Then θn → θ as n → ∞. It follows from (4.1) that θ < 1. Hence θn < 1 for n sufficientlylarge. Since εn → 0 as n → ∞, it follows from (4.10) that wn is a Cauchy sequence inH, that is wn+1 → w ∈ H as n → ∞.

From (4.9), we see that un is also Cauchy sequence in H, so there exists u ∈ H, suchthat un+1 → u as n → ∞.

Also from (4.4)-(4.6), we have

‖xn − xn−1‖ ≤ (1 + n−1)H(M(un), M(un−1)) ≤ (1 + n−1)δ‖un − un−1‖,

‖yn − yn−1‖ ≤ (1 + n−1)H(S(un), S(un−1)) ≤ (1 + n−1)γ‖un − un−1‖,

‖zn − zn−1‖ ≤ (1 + n−1)H(T (un), T (un−1)) ≤ (1 + n−1)ε‖un − un−1‖.

It follows that xn, yn and zn are also Cauchy sequences in H. Since H is complete,we may let xn → x, yn → y, zn → z as n → ∞. Further we have

d(x, M(u)) ≤ ‖x − xn‖ + d(xn, M(u))

≤ ‖x − xn‖ + H(M(un), M(u))

≤ ‖x − xn‖ + δ‖un − u‖ → 0 as n → ∞,

where d(x, M(u)) = inf‖u − p‖ : p ∈ M(u), and so we have d(x, M(u)) = 0. Hence wemust have x ∈ M(u). In a similar way, we can show y ∈ S(u) and z ∈ T (u).

Using the continuity of operators P, F, G, M, S, T, φ, J∂φ and Algorithm 3.4, we have

w = g(u)− η(P (x) − (Fy − Gz))

= J∂φ(.,u)η (w) − η(P (x) − (Fy − Gz)) ∈ H, η > 0 is a constant.

From Theorem 3.1, we see that u, w ∈ H, such that x ∈ M(u), y ∈ S(u) and z ∈ T (u)are solution of (2.2) and consequently, wn+1 → w, un+1 → u, xn+1 → x, yn+1 → y andzn+1 → z strongly in H. This completes the proof.

13

...GENERALIZED NONLINEAR VARIATIONAL.. 227

References

1. R. Ahmad, K. R. Kazmi and Salahuddin, Completely generalized nonlinear varia-tional inclusions involving relaxed Lipschitz and relaxed monotone mappings, Non-linear Anal. Forum, 5(2000), 61-69.

2. R. Ahmad, Salahuddin and S. Husain, Generalized nonlinear variational inclusionswith fuzzy mappings, Indian J. Pure Appl. Math. 32(6)(2001), 943-947.

3. R. Ahmad, Salahuddin and S. S. Irfan, Generalized quasi-complementarity problemswith fuzzy set valued mappings, Appearing in Int. J. Fuzzy Systems, September(2003).

4. H. Attouch, Variational convergence for function and operators, Pitman, London,1984.

5. H. Breziz, Operateurs Maximaux Monotones, North Holland, Amsterdam, 1973.

6. S. S. Chang, Coincidence theorems and variational inequalities for fuzzy mappings,Fuzzy Sets and Systems, 61(1994), 359-368.

7. S. S. Chang and N. J. Huang, Generalized complementarity problems for fuzzy map-pings, Fuzzy Sets and Systems, 55(1993), 227-234.

8. S. S. Chang and Y. G. Zhu, On variational inequalities for fuzzy mappings, FuzzySets and Systems, 32(1989), 359-367.

9. X. P. Ding and J. Y. Park, A new class of generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings, J. Comput. Appl. Math., 138(2002),243-257.

10. N. J. Huang, A general class of nonlinear variational inclusions for fuzzy mappings,Indian J. Pure Appl. Math., 29(9)(1998), 957-964.

11. U. Mosco, Implicit variational problems and quasi-variational inequalities, LectureNotes in Maths, Springer-Verlag, Berlin, 543(1976), 83-126.

12. Jr. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30(1969),475-488.

13. M. A. Noor, Variational inequalities for fuzzy mappings, Fuzzy Sets and System,56(1993), 309-312.

14. L. A. Zadeh, Fuzzy sets, Information and control, 8(1965), 338-353.

15. H. J. Zimmerman, Fuzzy sets theory and its applications, Kluwer Academic Pub-lishers, Boston, 1988.

14

R.P.AGARWAL ET AL228

Some Direct Results For The Iterative CombinationsOf The Second Kind Beta Operators

Zoltan FintaBabes - Bolyai University, Department of Mathematics and Computer Science

1, M. Kogalniceanu St., 400084 Cluj-Napoca, Romaniae-mail : fzoltan@math.ubbcluj.ro

Vijay GuptaSchool of Applied Sciences, Netaji Subhas Institute of TechnologySector 3 Dwarka, Azad Hind Fauj Marg, New Delhi 110045, India

e-mail : vijay@nsit.ac.in

Abstract : In the present paper, we study the second kind beta operators introducedby D. D. Stancu [8] and prove some local and global direct results for the iterativecombinations of these Stancu beta operators.

2000 AMS Subject Classification : 41A30 41A36

Key Words and Phrases : linear positive operators, iterative combinations, orderof approximation, modulus of smoothness of order m, K - functional, Ditzian - Totikmodulus of second order.

1 Introduction

The second kind beta operators Ln associating for n ∈ N = 1, 2, . . ., are definedby

Ln(f, x) ≡ (Lnf)(x) =1

B(nx, n + 1)

∫ ∞

0f(t)

tnx−1

(1 + t)nx+n+1dt (1)

The operators ( 1 ) were introduced by D. D. Stancu [8]. Recently U. Abel [1]estimated the complete asymptotic expansion for these operators. Another betaapproximating operators of second kind have been studied in [2], [3] and [4].

Alternatively we may rewrite ( 1 ) as

Ln(f, x) =∫ ∞

0Kn(x, t) f(t) dt, (2)

1

229JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,229-240,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

where

Kn(x, t) =1

B(nx, n + 1)· tnx−1

(1 + t)nx+n+1

It is easily verified the operators Ln are linear positive operators and preserve thelinear functions : Ln(1, x) = 1 and Ln(t, x) = x. Therefore we can take theiterative combinations of these operators easily.

It has been observed that the order of approximation by these operatorsLn is O(n−1), n → ∞. On the other hand C. A. Micchelli [7] offered an approachfor improving the order of approximation of Bernstein polynomials. It is interestingthat by considering the iterative combinations due to C. A. Micchelli [7], the higherorder of approximation may be achieved.

We consider here the iterative combinations of the Stancu beta operators.The iterative combinations Ln,k(f, x) of the operators Ln are defined as follows:

Ln,k(f, x) ≡ (Ln,kf)(x) = [I−(I−Ln)k](f, x) =k∑

r=1

(−1)r+1

(kr

)Lr

n(f, x) (3)

where Lrn denotes the r−th iterate ( superposition ) of the operator Ln.

In the present paper we prove an asymptotic formula and an error estimatein terms of higher order integral modulus of smoothness, for the iterative combina-tions of these Stancu beta operators of second kind. Further, in the last section wegive global direct results in terms of Ditzian - Totik modulus of smoothness.

2 Lemmas

In this section we present certain lemmas which are necessary to prove the mainresults of next two section.

Lemma 1 Let the function µn,m(x), m ∈ N ∪ 0, be defined as

µn,m(x) = Ln((t− x)m, x).

Then

µn,0(x) = 1, µn,1(x) = 0, µn,2(x) =x(x + 1)

n− 1

2

Z.FINTA,V.GUPTA230

andµn,m(x) = O

(n−[(m+1)/2]

), n →∞

where 0 ≤ x < ∞ and m ∈ N ∪ 0.

Proof. See [8, p. 234, Theorem 1 ].

For each m ∈ N ∪ 0 the m−th order moment µqn,m(x) for the operatorsLq

n is defined byµqn,m(x) = Lq

n((t− x)m, x) (4)

For q = 1, µ1n,m(x) ≡ µn,m(x).

Lemma 2 Let γ and δ be two positive numbers. Then for any m ∈ N there existsa constant C1 = C1(m) > 0 such that∥∥∥∥∥

∫|t−x|≥δ

Kn(x, t) tγ dt

∥∥∥∥∥C[a,b]

≤ C1 n−m

Proof. It follows easily by using Lemma 1.

Lemma 3 There holds the following relation

µq+1n,m (x) =

m∑j=0

(mj

) m−j∑i=0

1

i!Di(µqn,m−j(x)

)µn,i+j(x), D ≡ d

dx.

Proof. By ( 4 ), we have

µq+1n,m (x) = Ln (Lq

n((t− x)m, u), x)

=m∑

j=0

(mj

)Ln

((u− x)j Lq

n((t− u)m−j, u), x)

=m∑

j=0

(mj

)Ln

m−j∑i=0

(u− x)i+j

i!Di(µ

qn,m−j(x)), x

.

Now making use of Lemma 1, the conclusion of this lemma follows immediately.

3

...SECOND KIND BETA OPERATORS 231

Lemma 4 We have

µqn,m(x) = O(n−[(m+1)/2]

), n →∞ (5)

where 0 ≤ x < ∞ and m ∈ N ∪ 0.

Proof. For q = 1 the above result follows from Lemma 1. We shall prove the result( 5 ) by the principle of mathematical induction. Let the result be true for q and

we shall prove it for q + 1. Because µqn,m−j(x) = O

(n−[(m−j+1)/2]

)and µ

qn,m−j(x)

is a polynomial in x of degree ≤ m − j, it is obviously seen that Di(µqn,m−j(x)

)=

O(n−[(m−j+1)/2]

). Applying Lemma 3, we get

µq+1n,m (x) =

m∑j=0

(mj

) m−j∑i=0

1

i!O(n−[(m−j+1)/2]+[(i+j+1)/2]

)

= O

m∑j=0

m−j∑i=0

n−[(m+i+1)/2]

,

which implies equation ( 5 ). This completes the proof of lemma.

By direct application of Lemma 1, Lemma 3 and Lemma 4, we have

Ln,k((t− x)i, x) = O(n−k), n →∞,

where Ln,k is the iterative combination defined by ( 3 ).

3 Local Approximation

In this section we present a Voronovskaja type asymptotic formula and an errorestimate in terms of higher order modulus of smoothness, using the technique oflinear approximating method, namely Steklov mean’s.

Throughout this section let

Mγ[0,∞) = f : f be locally integrable on (0,∞) and

f(t) = O(tγ), t →∞ for some γ > 0 .

4

Z.FINTA,V.GUPTA232

For f ∈ Mγ[0,∞) we define the norm ‖ · ‖γ on Mγ[0,∞) by

‖f‖γ = sup0<x<∞

|f(x)| x−γ.

Theorem 1 Let f ∈ Mγ[0,∞). If the 2k−th derivative of f exists at a fixed pointx ∈ [0,∞) then

limn→∞

nk[Ln,k(f, x)− f(x)] =2k∑

j=2

Q(j, k, x) · f (j)(x)

j!(6)

where Q(j, k, x) are certain polynomials in x. Further, if f (2k−1) exists and is ab-solutely continuous over [0, b] and f (2k) ∈ L∞[0, b] then, for any [c, d] ⊂ (0, b), thereholds

‖Ln,kf − f‖C[c,d] ≤ C2 n−k‖f‖γ + ‖f (2k)‖L∞[0,b]

, (7)

where C2 is a constant independent of f and n.

Proof. First by Taylor’s expansion, we have

nk[Ln,k(f, x)− f(x)] =

= nk2k∑

j=1

f (j)(x)

j!Ln,k((t− x)j, x) + nk

k∑r=1

(−1)k

(kr

)Lr

n(ε(t, x)(t− x)2k, x)

= E1 + E2,

where ε(t, x) → 0 as t → x and |ε(t, x)| ≤ Mtγ, M > 0. Using ( 5 ), we have

E1 = nk2k∑

j=2

f (j)(x)

j!Ln,k((t− x)j, x) =

2k∑j=2

f (j)(x)

j!Q(j, k, x) + o(1).

If φδ(t) denotes the characteristic function of the interval (x− δ, x + δ) then

|E2| ≤ nkk∑

r=1

(kr

)Lr

n

(|ε(t, x)|(t− x)2kφδ(t), x

)+

+ nkk∑

r=1

(kr

)Lr

n

((t− x)2k(1− φδ(t)), x

)= E3 + E4.

5

...SECOND KIND BETA OPERATORS 233

Now we estimate E3. Since ε(t, x) → 0 as t → x therefore for given ε > 0 thereexists a δ > 0 such that |ε(t, x)| < ε whenever |t− x| < δ. Using Lemma 4, we get

E3 ≤(

sup|t−x|<δ

|ε(t, x)|)

nkk∑

r=1

(kr

)Lr

n((t− x)2s, x) < ε · C3

Next applying Lemma 2, for arbitrary p > 0, we obtain

E4 ≤ nkk∑

r=1

(kr

)Lr

n(Mtγ(t− x)2k(1− φδ(t)), x) < C4 n−p = o(1)

Due to the arbitrariness of ε > 0 we conclude that E2 → 0 as n →∞.

Combining the estimates of E1, E2, we get the required estimate ( 6 ).

To prove ( 7 ), we may write

Ln,k(f, x)− f(x) = Ln,k(f(t), x)− f(x)

= Ln,k(φ(t)(f(t)− f(x)), x) + Ln,k((1− φ(t))(f(t)− f(x)), x)

= E5 + E6,

where φ(t) denotes the characteristic function of the closed interval [0, b]. Proceedingalong the lines of the proof of E4, for all x ∈ [c, d], we have

E6 ≤ C5 n−k ‖f‖γ.

Again, for t ∈ [0, b] and x ∈ [c, d], we have, by given hypothesis

f(t)− f(x) =2k−1∑i=1

f (i)(x)

i!(t− x)i +

1

(2k − 1)!

∫ t

x(t− w)2k−1f (2k)(w) dw

Thus

E5 =2k−1∑i=1

f (i)(x)

i!Ln,k(φ(t)(t− x)i, x) +

+1

(2k − 1)!Ln,k

(φ(t)

∫ t

x(t− w)2k−1f (2k)(w) dw, x

)

=2k−1∑i=1

f (i)(x)

i!Ln,k((t− x)i, x) + Ln,k((φ(t)− 1)(t− x)i, x) +

+1

(2k − 1)!Ln,k

(φ(t)

∫ t

x(t− w)2k−1f (2k)(w) dw, x

)

=2k−1∑i=1

f (i)(x)

i!E7 + E8+ E9.

6

Z.FINTA,V.GUPTA234

By ( 6 ), we get E7 = O(n−k), uniformly for x ∈ [c, d]. Also as in the estimate ofE4, we have E8 = O(n−k). Finally, by ( 6 ) and Lemma 4, we get

E9 = ‖f (2k)‖L∞[0,b] ·O(n−k)

Combining the estimates E7, E8, E9, we obtain

‖E5‖C[c,d] ≤ C6

2k−1∑i=1

‖f (i)‖C[c,d] + ‖f (2k)‖L∞[0,b]

.

Finally, using the interpolation property due to S. Goldberg and V. Meir [6], weobtain the required result ( 7 ), which completes the proof of the theorem.

We now define the linear approximating function viz. Steklov’s mean, whichis the main tool to prove our next theorem.

Let f ∈ Mγ[0,∞), m ∈ N. Then the Steklov’s mean fη,m of m−th ordercorresponding to f, for sufficiently small η > 0 is defined by

fη,m = η−m∫ η/2

−η/2

∫ η/2

−η/2. . .∫ η/2

−η/2

f(t) + (−1)m−1∆m

u f(t)

dt1dt2 . . . dtm,

where u =∑m

i=1 ti, t ∈ [a, b] and ∆mη f(t) is the m−th forward difference with step

length η. We assume throughout this section that 0 < a1 < a2 < b2 < b1 < ∞. It iseasily verified ( see e.g. [9] ) that :

1. fη,m has continuous derivatives up to order m on [a1, b1]

2. ‖f (r)η,m‖C[a1,b1] ≤ C7 η−r ωr(f, η, a1, b1), r = 1, 2, . . . ,m

3. ‖f − fη,m‖C[a2,b2] ≤ C8 ωm(f, η, a1, b1)

4. ‖fη,m‖C[a2,b2] ≤ C9 ‖f‖γ

5. ‖f (m)η,m‖C[a2,b2] ≤ C10 ‖f‖γ,

where Ci’ s , i ∈ 7, 8, . . . , 10 are certain constants independent of f and η andωm(f, η, a, b) is the modulus of smoothness of order m corresponding to f :

ωm(f, η, a, b) = supa1≤x≤b1

sup0≤h≤η

|∆mh f(x)|.

7

...SECOND KIND BETA OPERATORS 235

Theorem 2 Let f ∈ Mγ[0,∞). Then, for sufficiently large n, there holds

‖Ln,kf − f‖C[a2,b2] ≤ C11

ω2k(f, n−1/2, a1, b1) + n−k‖f‖γ

,

where C11 is an absolute constant.

Proof. By linearity property, we have

‖Ln,kf − f‖C[a2,b2] ≤ ‖Ln,k(f − fη,2k)‖C[a2,b2] + ‖Ln,kfη,2k − fη,2k‖C[a2,b2] +

+ ‖fη,2k − f‖C[a2,b2] = H1 + H2 + H3.

Applying property 3. of Steklov’s mean, we get

H3 ≤ C12 ω2k(f, η, a1, b1)

Making use of Theorem 1, we have

H2 ≤ n−k C13

2k∑j=2

‖f (j)η,2k‖C[a2,b2] ≤ C14

‖fη,2k‖C[a2,b2] + ‖f (2k)

η,2k‖C[a2,b2]

,

where we have used the interpolation property due to S. Goldberg and V. Meir [6].Now, by using properties 4. and 5. of Steklov’s mean, we get

H2 ≤ C15 ‖f‖γ.

Setting a∗, b∗ satisfying a1 < a∗ < a2 < b2 < b∗ < b1 and let ξ(t) be the characteristicfunction of the closed interval [a∗, b∗], then, by Lemma 2 and property 3. of Steklov’smean, we obtain

H1 = ‖Ln,k(f(t)− fη,2k)‖C[a2,b2]

≤ ‖Ln,k(ξ(t) (f(t)− fη,2k))‖C[a2,b2] + ‖Ln,k((1− ξ(t))(f(t)− fη,2k))‖C[a2,b2]

≤ C16 ‖f − fη,2k‖C[a∗,b∗] + C17 n−m‖f‖γ

≤ C18 ω2k(f, η, a1, b1) + C19 ‖f‖γ.

Choosing m ≥ k and η = n−1/2 in the above estimate we get the required result.

8

Z.FINTA,V.GUPTA236

4 Global Approximation

In this section we establish direct global approximation theorems for Stancu betaoperators. Let CB[0,∞) be the space of all real valued continuous bounded functionsf on [0,∞) endowed with the norm ‖f‖ = sup0≤x<∞ |f(x)|. Our results are givenwith the aid of Ditzian - Totik modulus of smoothness of second order defined by

ω2ϕ(f, δ) = sup

0≤h≤δsup

x±hϕ(x)∈[0,∞)|f(x + hϕ(x))− 2f(x) + f(x− hϕ(x))|,

where ϕ(x) =√

x(1 + x), 0 ≤ x < ∞. The corresponding K−functional is

K2,ϕ(f, δ2) = infg∈W 2

∞

‖f − g‖+ δ2 ‖ϕ2g′′‖

,

where W 2∞(ϕ) = g ∈ CB[0,∞) : g′ ∈ ACloc[0,∞), ϕ2g′′ ∈ CB[0,∞) denote the

weighted Sobolev space.

Our first theorem in this section is :

Theorem 3 Let f ∈ CB[0,∞). Then

‖Lnf − f‖ ≤ C20 ω2ϕ

(f,

1√n− 1

), n = 2, 3, . . .

where C20 > 0 is an absolute constant.

Proof. Let g ∈ W 2∞(ϕ). By Taylor’s formula, we have

g(t) = g(x) + g′(x) (t− x) +∫ t

x(t− u) g′′(u) du.

In view of Lemma 1, we get

Ln(g, x)− g(x) = Ln

(∫ t

x(t− u) g′′(u) du, x

)

Hence

|Ln(g, x)− g(x)| ≤ Ln

(∣∣∣∣∣∫ t

x

|t− u|ϕ2(u)

· ϕ2(u)|g′′(u)| du

∣∣∣∣∣, x)

≤ Ln

(∣∣∣∣∣∫ t

x

|t− u|u(1 + u)

du

∣∣∣∣∣, x)· ‖ϕ2g′′‖

9

...SECOND KIND BETA OPERATORS 237

Furthermore, in view of [5, p. 141, ( 9.6.2 ) ] and Lemma 1, we have

|Ln(g, x)− g(x)| ≤ Ln

((t− x)2

x·(

1

1 + x+

1

1 + t

), x

)‖ϕ2g′′‖

=

1

x(1 + x)Ln((t− x)2, x) +

1

xLn

((t− x)2

1 + t, x

)‖ϕ2g′′‖

=

1

n− 1+

1

xLn

((t− x)2

1 + t, x

)‖ϕ2g′′‖ (8)

By direct computation we obtain

Ln

((t− x)2

1 + t, x

)= Ln

(t2

1 + t− 2x

t

1 + t+ x2 1

1 + t, x

)

=x(nx + 1)

nx + n + 1− 2x

nx

nx + n + 1+ x2 n + 1

nx + n + 1

=x(1 + x)

nx + n + 1

Hence, by ( 8 ), we have

|Ln(g, x)− g(x)| ≤(

1

n− 1+

1

n· x(1 + x)

nx + n + 1

)‖ϕ2g′′‖

=(

1

n− 1+

1 + x

nx + n + 1

)‖ϕ2g′′‖

≤(

1

n− 1+

1

n

)‖ϕ2g′′‖ ≤ 2

n− 1‖ϕ2g′′‖.

Also, Ln is a contraction, i.e.

‖Lnf‖ ≤ ‖f‖, f ∈ CB[0,∞). (9)

Thus

|Ln(f, x)− f(x)| ≤ |Ln(f − g, x)− (f − g)(x)|+ |Ln(g, x)− g(x)|

≤ 2 ‖f − g‖+2

n− 1‖ϕ2g′′‖

Taking the infimum on the right - hand side over all g ∈ W 2∞(ϕ), we get

|Ln(f, x)− f(x)| ≤ 2 K2,ϕ

(f,

1

n− 1

)(10)

10

Z.FINTA,V.GUPTA238

By [5, p. 11, Theorem 2.1.1 ], there exists an absolute constant C21 > 0 such that

K2,ϕ

(f,

1

n− 1

)≤ C21 ω2

ϕ

(f,

1√n− 1

), n > 1.

Hence, by ( 10 )

‖Lnf − f‖ ≤ C20 ω2ϕ

(f,

1√n− 1

), n = 2, 3, . . . ,

which was to be proved.

The next theorem is :

Theorem 4 Let f ∈ CB[0,∞). Then

‖Ln,kf − f‖ ≤ 2k k K2,ϕ

(f,

1

n− 1

), n = 2, 3, . . .

Proof. We have

Ln,k(f, x)− f(x) =

=k∑

r=1

(−1)r+1

(kr

)[Lr

n(f, x)− f(x)] +k∑

r=1

(−1)r+1

(kr

)f(x)− f(x)

=k∑

r=1

(−1)r+1

(kr

)[Lr

n(f, x)− f(x)] +k∑

r=0

(−1)r

(kr

)f(x)

=k∑

r=1

(−1)r+1

(kr

)[Lr

n(f, x)− f(x)] . (11)

On the other hand, by ( 9 ), we obtain

‖Lrnf − f‖ ≤ r ‖Lnf − f‖, r = 1, 2, . . .

Hence, by ( 11 ) and ( 10 ),

|Ln,k(f, x)− f(x)| ≤k∑

r=1

(kr

)|Lr

n(f, x)− f(x)| ≤k∑

r=1

(kr

)‖Lr

nf − f‖

≤k∑

r=1

(kr

)r ‖Lnf − f‖ = ‖Lnf − f‖

k∑r=1

k

(k − 1r − 1

)

= 2k−1 · k ‖Lnf − f‖ ≤ 2k · k K2,ϕ

(f,

1

n− 1

).

11

...SECOND KIND BETA OPERATORS 239

This completes the proof of Theorem 4 .

Corollary 1 There exists an absolute constant C22 = C22(k) > 0 such that

‖Ln,kf − f‖ ≤ C22 ω2ϕ

(f,

1√n− 1

),

for all f ∈ CB[0,∞) and n = 2, 3, . . .

Proof. It follows from Theorem 4 and [5, p. 11, Theorem 2.1.1 ].

References

[1] U. Abel, Asymptotic approximation with Stancu beta operators, Rev. Anal.Numer. Theorie Approx. 27 ( 1 ), 5 - 13 ( 1998 ).

[2] J. A. Adell, J. de la Cal, On a Bernstein - type operator associated with theinverse Polya - Eggenberger distribution, Rend. Circolo Matem. Palermo, Ser. II33, 143 - 154 ( 1993 ).

[3] J. A. Adell, J. de la Cal, Limiting properties of inverse beta and generalizedBleimann - Butzer - Hahn operators, Math. Proc. Cambridge Philos. Soc. 114 (3 ), 489 - 498 ( 1993 ).

[4] J. A. Adell, J. de la Cal, M. San Miguel, Inverse beta and generalized Bleimann- Butzer - Hahn operators, J. Approx. Theory 76 ( 1 ), 54 - 64 ( 1994 ).

[5] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer - Verlag, New York BerlinHeidelberg, 1987.

[6] S. Goldberg, V. Meir, Minimum moduli of ordinary differential operators, Proc.London Math. Soc. 23 ( 3 ), 1 - 15 ( 1971 ).

[7] C. A. Micchelli, The saturation class and iterates of the Bernstein polynomials,J. Approx. Theory 8, 1 - 18 ( 1973 ).

[8] D. D. Stancu, On the beta approximating operators of second kind, Rev. Anal.Numer. Theorie Approx. 24 ( 1 - 2 ), 231 - 239 ( 1995 ).

[9] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Hin-dustan Publ. Corp., Delhi, 1966.

12

Z.FINTA,V.GUPTA240

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TABLE OF CONTENTS,JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.2,2006 POWER SERIES OF FUZZY NUMBERS WITH CROSS PRODUCT AND APPLICATIONS TO FUZZY DIFFERENTIAL EQUATIONS, A.BAN,B.BEDE,………………………………………………………….125 ON A DELAY INTEGRAL EQUATION IN BIOMATHEMATICS, A.BICA,C.IANCU,………………………………………………………..153 SET-VALUED VARIATIONAL INCLUSIONS WITH FUZZY MAPPINGS IN BANACH SPACES, A.SIDDIQI,R.AHMAD,S.IRFAN,………………………………………..171 ON SOME PROPERTIES OF SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS IN ABSTRACT SPACES,C.GORI,V.OBUKHOVSKII, M.RAGNI,P.RUBBIONI,………………………………………………….183 COMPLETELY GENERALIZED NONLINEAR VARIATIONAL INCLUSIONS WITH FUZZY SETVALUED MAPPINGS,R.AGARWAL,M.KHAN, D.O’REGAN,SALAHUDDIN,…………………………………………….215 SOME DIRECT RESULTS FOR THE ITERATIVE COMBINATIONS OF THE SECOND KIND BETA OPERATORS, Z.FINTA,V.GUPTA,……………………………………………………….229

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On the uniform exponential stability ofevolution families in terms of the admissibility

of an Orlicz sequence space

Ciprian Preda, 1 Sever S. Dragomir 2 and Constantin Chilarescu 3

1 Department of Electrical Engineering, University of CaliforniaLos Angeles, CA 90095, U.S.A

e-mail: preda@ee.ucla.edu2 School of Computer Science and Mathematics Victoria University

PO Box 14428, Melbourne City, MC 8001, Australiae-mail: sever@matilda.vu.edu.au

3 Department of Mathematics, West University of Timisoara4 Parvan Blv., 300223, Timisoara, Romania

e-mail: cchilarescu@rectorat.uvt.ro

A characterization of the exponential stability for evolution families interms of the admissibility of some pairs of sequences spaces, is given.It is used the method of ”test function” using a very well-known classof sequence spaces (the so-called Orlicz sequence spaces ). Thus, areextended to the case of the abstract evolution families some resultsdue to Coffman, Schaffer, Przyluski, Rolewicz.1

1Mathematics Subject Classification: 34D05,47D06, 93D20.Key words and phrases: evolution families, exponential stability, Orlicz sequence spaces,

admissibility.

1

253JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,253-265,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

1 Introduction

Over the past ten years, the asymptotic theory of linear evolution equationsin Banach spaces has witnessed an explosive development. A number oflong-standing open problems have recently been solved and the theory seemsto have obtained a certain degree of maturity. An important role in theearly development of qualitative theory of differential systems, was played bythe paper ”Die stabilitatsfrage bei differentialgeighungen” [13],where Perrongave a characterization of exponential stability of the solutions to the lineardifferential equations

dx

dt= A(t)x, t ∈ [0, +∞), x ∈ IRn

where A(t) is a matrix bounded continuous function , in terms of the ex-

istence of bounded solutions of the equationsdx

dt= A(t)x + f(t), where f

is a continuous bounded function on IR+. After these seminal researches ofO. Perron, relevant results concerning the extension of Perron’s problem inthe more general framework of infinite-dimensional Banach spaces were ob-tained, in their pioneering monographs, by M. G. Krein, J. L. Daleckij, J. L.Massera and J. J. Schaffer. In the present there are different important char-acterizations of exponential stability or dichotomy for a strongly continuous,exponentially bounded evolution family, which can be found for instance inthe papers due to N. van Minh [11,12], Y. Latushkin[2,7,8], P. Randolph [8],P. Preda[10,15], R. Schnaubelt [11,17], S. Montgomery -Smith[7]. For thecase of discrete-time systems analogous results were firstly obtained by Ta Liin 1934 [see 18]. In Ta Li’s paper, can be found the same principal concernas in Perron’s work, but using discrete arguments. This approach was laterdeveloped for the discrete-time systems in the infinite-dimensional case byCh.V. Coffman and J.J. Schaffer in 1967 [3] and D. Henry in 1981[5]. Morerecently we refer the reader to the the papers due to A. Ben-Artzi[2], I. Go-hberg[2], M. Pinto[19], J. P. La Salle[8]. Applications of this ”discrete-timetheory” to stability theory of linear infinite-dimensional continuous-time sys-tems have been presented by Przyluski and Rolewicz in [16]. The first aimof this paper is to give a characterization of the exponential stability foran abstract evolution family (not necessary provided by a differential sys-tem) in terms of the admissibility of a Orlicz sequence space, so this work

2

254 PREDA ET ALL

fit into the context initiated by Ta Li. Roughly speaking, this means thatan evolution family, acting on a Banach space X, is uniformly exponentiallystable if and only if the corresponding inhomogeneous difference equationwith the inhomogeneous term from lΦ(X) admits a solution in lΦ(X), wherelΦ(X) = f : IN 7→ X : (||f(n)||)n∈IN is in the Orlicz sequence space lΦ.Also, we note that the technique used in this work does not require anycontinuity hypothesis about the evolution families (as the strongly continu-ity or partial strongly continuity, assumptions required in most of the abovecited works ). Classical examples of Orlicz sequence spaces are lp-spaces,p ∈ [1,∞), and there is also presented in the paper other example of a Orliczsequence spaces which is not lp (also, for well-chosen Young functions Φ othervarious examples of Orlicz sequence space can be found). So, this approachgeneralize the equivalence between (lp, lp)-admissibility and the exponentialstability of evolution families using the admissibility of a Orlicz sequencespace. Also, roughly speaking, we note that using this discrete-time the-ory we obtain here the uniform exponential stability of the continuous-timeevolution families , fact which is very useful. Moreover, this approach doesbreak a new ground and it can bring other useful situations (as Example 2.2),adding some nice twist to the subject concerning the connection between ad-missibility and exponential stability of evolution families.

2 Preliminaries

First, let us fix some standard notation. For X a Banach space we will denoteby lp(X) and l∞(X) the normed spaces,

lp(X) = f : N → X :∞∑

n=0

‖f(n)‖p < ∞, p ∈ [1,∞),

l∞(X) = f : N → X : supn∈N

‖f(n)‖ < ∞.

We note that lp(X), l∞(X) are Banach spaces endowed with the respectivelynoms

‖f‖p =( ∞∑

n=0‖f(n)‖p

)1/p

;

‖f‖∞ = supn∈N

‖f(n)‖.

3

255ON THE UNIFORM EXPONENTIAL...

For the simplicity of notations we denote by lp = lp(R), l∞ = l∞(R).For more convenience we will list in the next the definition of a Orlicz se-quence space. Let ϕ : IR+ → IR+ be a function which is non-decreasing, withϕ(t) > 0, for each t > 0. Define

Φ(t) =

t∫

0

ϕ(s)ds

A function Φ of this form is called an Young function. Forf : IN → IR a real sequence and the Young function Φ we define

mΦ(f) =∞∑

k=0

Φ(|f(k)|).

The set lΦ of all f for which there exists a j > 0 that mΦ(jf) < ∞ is easilychecked to be a linear space. Equipped with the Luxemburg norm

‖f‖Φ = inf

j > 0 : mΦ

(1

jf

)≤ 1

the space (lΦ, ‖ · ‖Φ) becomes a Banach space.

Remark 2.1. It is easy to check that χ0,...n ∈ lΦ and ||χ0,...n||Φ = 1Φ−1( 1

n+1),

for all n ∈ IN, where in general χA denotes the characteristic (indicator)function of the set A.

Example 2.1. Classical examples of Orlicz sequence spaces are the well-known lp-spaces, for all p ∈ [1,∞), (i.e. lΦ = lp if Φ(t) = tp).

Remark 2.2 If (lΦ, ‖ · ‖) = (lp, ‖ · ‖) then limt→0

Φ(t)tp

= 1.

Proof. If (lΦ, ‖ · ‖) = (lp, ‖ · ‖) then ‖χ0,...,n‖Φ = ‖χ0,...,n‖p, for all n ∈ IN ,which is equivalent with:

Φ−1(

1

n + 1

)=

(1

n + 1

) 1p

, for all n ∈ IN.

Let x ∈ (0, 1] and m =[

1x

]∈ N∗, where [a] denotes the greatest integer

which is less or equal than a. Using the fact that Φ−1 is nondecreasing we

4

256 PREDA ET ALL

have that

(1

m + 1

) 1p

= Φ−1(

1

m + 1

)≤ Φ−1(x) ≤ Φ−1

(1

m

)=

(1

m

) 1p

which implies that

1([

1x

]+ 1

)x

1p

≤ Φ−1(x)

x1p

≤ 1

x[

1x

]

1p

, for all x ∈ (0, 1].

Hence

limx→0

Φ−1(x)

x1p

= 1

and

limu→0

Φ(u)

up= lim

u→0

1[Φ−1(Φ(u))

(Φ(u))1p

]p = 1

Example 2.2. Consider ϕ, Φ : IR+ → IR+

ϕ(t) =∞∑

m=1

m√

t

m2, Φ(t) =

∫ t

0ϕ(s)ds =

∞∑

m=1

t1+ 1m

m(m + 1)

We pretend that lΦ 6= lp, for all p ∈ [1,∞). Indeed, limt→0

Φ(t)t

= 0 and

limt→0

Φ(t)tp

= ∞, for all p ∈ (1,∞) and using the above remark, our claim

follows easily.In what follows, if lΦ is a Orlicz sequence space we denote by

lΦ(X) = f : IN 7→ X : (||f(n)||)n∈IN is in lΦ.

Remark 2.3. lΦ(X) is a Banach space endowed with the norm

||f ||lΦ(X) = || ||f(·)|| ||Φ.

Remark 2.4. For any Orlicz sequence space lΦ we have that:i) lΦ ⊂ l∞;

5

257ON THE UNIFORM EXPONENTIAL...

ii)‖f‖∞ ≤ 1‖χ0‖Φ‖f‖Φ

Proposition 2.1. If Φ is an Young function of the Orlicz sequence space lΦ

then the followings statements hold:i) The map aΦ : IN → IR∗

+ given by aΦ(n) = (n + 1)Φ−1( 1n+1

) is anondecreasing one.

ii)n∑

k=0|f(k)| ≤ aΦ(n)||f ||Φ, for all t > 0, f ∈ lΦ.

6

258 PREDA ET ALL

Proof.i) First let us prove that the map b : IR∗

+ → IR∗+, given by b(u) = Φ(u)

uis

nondecreasing. If 0 < u1 ≤ u2 then

Φ(u1)

u1

=1

u1

u1∫

0

ϕ(s)ds =1

u1

u2∫

0

ϕ(u1

u2

v)u1

u2

dv =

=1

u2

u2∫

0

ϕ(u1

u2

v)dv ≤ 1

u2

u2∫

0

ϕ(v)dv =Φ(u2)

u2

.

In order to prove that aΦ is nondecreasing let n ∈ IN. It follows that 0 <w2 := Φ−1( 1

n+2) ≤ Φ−1( 1

n+1) := w1 and so b(w2) ≤ b(w1). Having in mind

that

b(w1) =1

aΦ(n + 1)and b(w2) =

1

aΦ(n),

it results that aΦ is a nondecreasing function.ii) Consider f ∈ lΦ, n ∈ IN∗, c > 0 such that mΦ(1

cf) ≤ 1. Then we have

that

Φ

(1

c(n + 1)

n∑

k=0

|f(k)|)≤ 1

n + 1

n∑

k=0

Φ(

1

c|f(k)|

)≤ 1

n + 1,

and so

n∑

k=0

|f(k)| ≤ (n + 1)Φ−1(

1

n + 1

)c

which implies that

n∑

k=0

|f(k)| ≤ (n + 1)Φ−1(

1

n + 1

)||f ||Φ = aΦ(n)||f ||Φ,

for all n ∈ IN, f ∈ lΦ.

Remark 2.5. Using a simple translation argument we may state that

n0+n∑

k=n0

|f(k)| ≤ aΦ(n)||f ||Φ,

7

259ON THE UNIFORM EXPONENTIAL...

for all n0, n ∈ IN, f ∈ LΦ.

Definition 2.1. A family of bounded linear operators acting on X denotedby U = U(t, s)t≥s≥0 is called an evolution family if the following statementshold:

e1) U(t, t) = I (where I is the identity operator on X), for all t ≥ 0;e2) U(t, s) = U(t, r)U(r, s), for all t ≥ r ≥ s ≥ 0;e3) there exist M > 0, ω > 0 such that

||U(t, s)|| ≤ Meω(t−s) for all t ≥ s ≥ 0.

Definition 2.2. The evolution family U = U(t, s)t≥s≥0 is called uniformlyexponentially stable (u.e.s) if there exist two strictly positive constants N, νsuch that the following statement hold:

‖U(t, s)‖ ≤ Ne−ν(t−s).

If lΦ is a Orlicz sequence space we give the following:

Definition 2.3. lΦ is said to be admissible to the evolution family U if forall f ∈ lΦ(X) the application xf : IN → X defined by

xf (n) =n∑

k=0

U(n, k)f(k) lies in lΦ(X).

3. The main result

Let lΦ be a Orlicz sequence space.

Lemma 3.1. If lΦ is admissible to U then there is K > 0 such that

||xf ||lΦ(X) ≤ K||f ||lΦ(X)

Proof. We define the linear operator, acting on lΦ(X), denoted byT : lΦ(X) → lΦ(X), and given by

(Tf)(m) =m∑

k=0

U(m, k)f(k).

8

260 PREDA ET ALL

Consider gnn∈IN ⊂ lΦ(X), g ∈ lΦ(X), h ∈ lΦ(X) such that

gnlΦ(X)−→ g, Tgn

lΦ(X)−→ h

Then

||(Tgn)(m)− (Tg)(m)|| ≤m∑

k=0

||U(m, k)(gn(k)− g(k))|| ≤

≤m∑

k=0

Meωm||gn(k)− g(k)|| ≤ MeωmaΦ(m)||gn − g||lΦ(X),

for all m, n ∈ IN .It follows, using again the Remark 2.4 , that Tg = h, and hence T is closedand so, by the Closed-Graph Theorem it is also bounded.So we obtain that

||xf ||lΦ(X) = ||Tf ||lΦ(X) ≤ ||T || ||f ||lΦ(X), for all f ∈ lΦ(X) as required.

Lemma 3.2. Let g : (t, t0) ∈ R2 : t ≥ t0 ≥ 0 → R+ be a function suchthat the following properties hold.

1) g(t, t0) ≤ g(t, s)g(s, t0) , for all t ≥ s ≥ t0 ≥ 0;

2) sup0≤t0≤t≤t0+1

g(t, t0) < ∞;

3) there exist a sequence h : IN → IR+, limn→∞h(n) = 0 and g(m+n, n) ≤

h(m) , for all m,n ∈ N.Then there exist two constants N, ν > 0 such that

g(t, t0) ≤ Ne−ν(t−t0) , for all t ≥ t0 ≥ 0

Proof. Let a = sup0≤t0≤t≤t0+1

g(t, t0),m0 = minm ∈ N∗ : h(m) ≤ 1

e

.

Conditions 1) and 2) imply that sup0≤t0≤t≤t0+2m0

g(t, t0) ≤ a2m0 .

Fix t0 ≥ 0, t ≥ t0 + 2m0,m =[

tm0

], n =

[t0m0

]where [s] is the largest integer

9

261ON THE UNIFORM EXPONENTIAL...

equal or less than s ∈ R. One can see that m0m ≤ t < m0(m + 1),m0n ≤t0 < m0(n + 1),m ≥ n + 2, and so,

g(t, t0) ≤ g(t,m0m)g(m0m, m0(n + 1))g(m0(n + 1), t0) ≤

≤ a4m0

m∏

k=n+2

g(m0k, m0(k − 1)) ≤ a4m0

m∏

k=n+2

h(m0)

≤ a4m0e−(m−n−1) ≤ a4m0e− t−t0

m0+2

.

If we note that

g(t, t0) ≤ a2m0 ≤ a2m0e2e− t−t0

m0 , for all t0 ≥ 0, t ∈ [t0, t0 + 2m0]

we obtain easily that

g(t, t0) ≤ Ne−ν(t−t0) , for all t ≥ t0 ≥ 0 , where

N = maxa4m0e2, a2m0e2, ν =1

m0

.

Theorem 3.1. U is u.e.s. if and only if there exists a Orlicz sequence spacelΦ admissible to U .

Proof.Necessity. It follows easily from (Definition 2.3) that the space l1 is admissibleto U .Sufficiency. Let m ∈ N, x ∈ X, and f : N → X, f = χmx. It is easy to

verify that f ∈ lΦ(X) and ‖f‖Φ = ‖χ0‖Φ‖x‖ and

(xf )(k) =k∑

j=0

U(k, j)f(j) =

U(k, m)x , k ≥ m0 , k < m

and so ‖U(k,m)x‖ ≤ ‖xf‖∞ ≤ 1‖χ0‖Φ‖xf‖Φ ≤ K 1

‖χ0‖Φ‖f‖Φ = K‖x‖,for all k ≥ m.

10

262 PREDA ET ALL

Let n0,m ∈ N, x ∈ X, f : N → X, given by

f(n) =

U(n, n0)x , n ∈ n0, ..., n0 + m0 , n 6∈ n0, ..., n0 + m

Then f ∈ lΦ(X), ‖f‖lΦ(X) ≤ K 1Φ−1( 1

m+1)‖x‖. It follows that

(xf )(n) =n∑

k=0

U(n, k)f(k) =

0 , n < n0

(n− n0 + 1)U(n, n0)x , n ∈ n0, ..., n0 + m(m + 1)U(m,n0)x , n ≥ n0 + m + 1

and so

(m + 1)(m + 2)

2‖U(m + n0, n0)x‖ =

n0+m∑n=n0

(n− n0 + 1)‖U(m + n0, n0)x‖

≤ Kn0+m∑n=n0

(n− n0 + 1)‖U(n, n0)x‖ = Kn0+m∑n=n0

‖xf (n)‖ ≤ KaΦ(m)‖xf‖lΦ(X)

≤ KaΦ(m)‖f‖lΦ(X) ≤ K3‖x‖(m + 1).

We obtain that

‖U(m + n0, n0)‖ ≤ 2K3

m + 2‖x‖ , for all m,n0 ∈ N

By Lemma 3.2 it results that there exist two constants N, ν > 0 such that

‖U(t, t0)‖ ≤ Ne−ν(t−t0) , for all t ≥ t0 ≥ 0 .

3 References

[1] A. Ben-Artzi, I. Gohberg, Dichotomies of systems and invertibility oflinear ordinary differential operators, Oper. Theory Adv. Appl., 56,1992 ,90-119.

[2] C. Chicone, Y. Latushkin, Evolution semigroups in dynamicalsystems and differential equations, Mathematical Surveys and Mono-graphs, vol. 70, Providence, RO: American Mathematical Soxiety,1999.

11

263ON THE UNIFORM EXPONENTIAL...

[3] C. V. Coffman, J.J. Schaffer, Dichotomies for linear difference equa-tions, Math. Annalen 172, 1967, 139-166.

[4] J.L Daleckij, M.G. Krein, Stability of differential equations inBanach space, Amer. Math. Scc, Providence, R.I. 1974

[5] D. Henry, Geometric theory of semilinear parabolic equations, SpringerVerlag, New-York, 1981

[6] J. P. La Salle, The stability and control of discrete processes, SpringerVerlag, Berlin, 1990.

[7] Y. Latushkin, S. Montgomery-Smith, Evolutionary semigroups andLyapunov theorems in Banach spaces, J. Funct. Anal., 127 (1995),173-197.

[8] Y. Latushkin, T. Randolph, Dichotomy of differential equations on Ba-nach spaces and an algebra of weighted composition operators, IntegralEquations Operator Theory, 23 (1995), 472-500.

[9] J.L. Massera, J.J. Schaffer, Linear Differential Equations and FunctionSpaces, Academic Press, New York, 1966.

[10] M.Megan , P. Preda, Admissibility and dichotomies for linear discrete-time systems in Banach spaces,An. Univ. Tim., Seria St. Mat.,vol.26 (1), 1988, 45-54.

[11] N. van Minh, F. Rabiger, R. Schnaubelt, Exponential stability, expo-nential expansiveness and exponential dichotomy of evolution equationson the half-line, Int. Eq. Op. Theory, 32 (1998), 332-353.

[12] N. van Minh, On the proof of characterizations of the exponential di-chotomy, Proc. Amer. Math. Soc., 127 (1999), 779-782.

[13] O. Perron, Die stabilitatsfrage bei differentialgeighungen, Math. Z., 32(1930), 703-728.

[14] M. Pinto, Discrete dichotomies, Computers Math. Applic., vol. 28,1994, 259-270.

12

264 PREDA ET ALL

[15] P.Preda, On a Perron condition for evolutionary processes in Banachspaces, Bull. Math. de la Soc. Sci. Math. de la R.S. Roumanie, T 32(80), no.1, 1988, 65-70.

[16] K.M. Przyluski, S. Rolewicz, On stability of linear time-varying infinite-dimensional discrete-time systems, Systems Control Lett. 4, 1994, 307-315.

[17] R. Schnaubelt, Sufficient conditions for exponential stability and di-chotomy of evolution equations, Forum Mat., 11 (1999), 543-566.

[18] L.Ta, Die stabilitatsfrage bei differenzengleichungen, Acta Math. 63,1934, 99-141.

[19] A. C. Zaanen, Integration, North-Holland, Amsterdam, 1967.

13

265ON THE UNIFORM EXPONENTIAL...

266

Generalized Quasilinearization Technique for the SecondOrder Differential Equations with General Mixed Boundary

Conditions

Bashir Ahmada and Rahmat A. Khanb

aDepartment of Mathematics, Faculty of Science, King Abdulaziz University,P.O.Box 80203, Jeddah-21589, Saudi Arabia

E-mail: bashir−qau@yahoo.combDepartment of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

E-mail: rahmat−alipk@yahoo.com

Abstract

The generalized quasilinearization method for a second order nonlineardifferential equation with general type of boundary conditions is developed. Amonotone sequence of approximate solutions converging uniformly and rapidlyto a solution of the problem is obtained. Some special cases have also beendiscussed.

Keywords and Phrases: Quasilinearization, mixed boundary conditions, rapidconvergence.AMS Subject Classifications (2000): 34A45, 34B15.

1. Introduction

The method of generalized quasilinearization introduced by Lakshmikantham [4,5]has been effectively applied to find a sequence of approximate solutions of thenonlinear initial and boundary value problems, see, for example, [ 6-9]. In reference[2], the generalized quasilinearization method was discussed for a nonlinear Robinproblem.In this paper, we develop a generalized quasilinearization technique for a second or-der nonlinear differential equation subject to general mixed boundary conditions andobtain a sequence of approximate solutions converging monotonically and rapidlyto a solution of the problem. Some interesting observations have also been presented.

2. Some Basic Results

Consider the nonlinear problem

−u′′(t) = f(t, u), t ∈ J = [0, π], (1)

pou(0)− qou′(0) = c1, pou(π) + qou(π) = c2,

where f ∈ C[J ×R,R], po, qo, c1, c2 ∈ R with po, qo > 0 (2po > qo) and c1, c2 ≥ 0.

267JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,267-276,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

We know that the linear problem

−u′′(t) = λu(t), t ∈ J,

pou(0)− qou′(0) = 0, pou(π) + qou

′(π) = 0,

has a trivial solution for all real values of λ except for those values of λ which arethe roots of the equation

tan√

λπ =2√

λpoqo

(q2oλ− p2

o).

Consequently, for all such values of λ for which the homogeneous linear problem hasa trivial solution, the corresponding non homogeneous problem

−u′′(t)− λu(t) = σ(t), t ∈ J,

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2,

has a unique solution

u(t) =∫ π

0Gλ(t, s)σ(s)ds + c2

d

dsGλ(t, s)|s=π − c1

d

dsGλ(t, s)|s=0,

where Gλ(t, s) is the Green’s function of the associated homogeneous problem andis given by

Gλ(t, s) =1

2poqoλ cos√

λπ + (p2o

√λ− q2

oλ32 ) sin

√λπ

×

(po sin

√λt+qo

√λ cos

√λt)

[qo

√λ cos

√λ(π−s)+po sin

√λ(π−s)]

−1 , if 0 ≤ t ≤ s ≤ π,

(po sin√

λs+qo

√λ cos

√λs)

[qo

√λ cos

√λ(π−t)+po sin

√λ(π−t)]

−1 , if 0 ≤ s ≤ t ≤ π (λ > 0),

Gλ(t, s) =1

po(2qo + poπ)

(qo + pot)(qo + po(π − s), if 0 ≤ t ≤ s ≤ π,(qo + pos)(qo + po(π − t), if 0 ≤ s ≤ t ≤ π (λ = 0),

Gλ(t, s) =−1

2poqo(−λ) cosh√−λπ + (p2

o

√−λ + q2

o(−λ)32 sinh

√−λπ

×

(po sinh

√−λt+qo

√−λ cosh

√−λt)

[qo√−λ cosh

√−λ(π−s)+po sinh

√−λ(π−s)]

−1 , if 0 ≤ t ≤ s ≤ π,

(po sinh√−λs+qo

√−λ cosh

√−λs)

[qo√−λ cosh

√−λ(π−t)+po sinh

√−λ(π−t)]

−1 , if 0 ≤ s ≤ t ≤ π (λ < 0).

We observe that Gλ ≥ 0 for λ < 1/4. Now, we state the following lemma ( for proof,see [2]).

2

268 AHMAD,KHAN

Lemma 2.1. (Comparison Result) Let λ < 1/4, σ(t) ≥ 0 and u ∈ C2(J). Thenthe nonhomogeneous problem

−u′′(t)− λu(t) = σ(t), t ∈ J,

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2,

has a nonnegative solution.We shall say that α(t) ∈ C2(J) is a lower solution of (1) if

−α′′(t) ≤ f(t, α(t)), t ∈ J,

poα(0)− qoα′(0) ≤ c1, poα(π) + qoα

′(π) ≤ c2.

Similarly, β ∈ C2(J) is an upper solution of (1) if

−β′′(t) ≥ f(t, β(t)), t ∈ J,

poβ(0)− qoβ′(0) ≥ c1, poβ(π) + qoβ

′(π) ≥ c2.

Now, we state some theorems (without proof) which can be proved using the stan-dard arguments [2, 9].Theorem 2.2. Let f ∈ C2[J ×R,R] be nonincreasing in u for each t ∈ J and α, βare lower and upper solutions of (1) respectively. Further, there exists a constantL ≥ 0 such that

f(t, u1)− f(t, u2) ≤ L(u1 − u2),

whenever u1 ≥ u2. Then α(t) ≤ β(t) on J.Theorem 2.3. Let α, β be lower and upper solutions of (1) respectively such thatα(t) ≤ β(t) on J and f ∈ C2[J × R,R]. Then there exists a solution u(t) of (1)such that α(t) ≤ u(t) ≤ β(t), t ∈ J.

3. Generalized Quasilinearization Technique

Theorem 3.1. Assume that

(A1) α and β ∈ C2(J) are lower and upper solutions of (1) respectively such thatα(t) ≤ β(t) on J.

(A2)∂f∂u

(t, u), ∂2f∂u2 (t, u) exist and are continuous such that ∂f

∂u(t, u) < 1/4 and

∂2

∂u2 (f(t, u) + φ(t, u)) ≥ 0, where φ ∈ C[J × R,R] such that ∂2φ∂u2 (t, u) ≥ 0

for every (t, u) ∈ J ×R.

Then there exists a monotone sequence wn of solutions converging uniformly andquadratically to the unique solution of the problem.Proof. We define

F (t, u) = f(t, u) + φ(t, u), t ∈ J.

3

269GENERALIZED QUASILINEARIZATION...

Clearly F (t, u) ∈ C[J × R,R]. In view of (A2) and the generalized mean valuetheorem, we have

F (t, u) ≥ F (t, v) + Fu(t, v)(u− v)− φ(t, u),

where α ≤ v ≤ u ≤ β on J (u, v ∈ R ). Define

g(t, u, v) = F (t, v) + Fu(t, v)(u− v)− φ(t, u), (2)

and observe that

g(t, u, v) ≤ F (t, u), g(t, u, u) = F (t, u). (3)

Also, it can easily be seen that g(t, u, v) is nonincreasing in u for each fixed (t, v).Further, using the mean value theorem repeatedly, we have

g(t, u, v1)− g(t, u, v2) ≥∂2

∂u2F (t, ξ1)(ξ − v2)(v1 − v2),

(v2 ≤ ξ1 ≤ ξ, v2 ≤ ξ ≤ v1, t ∈ J), which in view of (A2) implies that g(t, u, v) isnondecreasing in v for each fixed (t, u). Also, g(t, u, v) satisfies Lipschitz condition:

g(t, u, v1)−g(t, u, v2) = (Fu(t, v)−φu(t, η))(u1−u2) ≤ fu(t, u)(u1−u2) ≤ L(u1−u2),

where L > 0, and u1 ≥ u2.Now, we set wo = α and consider

−u′′(t) = g(x, u(t), wo(t)), t ∈ J, (4)

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2.

Using (A1), (3) and the nondecreasing nature of g(t, u, v) in v, we get

−w′′o(t) ≤ f(t, wo(t)) = g(t, wo(t), wo(t)), t ∈ J,

powo(0)− qow′o(0) ≤ c1, powo(π) + qow

′o(π) ≤ c2,

and−β′′(t) ≥ f(t, β(t)) ≥ g(t, wo, β), t ∈ J

poβ(0)− qoβ′(0) ≥ c1, poβ(π) + qoβ

′(π) ≥ c2,

which imply that wo and β are lower and upper solution of (4) respectively. Hence,by Theorem 2.3, there exists a solution w1 of (4) such that wo ≤ w1 ≤ β on J.Now, consider the problem

−u′′(t) = g(t, u(t), w1(t)), t ∈ J, (5)

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2.

4

270 AHMAD,KHAN

From−w′′1(t) = g(t, w1(t), wo(t)) ≤ g(t, w1, w1), t ∈ J,

pow1(0)− qow′1(0) = c1, pow1(π) + qow

′1(π) = c2,

and−β′′(t) ≥ f(t, β(t)) ≥ g(t, wo, β), t ∈ J,

poβ(0)− qoβ′(0) ≥ c1, poβ(π) + qoβ

′(π) ≥ c2,

it follows that w1 and β are lower and upper solutions of (5) respectively. Again,by Theorem 2.3, there exists a solution w2 of (5) such that w1 ≤ w2 ≤ β onJ. Continuing this process successively, we obtain a monotone sequence wn ofsolutions on J satisfying

wo ≤ w1 ≤ w2 ≤ w3 ≤ ... ≤ wn−1 ≤ wn ≤ β,

where the element wn of the sequence is the solution of the problem

−u′′(t) = g(x, u(t), wn−1(t)), t ∈ J,

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2.

Since the sequence wn is monotone, it has a pointwise limit w. To show that w isin fact a solution of (1), we note that

wn(t) =∫ π

0G0(t, s)σn(s)ds + c2

d

dsGo(t, s)|s=π − c1

d

dsGo(t, s)|s=0,

(G0(t, s) is given in section 2) is a solution of the following problem

−u′′(t) = σn(t), t ∈ J,

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2,

whereσn(t) = F (t, wn−1) + Fu(t, wn−1)(wn − wn−1)− φ(t, wn).

Since σn is continuous for each n ∈ N and α ≤ wn ≤ β (n = 1, 2, 3...), it follows thatσn is bounded in C(J). Also, limn→∞σn(t) = f(t, w). Thus, the sequence wnis bounded in C2(J) and hence wn w uniformly on J. Passing onto the limitn →∞, we get

w =∫ π

0G0(t, s)f(s, u)ds + (c2 + c1)

qo

(2qo + poπ).

This shows that w is a solution of (1).Now we show that the convergence is quadratic. For that, we set en = u−wn, n =

5

271GENERALIZED QUASILINEARIZATION...

1, 2, 3... and note that en ≥ 0, poen(0) − qoe′n(0) = 0 and poen(π) + qoe

′n(π) = 0.

Using the definition of g(t, u, v) and the generalized mean value theorem, we get

−e′′n(t) = −u′′(t) + w′′n(t)

= [∂F

∂u(t, ξ1)−

∂F

∂u(t, , un−1)](u− un−1)

− φu(t, ξ2)(u− un) +∂F

∂u(t, , un−1)(u− un)

=∂2F

∂u2(t, ξ3)(ξ1 − un−1)(u− un−1) + [

∂F

∂u(t, , un−1)− φu(t, ξ2)](u− un)

≤ Ce2n−1 +

∂f

∂u(t, , un−1)en ≤ Ce2

n−1 + an(t)en,

where wn−1(x) ≤ ξ3 ≤ ξ1 ≤ u, un ≤ ξ2 ≤ u and an = ∂f∂u

(t, un−1). Sincelimn→∞an(t) = fu(t, w) < 1/4, therefore we can choose λ < 1/4 and no ∈ Nsuch that for n ≥ no, an(t) < λ, t ∈ J. Thus, for some bn(t) ≤ 0, the error functionen(t) satisfies the problem

−e′′n(t)− λ(t)en(t) = (an(t)− λ)en(t) + ce2n−1 + bn(t), t ∈ J,

poen(0)− qoe′n(0) = 0, poen(π) + qoe

′n(π) = 0,

whose solution is

en(t) =∫ π

0Gλ(t, s)[(an(s)− λ)en(s) + ce2

n−1 + bn(s)]ds, t ∈ J.

Since an(t)− λ ≤ 0, en = w − wn ≥ 0, bn ≤ 0, therefore

(an(t)− λ)en(t) + bn(t) + ce2n−1 ≤ ce2

n−1.

Consequently, we have

en(t) ≤ c∫ π

0Gλ(t, s)e

2n−1(s)ds, n ≥ no,

which, on taking the maximum, can be written as

||en|| ≤ δ||en−1||2,

where δ provides a bound on c∫ π0 Gλ(t, s)ds and

||u|| = max|u(t)|, t ∈ J

.Theorem 3.2. (Rapid Convergence) Assume that

(B1) α, β ∈ C2(J) are lower and upper solutions of (1) respectively such that α ≤ βon J.

6

272 AHMAD,KHAN

(B2)∂if∂xi (t, u), i = 1, 2, 3, ..., k exist and are continuous on Ω = (t, u) ∈ J×R such

that ∂f∂u

(t, u) < 14, ∂k

∂uk (f(t, u)+φ(t, u)) ≥ 0 for some function φ ∈ C0,(k−1)[J ×R,R] such that ∂kφ

∂uk (t, u) ≥ 0.

Then there exists a monotone sequence wn of solutions converging uniformly andrapidly with the order of convergence k (k ≥ 2) to a solution of (1).Proof. Define

F (t, u) = f(t, u) + φ(t, u), t ∈ J.

Using (B2) and generalized mean value theorem, we have

f(t, u) ≥k−1∑i=0

∂iF

∂ui(t, v)

(u− v)i

(i)!− φ(t, u).

Now, we set

K(t, u, v) =k−1∑i=0

∂iF

∂ui(t, v)

(u− v)i

(i)!− φ(t, u)

=k−1∑i=0

∂if

∂ui(t, v)

(u− v)i

(i)!− ∂kφ

∂uk(t, ξ)

(u− v)k

(k)!,

where v ≤ ξ ≤ u and α ≤ v ≤ u ≤ β on J.Observe that

K(t, u, v) ≤ F (t, u), K(t, u, u) = F (t, u). (6)

Fix wo = α and consider the problem

−u′′(t) = K(t, u(t), wo(t)), t ∈ J, (7)

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2.

Using (B1) and (6), we get

−w′′o(t) ≤ f(t, wo(t)) = K(t, wo(t), wo(t)), t ∈ J,

powo(0)− qow′o(0) ≤ c1, powo(π) + qow

′o(π) ≤ c2,

and−β′′(t) ≥ f(t, β(t)) ≥ K(t, wo, β), t ∈ J,

poβ(0)− qoβ′(0) ≥ c1, poβ(π) + qoβ

′(π) ≥ c2,

which imply that wo and β are lower and upper solution of (7) respectively. Hence,by Theorem 2.3, there exists a solution w1 of (7) such that wo ≤ w1 ≤ β on J.Similarly, we can show that the following problem

−u′′(t) = K(t, u(t), w1(t)), t ∈ J, (8)

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2,

7

273GENERALIZED QUASILINEARIZATION...

has a solution w2 such that w1 ≤ w2 ≤ β on J, where w1 and β are lower and uppersolution of (8) respectively.Continuing this process successively, we obtain a monotone sequence wn of solu-tions satisfying

wo ≤ w1 ≤ w2 ≤ w3 ≤ ... ≤ wn−1 ≤ wn ≤ β

on J, where the element wn of the sequence is the solution of the problem

−u′′(t) = K(t, u(t), wn−1(t)), t ∈ J,

pou(0)− qou′(0) = c1, pou(π) + qou

′(π) = c2.

Employing the procedure used in the preceding theorem, we can show that thesequence wn converges to the unique solution w of the boundary value problem (1).

To show that the convergence of the sequence is of order k (k ≥ 2), we set en = w−wn, an = wn+1−wn, n = 1, 2, 3.... Clearly, an ≥ 0, en ≥ 0, en−an = en+1, an ≤ en

and akn ≤ ek

n. Using the mean value theorem repeatedly, we have

−e′′n(t) = w′′n(t)− w′′(t)

=k−1∑i=0

∂if

∂ui(t, wn−1)

(ein−1 − ai

n−1)

(i)!

+∂kf

∂uk(t, ζ(t))

ekn−1

k!+

∂kφ

∂uk(t, ζ(t))

akn−1

k!

≤ (k−1∑i=0

∂if

∂ui(t, wn−1)

1

(i)!

i−1∑j=0

ej−1n−1a

i−j−1n−1 )en

+ [∂kf

∂uk(t, ζ(t)) +

∂kφ

∂uk(t, ζ(t))]

ekn−1

k!≤ qn(t)en + Nek

n−1,

where

qn(t) =k−1∑i=1

∂i

∂uif(t, wn−1)

1

(i)!

i−1∑j=0

ei−1−jn−1 aj

n−1,

and N > 0 provides bound for ∂kf∂uk (t, wn−1(t)) on Ω. As limn→∞qn(t) = fu(t, w) <

1/4, we can choose λ < 1/4 and no ∈ N such that for n ≥ no, qn(t) < λ, we have

−e′′n(t)− λ(t)en(t) ≤ (qn(t)− λ)en(t) + Nekn−1 ≤ Nek

n−1,

poen(0)− qoe′n(0) = 0, poen(π) + qoe

′n(π) = 0,

whose solution isen(t) =

∫ π

0Gλ(t, s)Nek

n−1ds, t ∈ J.

Taking maximum over [0, π], we obtain

||en|| ≤ C||en−1||k,

8

274 AHMAD,KHAN

where C provides a bound on N∫ π0 Gλ(t, s)ds.

4. Concluding Remarks

In this paper, we have developed a generalized quasilinearization method with rapidconvergence for a somewhat more general type of boundary value problem. Someinteresting cases can be obtained by assigning particular values to the parametersinvolved in the boundary conditions and these are given below:

(i) The results of references [10] and [3] can be recorded as a special case by takingpo = 1, qo = 0, c1 = 0, c2 = 0 for φ = Mu2 and φ = Muk respectively.

(ii) For po = 0, qo = 1, c1 = 0, c2 = 0, we recover the results presented inreference [1].

(iii) The results of reference [2] appear as a special case by taking po = 1, qo =1, c1 = 0, c2 = 0.

References

[1] Bashir Ahmad, J.J. Nieto and N. Shahzad, The Bellman-Kalaba-Lakshmikantham quasilinearization method for Neumann problems, J. Math.Anal. Appl., 257(2001) 356-363.

[2] Bashir Ahmad, Rahmat A. Khan and S. Sivasundaram, Generalized quasilin-earization method for nonlinear boundary value problems, Dynamic SystemsAppl., 11(2002) 359-370.

[3] Albert Cabada, J.J. Nieto and Rafael Pita-da-veige, A note on rapid conver-gence of approximate solutions for an ordinary Dirichlet problem, Dyna. Contin.Discrete Impuls. Syst., 4(1998) 23-30.

[4] V. Lakshmikantham, An extension of the method of quasilinearization, J. Op-tim. Theory Appl., 82(1994) 315-321.

[5] V. Lakshmikantham, Further improvement of generalized quasilinearization,Nonlinear Anal., 27(1996) 315-321.

[6] V. Lakshmikantham, S. Leela and F.A. McRae, Improved generalized quasilin-earization method, Nonlinear Anal., 24(1995) 1627-1637.

[7] V. Lakshmikantham and N. Shahzad, Further generalization of generalizedquasilinearization method, J. Appl. Math. Stoch. Anal., 7(1994), 545-552.

[8] V. Lakshmikantham, N. Shahzad and J.J. Nieto, Method of generalized quasi-linearization for periodic boundary value problems, Nonlinear Anal., 27(1996)143-151.

9

275GENERALIZED QUASILINEARIZATION...

[9] V. Lakshmikantham and A.S. Vatsala, Generalized Quasilinearization for Non-linear Problems, Kluwer Academic Publishes, Dordrecht, 1998.

[10] J.J. Neito, Generalized quasilinearization method for a second order ordinarydifferential equation with Dirichlet boundary conditions, Proc. Amer. Math.Soc., 125(1997) 2599-2604.

10

276 AHMAD,KHAN

Existence and Construction of Finite Tight

Frames

Peter G. Casazza, Manuel T. Leon∗

Department of MathematicsUniversity of MissouriColumbia, MO 652511

pete@math.missouri.edu

February 13, 2006

Abstract

The space of finite tight frames of M vectors in RN with prescribednorms bjM

j=1 and frame constant A corresponds to the first N columnsof matrices in O(M) (the orthogonal group) with the property that thenorms of the first N elements of their rows equal the values aj = bj/

√A.

Then a2j ≤ 1 and

j=MPj=1

a2j = N .

For any sequence ajMj=1 of positive real numbers such that a2

j ≤ 1

andj=MPj=1

a2j = N , correspond to the first N rows of matrices in O(M) (the

orthogonal group) with the property that the norms of the first N elementsof their rows equal the values aj . we prove existence of such frames. Theproof is constructive, giving an embedding of O(N) × O(M) into thespace of such frames. In addition, for any such ajM

j=1, all solutions

can be factorised into M∗(M−1)2

− N∗(N−1)2

parameters and one matrixRN ∈ O(N), and each different solution provides a different embedding ofO(N) into the space of such frames. Replacing RN by any other element

of O(M), N∗(N−1)2

different solutions are obtained. A simple and fastalgorithm providing particular solutions is given. In particular, Parseval

frames correspond to sequences ajMj=1 such that aj =

qNM

. The results

provide an independent proof of some of the existence results in [CKLT02].All proofs are constructive. A MATLAB toolbox implementing all resultsis freely distributed by the authors.

∗P.G. Casazza and M. T. Leon were supported by NSF DMS 0102686

1

277JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,277-289,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

1 Introduction

Frames for Hilbert spaces play a fundamental role in a variety of importatnt ar-eas including signal/image processing [Cas00], multiple antenna coding [GKV98],perfect reconstruction filter banks [DGK02], quantum theory [EF01], [S] andmuch more. For applications, tight frames are preferred since they alow simplereconstruction formulas. A common problem in applications of frames is to finda tight frame ϕjM

j=1 for an N -dimensional Hilbert space HN with ‖ϕj‖Mj=1

prescribed in advance. In [CKLT02] the authors give necessary and sifficientconditions on ajM

j=1 so that there exists a tight frame ϕjMj=1 for HN with

‖ϕj‖ = aj , for all j = 1, . . . ,M . The main tool used in [CKLT02] is the notionof frame potentials introduced by Benedetto and Fickus [BF02]. However, thisis an existence proof while for applications we need an exact representation forthe frame. In this paper we will give an algorithm for constructing a family oftight frame vectors ϕjM

j=1 for HN with ‖ϕj‖ = aj , for all j = 1, . . . ,M . Thisprovides an independent proof of some of the results in [CKLT02] while at thesame time allowing exact constructions for the necessary frames. A MATLABtoolbox implementing all these results is freely distributed by the authors.

2 Definitions

Througout the paper we let N < M be fixed positive integers. First we recallthe well known fact that finite normalized tight frames correspond to the first Ncolumns of matrices in O(M), the M ×M orthonormal matrices. Backgrounddefinitions and a detailed proof of this fact is found in [GK01] and [BF02].

For the purposes of this paper it suffices to recall that ϕjMj=1 is a (finite)

tight frame in RN with frame constant A if for every vector y ∈ RN

Sy =j=M∑j=1

〈y, ϕj〉ϕj = Ay

The ”analysis frame operator” L of the frame is given by

Ly = 〈y, ϕj〉j=Mj=1

with adjoint operator L∗ given by

anj=Mj=1 −→

j=M∑j=1

ajϕj .

We have the diagram

RN L−→ l2(M) L∗−→ RM L−→ l2(M)

278 CASAZZA,LEON

Then the frame operator S and grammian operator G equal S = L∗L andG = LL∗, where L is an M ×N matrix and L∗ is an N ×M matrix given by

L =

ϕ∗1...ϕ∗M

, L∗ =(ϕ1| . . . |ϕM

)where ϕ∗j are row vectors and ϕj are column vectors. The tight frame conditiongives L∗L = AIN×N and the diagonal of the grammian is ‖ϕ1‖2

, . . . , ‖ϕM‖2.Now extend L√

Ato a matrix U ∈ O(M),

U =

ϕ1√

A| .

... |...

ϕM√A

| .

In [CKLT02] it is proved that for aj ≥ 0j=M

j=1 the conditionj=M∑j=1

a2j ≥ Na2

1

(suppose a1 is the largest aj) is sufficient for the existence of a tight frameϕjj=M

j=1 with ‖ϕj‖ = aj . The associated matrix U ∈ O(M) satisfies thecondition

N =j=M∑j=1

∥∥∥∥ ϕj√A

∥∥∥∥2

, and∥∥∥∥ φj√

A

∥∥∥∥2

≤ 1,

where A is the tight frame bound of ϕjj=Mj=1 .

The main result in this paper is then:

Theorem 2.1. For a sequence ajMj=1 of positive real numbers such that a2

j ≤ 1

andj=M∑j=1

a2j = N a matrix U ∈ O(M) exists such that, in squared norms, the

first N columns are a21 | .... |

...a2

M | .

.

We proceed to obtain such matrices. Their construction provides an inde-pendent proof of some of the results on existence of tight frames in [CKLT02].

Any sequence ajMj=1 of positive real numbers such that a2

j ≤ 1 andj=M∑j=1

a2j =

N will be called admissible. For RM ∈ O(M) let rleft,j denote the first N terms

of its jth row. Thenj=M∑j=1

‖ rleft,j‖2 = N . We will say that RM is a solution for

the admissible sequence ajMj=1 if ‖rleft,j‖ = aj for j = 1, . . . ,M .

279...FINITE TIGHT FRAMES

3 Factorisation of O(M)

The following factorisation is similar, with minor changes, to that described in[Vai93], p. 747. Every matrix in O(M) is obtained as a product of Givensrotations θ(t, j, k) ∈ O(M),j < k, where

θ(t, j, k) =

Ij−1,j−1 0 0 0 0

0 cos(t) 0 sin(t) 00 0 IM−j−k−2,M−j−k−2 0 00 − sin(t) 0 cos(t) 00 0 0 0 Ik−1,k−1

It is clear that

θ(t, j, k)−1 = θ(−t, j, k)

To begin with, every R2 ∈ O(2) is obtained as

R2 =(

cos(t) sin(t)− sin(t) cos(t)

) (µ 00 1

), µ =+

− 1

and thus can be stored with one parameter t and a sign +−1 (taking account of

the determinant ).Let RM ∈ O(M), and we will see how to factor RM . Let

tM−1M = π/2 if RM (M,M) = 0,

and

tM−1M = arg(RM (M,M)− iRM (M − 1,M)) otherwise. (1)

Then, if T = θ(tM−1M ,M − 1,M)RM , we have that T (M − 1,M) = 0 and

T (M,M) ≥ 0 . That is,

T =

. . . | T (1,M). . . | . . .. . . | T (M − 2,M). . . | 0. . . | T (M,M)

Now repeat the process on the M − 2 term of Mth column of T , thus obtainingtM−2M . The matrix

S = θ(tM−2M ,M − 2,M)T = θ(tM−2

M ,M − 2,M)θ(tM−1M ,M − 1,M)RM

is such that S(M − 2,M) = S(M − 1,M) = 0 and S(M,M) ≥ 0.

S =

. . . | S(1,M). . . | . . .. . . | S(M − 3,M). . . | 0. . . | 0. . . | S(M,M)

280 CASAZZA,LEON

Iterating the process we obtain t1M , . . . , tM−1M such that if

QM = θ(t1M , 1,M)θ(t2M , 2,M) . . . θ(tM−2M ,M − 2,M)θ(tM−1

M ,M − 1,M)RM

then

QM (1,M) = QM (2,M) = · · · = QM (M − 1,M) = 0

and QM (M,M) ≥ 0. Since QM ∈ O(M), we now have QM (M,M) = 1. Alsoand since QM (M,M) = 1, it follows that

QM (M, 1) = QM (M, 2) = · · · = QM (M,M − 1) = 0

This means that

QM =(RM−1 0

0 1

)(2)

for some matrix RM−1, and since QM ∈ O(M), we have RM−1 ∈ O(M − 1).Therefore,

RM = θ(−tM−1M ,M − 1,M)θ(−tM−2

M ,M − 1,M) . . . θ(−t2M , 2,M)θ(−t1M , 1,M)QM

for some matrix QM as in ( 2 ).Any choice of M − 1 parameters t1M , . . . , tM−1

M will induce embeddings

θt1M ,...,tM−1M

: O(M − 1) −→ O(M) (3)

taking RM−1 ∈ O(M − 1) into

θ(tM−1M ,M − 1,M)θ(tM−2

M ,M − 1,M) . . . θ(t2M , 2,M)θ(t1M , 1,M)QM

for QM as in ( 2 ). Multiplication by any of the matrices θ(t, j, k) will onlyintroduce changes in rows j and k. Parameters defined in ( 1 ) are defined upto integer multiples of 2π. Therefore for different (up to integer multiples of 2π)M − 1 parameters s1M , . . . , sM−1

M we obtain different embeddings θs1M ,...,sM−1

M.

The process can be iterated for RM−1. And composing such embeddings( 3 ) we obtain embeddings

O(N) −→ O(N + 1) −→ . . .O(M − 1) −→ O(M)

The resulting parametrisation and the embeddings of O(N) into O(M) willplay a significant role and we record them as follows.

Lemma 3.1. If N < M , then

(a) Every matrix in O(M) is parametrised by (M − 1) + (M − 2) + · · ·+2 + 1 = M(M−1)

2 parameters and one sign.

(b) Every matrix in O(M) is parametrised by (M − 1) + (M − 2) + · · ·+(N + 1) = M(M−1)

2 − N(N−1)2 parameters and a matrix RN ∈ O(N).

281...FINITE TIGHT FRAMES

(c) Both of these parametrisations are unique (up to integer multiples of2π for the real parameters).

Proof. Parts (a) and (b) are the well known representation of matrices in O(M)described above. For part (c), the sign of the decomposition is the determinantof the matrix. The real parameters t in ( 3 ) are defined up to integer multiplesof 2π.

The factorisation reformulates, for storage and computation purposes, thewell known fact that as a Lie Group, O(M) has dimension M(M−1)

2 and twocomponents, corresponding to the two possible values of the determinant.

Whenever we refer to a matrix in O(M) by giving the M ∗ (M − 1)/2 pa-rameters and a sign we refer to the decomposition of the Lemma 3.1 . For laterreference we now record both factorisations of arbitrary RM ∈ O(M).

RM = θ(tM−1M ,M − 1,M)θ(tM−2

M ,M − 1,M) . . . θ(t2M , 2,M)θ(t1M , 1,M)

θ(tM−2M−1,M−2,M−1)θ(tM−3

M−1,M−2,M−1) . . . θ(t2M−1, 2,M−1)θ(t1M−1, 1,M−1)

. . . θ(t23, 2, 3)θ(t13, 1, 3)

cos(t12) sin(t12)− sin(t12) cos(t12)

IM−2,M−2

µ 00 1

IM−2,M−2

,

µ =+− 1 (4)

and if any QN ∈ O(M) is given by

QN = θ(tN−1N , N − 1, N)θ(tN−2

N , N − 2, N) . . . θ(t2N , 2, N)θ(t1N , 1, N − 1)

. . . θ(t23, 2, 3)θ(t13, 1, 3)

cos(t12) sin(t12)− sin(t12) cos(t12)

IM−2,M−2

µ 00 1

IM−2,M−2

,

µ =+− 1

then

QN =(RN 00 IM−N,M−N

)(5)

where RN ∈ O(N), and

RM = θ(tM−1M ,M − 1,M)θ(tM−2

M ,M − 1,M) . . . θ(t2M , 2,M)θ(t1M , 1,M) . . .

θ(tNN+1, N,N + 1)θ(tN−1N+1, N − 1, N + 1) . . . θ(t1N+1, 1, N + 1)QN = ΨQN (6)

where

Ψ = θ(tM−1M ,M − 1,M)θ(tM−2

M ,M − 2,M) . . . θ(t2M , 2,M)θ(t1M , 1,M)

. . . θ(tNN+1, N,N + 1)θ(tN−1N+1, N − 1, N + 1) . . . θ(t1N+1, 1, N + 1)

282 CASAZZA,LEON

4 Existence and Construction Results

Lemma 4.1. Let RM ∈ O(M) be a solution for the admissible sequence ajMj=1,

and let RM be factorised as in ( 6 ) by means of M(M−1)2 − N(N−1)

2 parametersand a matrix RN ∈ O(N) as in ( 5 ). For any matrix SN ∈ O(N), let

PN =(SN 00 IM−N,M−N

)Then

SM = θ(tM−1M ,M − 1,M)θ(tM−2

M ,M − 2,M) . . . θ(t2M , 2,M)θ(t1M , 1,M) . . .

θ(tNN+1, N,N + 1)θ(tN−1N+1, N − 1, N + 1) . . . θ(t1N+1, 1, N + 1)PN = ΨPN

is a solution for the admissible sequence ajMj=1. This provides an embedding

of O(N) into the space of solutions for the admissible sequence ajMj=1.

Proof. We will show that the values ajMj=1 depend on the parameters

tM−1M , tM−2

M , . . . t2N+1, t1N+1 (7)

and are independent of the particular choice of RM ∈ O(N).Let rl be the lth row of RN for l = 1, . . . , N . Then 〈rl, rk〉 = δl,k. Let xj be

the first N terms of the jth row of RM for j = 1, . . . ,M . Multiplication by aGivens rotation will replace the first N terms of any row by a linear combinationof rlM

l=1. At the end of the process

xj =l=N∑l=1

λjl rl

where the coefficients λl depend only on the M(M−1)2 − N(N−1)

2 parameters in( 7 ) and are independent of the rl given by RN . Since the vectors rl areorthonormal,

‖xj‖2 = 〈xj , xj〉 = 〈l=N∑l=1

λjl rl,

l=N∑l=1

λjl rl〉 =

l=N∑l=1

(λjl )

2

Thus under replacement of RN by another matrix in O(N) the values of ‖xj‖2

do not change. Again from ( 6 ) it follows that the parameters in ( 7 ) inducean embedding of O(N) into the space of solutions for the admissible sequenceajM

j=1.

Remark 4.2. The equations giving λjl in terms of ( 7 ) are in practical terms

unmanageable. They are given in terms of the M(M−1)2 − N(N−1)

2 parameters.

Theorem 4.3. Let N < M be fixed. If ajMj=1 is an admissible sequence, then

283...FINITE TIGHT FRAMES

(a) There is a solution for ajMj=1

(b) Moreover, there is an embedding from O(N)×O(M −N)

O(N)×O(M −N)

into the space of all solutions for ajMj=1. The embedding will take the

pair of matrices RN ∈ O(N) and RM−N ∈ O(M −M) into

Ψ(RN 00 RM−N,M−N

)

where the matrix Ψ is determined by the admissible sequence ajMj=1.

Proof. The proof is just an algorithm. Let Ψ = IM,M We start with the matrixO ∈ O(M) given by

O =(RN 00 RM−N

)where RN and RM−N are arbitrary matrices in O(N) and O(M −N) respec-tively. The fact that there is no other requirement on RN and RM−N willprovide the clue for the proof of the second claim of the lemma. We will con-struct matrices On ∈ O(M) for n = 1, . . . ,M − 1. We will denote

onleft,j = first N terms of jth row of On and

onright,j = last M −N terms of jth row of On

Once the values a21, . . . , a

2M−1 have been attained the problem is solved since

each On is an orthogonal matrix. The solution will be OM−1. Let

t1 = arg(a1 + i√

1− a21)

and rename

Ψ = θ(t1, 1,M)Ψ.

Let

O1 = θ(t1, 1,M)O

Then O1 = ΨO, and∥∥o1left,1

∥∥2= a2

1, and∥∥o1left,M

∥∥2= 1− a2

1.

Also o1left,M remains orthogonal to o1left,j for j ≥ 2 and o1right,M remains or-thogonal to o1right,j for j ≤ M − 1. The values of the squared norms of o1left,j

284 CASAZZA,LEON

and o1right,j are described by the diagram:

a21 | 1− a2

1

1 | 01 | 0... |

...1 | 0

−−− | − −−−0 | 1... |

...0 | 1

1− a21 | a2

1

Now one of the following applies:

(CASE I) 1 ≤ a21 + a2

2, or

(CASE II) 1 > a21 + a2

2

In CASE I, take t2 a solution of

sin(t2)2 =1− a2

2

a21

and O2 = θ(t2, 2,M)O1

Then∥∥o2left,2

∥∥2=

∥∥cos(t2)o1left,2 + sin(t2)o1left,M

∥∥2 (1)=

∥∥cos(t2)o1left,2

∥∥2

+∥∥sin(t2)o1left,M

∥∥2= cos(t2)2

∥∥o1left,2

∥∥2+ sin(t2)2

∥∥o1left,M

∥∥ (2)= cos(t2)2

+ sin(t2)2(1− a21) = 1− sin(t2)2(a2

1) = 1− 1− a22

a21

a21 = a2

2

where equality (1) holds because o1left,2 and o1left,M are orthogonal and equality(2) holds because o1left,2 is a unitary vector (in RN ). The values of the squarednorms of o2left,j and o2right,j are described by the diagram:

a21 | 1− a2

1

a22 | 1− a2

2

1 | 0... |

...1 | 0

−−−−− | − −−−−−0 | 1... |

...0 | 1

2− a21 − a2

2 | a21 + a2

2 − 1

285...FINITE TIGHT FRAMES

Now for j = 3, . . . , N , we have o2left,j = o1left,j (they come from RN ) so theyare unitary and they remain orthogonal to o2left,M since the latter has beenobtained as a linear combination of o1left,j for j ≤ 2.

Similarly, for j = N + 1, . . . ,M − 1, we have o2right,j = o1right,j (they comefrom RM ) so they are unitary and remain orthogonal to o2right,M since the latteris just a scalar multiple of o1right,M .

In CASE II, take t2 a solution of

sin(t2)2 =a22

1− a21

and let O2 = θ(t2,M − 1,M)O1

Then∥∥o2right,M−1

∥∥2=

∥∥cos(t2)o1right,M−1 + sin(t2)o1right,M

∥∥2 (1)=

∥∥cos(t2)o1right,M−1

∥∥2

+∥∥sin(t2)o1right,M

∥∥2= cos(t2)2

∥∥o1right,M−1

∥∥2+ sin(t2)2

∥∥o1right,M

∥∥(2)= cos(t2)2 + sin(t2)2a2

1 = 1− sin(t2)2 + sin(t2)2a21

= 1 + sin(t2)2(a21 − 1) = 1 +

a22

1− a21

(a21 − 1) = 1− a2

2

where equality (1) holds because o1right,M−1 and o1right,M are orthogonal andequality (2) holds since o1right,M−1 is a unitary vector (in RM−N ). The valuesof the squared norms of o2left,j and o2right,j are described by the diagram:

a21 | 1− a2

1

1 | 01 | 0. . . | . . .1 | 0

−−−−− | − −−−−0 | 1. . . | . . .0 | 1a22 | 1− a2

2

1− a21 − a2

2 | a21 + a2

2

Now for j = 2, . . . , N , we have o2left,j = o1left,j (they come from RM−N ).

Hence, they are unitary and they remain orthogonal to o2left,M since the latterhas been obtained as a linear combination of o1left,j for j < 2.

Similarly, for j = N + 1, . . . ,M − 2, we have o2right,j = o1right,j (they comefrom RM ). So they are unitary and remain orthogonal to o2right,M since thelatter has been obtained as a linear combination of o1right,j for j > M − 2.

In either case, by renaming

Ψ = θ(t2,m,M)Ψ

286 CASAZZA,LEON

where m is given by the CASE used, we have

O2 = ΨO

After k applications of CASE I and l applications of CASE II we obtainthe matrix Ok+l+1.The values of the squared norms of ok+l+1

left,j and ok+l+1right,j are

described (up to reordering of ajMj=1) by the diagram:

a21 | 1− a2

1

a22 | 1− a2

2

. . . | . . .a2

k+1 | 1− a2k+1

1 | 0. . . | . . .1 | 0

−−−−−−−−−− | − −−−−−−−−0 | 1. . . | . . .0 | 1. . . | . . .

a2k+l+1 | 1− a2

k+l+1

. . . | . . .a2

k+3 | 1− a2k+3

a2k+2 | 1− a2

k+2

k + 1− a21 − · · · − a2

k+l+1 | a21 + · · ·+ a2

k+l+1 − k

Now the two cases correspond to :

(CASE I) k + 1 ≤ a21 + · · ·+ a2

k+l+1 + a2k+l+2, or

(CASE II) k + 1 > a21 + · · ·+ a2

k+l+1 + a2k+l+2

In CASE I, take tk+l+2 a solution of

sin(tk+l+2)2 =1− a2

k+l+2

a21 + · · ·+ a2

k+l+1 − k

and

Ok+l+2 = θ(tk+l+2, k + 2,M)Ok+l+1

In CASE II, take tk+l+2 a solution of

sin(tk+l+2)2 =a2

k+l+2

k + 1− a21 − · · · − a2

k+l+1

and

Ok+l+2 = θ(tk+l+2,M − l − 1,M)Ok+l+1

287...FINITE TIGHT FRAMES

In either case, by renaming

Ψ = θ(tk+l+2,m,M)Ψ

where m is given by the CASE used, we have

Otk+l+2 = ΨO.

Now it remains to prove that CASE I eventually takes place whenever k <N − 1 and when k = N − 1 only CASE II can apply.

Suppose k < N − 1. Then k + 1 ≤ N − 1. At the same time

a21 + · · ·+ a2

k+l+1 + a2M−1 ≥ N − 1

giving

k + 1 ≤ N − 1 ≤ a21 + · · ·+ a2

k+l+1 + a2M−1

so CASE I will eventually apply .If k = N − 1 and CASE I applies at some step,

k + 1 = N ≤ a21 + · · ·+ a2

k+l+2 ≤ N

This could only happen if N = a21 + · · · + a2

k+l+2, so ak+l+3 = · · · = aM−1 =aM = 0. In this case Ok+l+1 is a solution for ajM

j=1. It is possible to reorderthe sequence aj so that in the algorithm first take place N − 1 times CASE Iand then It is clear from the construction that

OM−1 = ΨO.

Finally, the parameters t1, . . . , tM−1 and the matrix Ψ were determined bythe norms a2

1, . . . , a2M−1 and so is Ψ. Replacement of O by any other

O′ =(R′N 00 R′M−N

)would have lead to another solution

O′M−1 = ΨO′

It is clear that multiplication by a unitary matrix is an injection. The claim isthus proved.

It must be pointed out that the embedding of O(N)×O(M−M) into O(M)does not fill out the space of solutions. Using alternative algorithms it is possibleto construct solutions which do not belong to the image of the embedding, thatis, can not be factorised as

ψO = ψ

(RN 00 RM−N

)

288 CASAZZA,LEON

References

[BF02] J.J. Benedetto and M. Fickus. Finite normalized tight frames.2002. Preprint.

[Cas00] P.G. Casazza. The art of frame theory. Taiwan Jour. of Math.,4(2):129-201, 2000. To appear.

[CKLT02] P.G. Casazza, J. Kovacevic, M.T. Leon, and J.C. Tremain. Cus-tom built tight frames. 2002. Preprint.

[DGK02] P.L. Dragotti, V.K. Goyal, and J. Kovacevic. Filter bank frameexpansions with erasures. IEEE Trans. Inform. Th., special is-sue in Honor of Aaron D. Wyner, 46(6):1439-1450, 2002, InvitedPaper.

[EF01] Y. Eldar and G.D. Forney. Optimal tight frames and quantummeasurement. 2001. Preprint.

[GK01] V.K. Goyal and J. Kovacevic. quantized frame expansions witherasures. Appl. Comput. Harmon. Anal., 10(3):203-233, 2001. Toappear.

[GKV98] V.K. Goyal, J. J. Kovacevic, and M. Vetterli. Multiple descrip-tion transform coding: Robustness to erasures using tight frameexpansions. Proc. IEEE Int. Symp. on Inform. Th., 1998. Cam-bridge, MA.

[Vai93] P.P. Vaidyanathan. Multirate systems and filters banks. PrenticeHall, Englewood Cliffs, N.J., 1993.

[S] E. Soljanin. Tight frames in quantum information the-ory. DIMACS Workshop on Source Coding and Har-monic Analysis. Rutgers, New Jersey, May 2002,http://dimacs.rutgers.edu/Workshops/Modern/abstracts.html.

289...FINITE TIGHT FRAMES

290

Semigroups and Genetic Repression

Genni FragnelliDipartimento di Ingegneria dell’Informazione,

Universita degli Studi di Siena,Via Roma 56, 53100 Siena, Italy

fragnelli@dii.unisi.it

Abstract

In this article we start from a model presented in [5], [16] and in [21,Introduction]. We explain why this model is unrealistic and propose a sys-tem of modified equations. Our approach to the study of these equationsis based on semigroup techniques. In particular we use the theory devel-oped in [9] and [10]. The basic idea is to rewrite the equations that expressthe biological model as a delay equation with nonautonomous past on asuitable state space and then to study their well-posedness and stability.

2000 Mathematics Subject Classification:34G10, 35B40, 47A10, 47D06, 47N60.Key words and phrases: Genetic repression, partial differential equations, semi-groups, stability.

1 Introduction

Systems of delayed reaction diffusion equations have been used frequently inmodeling genetic repression, (see, e.g., J. Wu [21, Introduction]). The studyof these equations goes back to the 60’s with B.C. Goodwin, F. Jacob and J.Monod (see, e.g.,[12], [13] or [14]).

In particular, Goodwin suggested that time delays caused by the processesof transcription and translation as well as spatial diffusion of reactants couldplay a role in the behavior of this system. The later studies of these modelsincluded either time delays (see, e.g., [2], [15] or [20]) or the spatial diffusion(see, e.g. [17]).

In this paper we study the one-dimensional model presented, e.g., in [5],[16], and in [21, Introduction] which includes spatial diffusion and time delay.According to this model the repressor in the cytoplasm at time t and positionx depends on the mRNA (messenger ribo nuclein acid) that was at time t − rat the same position x, where r > 0. This assumption, however, is unrealistic.For this reason we present a system of modified equations, which take intoaccount the diffusion in the past of the mRNA contained in the cytoplasm. Inparticular we modify the equations associated to this genetic repression in such

291JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,291-305,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

a way and then study them with the theory developed in [9] and in [10]. Inparticular in Section 2 we explain this model following [16] or [21, Introduction]and then justify the modifications of these equations. In Section 3 we considerthe model without delay. In Section 4 we rewrite the biological model as adelay equation with nonautonomous past, studying its well-posedness. The lastsection is dedicated to a stability analysis.

2 The Biological Model

2.1 The Classical Equations

In this section we briefly explain the biological model and rewrite it as a systemof equations following, e.g., [5], [16] and [21, Introduction].

The eucharyotic cell Ω consists of two compartments where the most im-portant chemical reactions happen. Such compartments are enclosed withinthe cell wall, unpermeable to the mRNA and to the repressor, and separatedby the permeable nuclear membrane. The first compartment ω is the nucleuswhere mRNA is produced. The second compartment, denoted by Ω \ ω, isthe cytoplasm in which the ribosomes are randomly dispersed. The process oftranslation and the production of the repressor occurs here.

We denote by ui and vi the concentrations of mRNA and of the repressor,respectively, in ω if i = 1 and in Ω \ ω if i = 2. These two species interactto control each other’s production. In the nucleus ω, mRNA is transcribedfrom the gene at a rate depending on the concentration of the repressor v1. ThemRNA leaves ω and enters the cytoplasm Ω\ω where it diffuses and reacts withribosomes. Through the delayed process of translation, a sequence of enzymesis produced which in turn produce a repressor v2. This repressor comes back toω where it inhibits the production of u1. This process can be written, in onedimension, as the following system of equations.

du1(t)dt

= h(v1(t + r1))− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t + r2, x), t ≥ 0, x ∈ (0, 1],

(1)with boundary conditions

∂u2(t, 0)∂x

= −β1(u2(t, 0)− u1(t)), t ≥ 0,

∂v2(t, 0)∂x

= −β∗1(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, 1)∂x

=∂v2(t, 1)

∂x= 0, t ≥ 0,

(2)

292 FRAGNELLI

and initial conditions u1(s) = f1(s),v1(s) = g1(s),u2(s, x) = f2(s, x),v2(s, x) = g2(s, x),

(3)

for x ∈ (0, 1] and s ∈ [−ri, 0], i = 1, 2. As in [16], the interval (0, 1] correspondsto the cytoplasm Ω \ ω and the nucleus ω is localized in 0. The constants bi

are kinetic rates of decay, ai are rates of transfer between ω and Ω \ ω andare directly proportional to the concentration gradient. The constants Di arethe diffusivity coefficients, and the constant c0 is the production rate for therepressor. The function h is a decreasing function and represents the production

of mRNA. It is of the form h(x) =1

1 + kxρ, where k is a kinetic constant and ρ

is the Hill coefficient. The delay −r1 ≥ 0 is the transcription time, i.e., the timenecessary to the transcription reaction, and −r2 ≥ 0 is the translation time. Theconstants β1 and β∗1 are the constants of Fick’s law (see, e.g., [1, Chapter VI]or [19, Chapter V]). We have to underline the fact that all biological constantsare positive.

2.2 A better model

According to model (1), the variation of the repressor v2(t, x) at time t andposition x depends on the mRNA which was at time t+ r2 at the same positionx. Clearly, this is unrealistic because the mRNA is subject to a diffusion processin the time interval [t + r2, t], where we recall that r2 < 0.

To include such a phenomenon in our model, we suppose, for simplicity, thatthis migration is given by a diffusion of the form et∆D , where ∆D := d2

dx2 is theLaplacian with Dirichlet boundary conditions.

To be more precise, we take the Laplacian ∆D with domain

D(∆D) := f ∈ W 2,1[0, 1] : f(0) = f(1) = 0

on the Banach space L1[0, 1].Then the evolution family U :=(U(t, s))−1≤t≤s≤0 solving the corresponding

Cauchy problem (see [9, Example 6.1]) is

U(t, s) := T (s− t), −1 ≤ t ≤ s ≤ 0, (4)

where (T (t))t≥0 = (et∆D )t≥0 is the heat semigroup on L1[0, 1]. Observe thatthe growth bound of the evolution family (U(t, s))−1≤t≤s≤0 and of the heatsemigroup (T (t))t≥0 coincide and are negative, i.e.

−π2 = ω0(U) = ω0(T (·)) < 0. (5)

Thus, assuming that the mRNA in the cytoplasm is subject to a diffusionin the past of the form et∆D , the term u2(t + r2, x) must be modified. Let

293SEMIGROUPS AND GENETIC...

u2(t + r2, x) be the modification of u2(t + r2, x) governed by (U(t, s))−1≤t≤s≤0,i.e.

u2(t + r2, x) : =

U(r2, 0)u2(t + r2, x), 0 ≤ r2 + t,U(r2, t + r2)f2(t + r2, x), r2 + t ≤ 0,

=

T (−r2)u2(t + r2, x), 0 ≤ r2 + t,T (t)f2(t + r2, x), r2 + t ≤ 0.

(6)

Then, the system (1) becomes

du1(t)dt

= h(v1(t + r1))− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t + r2, x), t ≥ 0, x ∈ (0, 1].

(7)

Remark 2.1. If ∆D is substituted by ∆N , i.e. the Neumann Laplacian withdomain

D(∆N ) := f ∈ W 2,1[0, 1] : f ′(0) = f ′(1) = 0,

then for the growth bound of the associated heat semigroup (T (t))t≥0 we onlyhave ω0(T ) ≤ 0.

3 Semigroup without Delay

We want to study the system (7) with the theory developed in [9] and [10]. Tothis aim we have to rewrite the genetic repression as a delay equation of theform

(NDE)

W (t) = BW (t) + ΦWt, t ≥ 0W (0) = y ∈ X

W0 = f ∈ Lp([−1, 0], X),

on some Banach space X, where (B, D(B)) is the generator of a strongly continu-ous semigroup (S(t))t≥0 on X, the delay operator Φ : W 1,p([−1, 0], X) → X isa linear operator, p ≥ 1 and the modified history function Wt : [−1, 0] → X(see [9, Definition 3.2]) is given by

Wt(τ) :=

U(τ, 0)W (t + τ) for 0 ≤ t + τ,

U(τ, t + τ)f(t + τ) for t + τ ≤ 0(8)

for some evolution family (U(t, s))−1≤t≤s≤0.

In the definition of the modified history function Wt two time variables tand τ appear. The variable t can be interpreted as the absolute time and τ asthe relative time. The meaning of (NDE) is that if we start with the historyfunction f , this function is shifted to the left by −t, and for values greater than

294 FRAGNELLI

−t the value of the solution is given by the delay operator Φ applied to theshifted function.

In order to rewrite (7) as a delay equation with nonautonomous past weproceed as follows.

As a first step we ignore the terms with delay, i.e. we consider the system

du1(t)dt

= −b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x), t ≥ 0, x ∈ (0, 1],

(9)

with initial conditions (3).To rewrite (9) as an ordinary differential equation of the form

W (t) = BW (t), t ≥ 0,W (0) = y ∈ X,

(10)

for a linear operator (B, D(B)) on a Banach space X, we take as X the spaceX := R2 × (L1[0, 1])2 and as B the operator

B :=

−b1 − a1 0 a1δ0 0

0 −b2 − a2 0 a2δ0

0 0 D1∆− b1 00 0 0 D2∆− b2

, (11)

with domain

D(B) :=

(xyfg

)∈ R2 × (W 2,1[0, 1])2 : L

(fg

)=(

xy

)and f ′(1) = g′(1) = 0

where δ0f := f(0) for f ∈ C[0, 1], ∆ := d2

dx2 and the operator L : (W 2,1[0, 1])2 →R2 is defined by

L

(fg

)=

(f ′(0)β1

+ f(0)g′(0)β∗1

+ g(0)

). (12)

Remark 3.1. The Laplacian ∆ in the definition of the operator matrix B has”maximal” domain W 2,1[0, 1] and thus is different from the Dirichlet Laplacian∆D that governs the diffusion in the past.

Let nowW (t) := (u )1 (t)v1(t)u2(t)v2(t). (13)

Then the ordinary differential equation (10) and the system (9) are ”equivalent”.Thus, to prove the existence of a solution of (9) is sufficient to prove that(B, D(B)) is the generator of a strongly continuous semigroup (S(t))t≥0 . In

295SEMIGROUPS AND GENETIC...

this case the solution of (10) is given by W (t) = S(t)y. To this aim we rewriteB in the form

B = B0 + Θ−Π :=

0 0 0 00 0 0 00 0 D1∆ 00 0 0 D2∆

+

0 0 a1δ0 00 0 0 a2δ0

0 0 0 00 0 0 0

−

b1 + a1 0 0 0

0 b2 + a2 0 00 0 b1 00 0 0 b2

,

with domains D(B0) = D(B), D(Θ) = R2 × (W 1,1[0, 1])2 and Π ∈ L(X).

Proposition 3.2. The operator (B0, D(B0)) generates an analytic semigroup.

Proof. Let

Bm :=(

D1∆ 00 D2∆

)with the maximal domain D(Dm) = (W 2,1[0, 1])2 and B0 := Bm

|Ker L. Then,

by [8, Chapter VI, Section 4], B0 generates an analytic positive semigroup(S0(t))t≥0.

Applying [6, Corollary 2.8] to the operator (B0, D(B0)), the assertion follows.

The next definition will be needed to state Lemma 3.4.

Definition 3.3 (see [8], Definition III.2.1). Let A : D(A) ⊂ X → X be a linearoperator on the Banach space X. An operator B : D(B) ⊂ X → X is calledA−bounded if D(A) ⊆ D(B) and if there exist constants a, b ∈ R+ such that

‖Bx‖ ≤ a‖Ax‖+ b‖x‖ (14)

for all x ∈ D(A). The A− bound of B is

a0 := infa ≥ 0 : there exists b ∈ R+ such that (14) holds.

Lemma 3.4. The operator Θ is B0−bounded with B0−bound a0 = 0.

Proof. Obviously, D(B0) ⊆ D(Θ) and for

V =

(xyfg

)∈ D(B0),

we have

‖ΘV‖ =

∥∥∥∥∥∥∥

a1f(0)a2g(0)

00

∥∥∥∥∥∥∥ ≤ ‖a1f(0)‖+ ‖a2g(0)‖.

296 FRAGNELLI

Since

‖B0V‖ =

∥∥∥∥∥∥∥

00f ′′

g′′

∥∥∥∥∥∥∥ ,

it suffices to prove that for arbitrary small a, b > 0 there exist constants c, d ∈R+, such that ‖f(0)‖ ≤ a‖f ′′‖1 + b‖f‖1 and ‖g(0)‖ ≤ c‖f ′′‖1 + d‖f‖1.

Using the fundamental theorem of calculus and the fact that the operatord

dxis

d2

dx2−bounded with

d2

dx2−bound 0 (see [8, Example III.2.2]), the assertion

follows.

The following proposition is an easy consequence of Proposition 3.2 andLemma 3.4.

Proposition 3.5. The operator (B, D(B)) generates a positive analytic semi-group (S(t))t≥0 on X.

Proof. By the previous proposition, the operator Θ and hence also Θ + Π isB0−bounded with B0−bound a0 = 0. Using [8, Theorem III.2.10], one has thatB0 + Θ + Π generates an analytic semigroup. Moreover, by [6, Proposition 5.2],this semigroup is positive as well.

4 The Genetic Repression as a Delay Equationwith Nonautonomous Past

As we mentioned at the beginning of Section 3, we want to rewrite (7) as a delayequation with nonautonomous past, i.e. in the form (NDE) (see the previoussection). To this end and for the sake of simplicity, assume r1 = r2 = −1 and,instead of h(x) = 1

1+kxρ , consider its linearization, i.e. the function h : R → Rgiven by

h(x) = − kρ(ve1)

ρ−1

(1 + k(ve1)ρ)2

x, x ∈ R. (15)

Here ve1 is such that (ue

1, ve1, u

e2, v

e2)

T is one of the steady-state solutions of (7) (see[16, Section 5]). Hence, instead of (7) we consider the simplified and linearizedsystem

du1(t)dt

= cv1(t− 1)− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t− 1, x), t ≥ 0, x ∈ (0, 1],

(16)

297SEMIGROUPS AND GENETIC...

where c := − kρ(ve1)ρ−1

(1+k(ve1)ρ)2 . Let Φ : W 1,p([−1, 0], X) → X be the delay operator

given by

Φ :=

0 cδ−1 0 00 0 0 00 0 0 00 0 c0δ−1 0

, (17)

where δ−1f = f(−1) for f ∈ W 1,p([−1, 0], R) → R and take

U(t, s) := ( I ) dIdU(t, s)Id = ( I ) dIdT (s− t)Id for − 1 ≤ t ≤ s ≤ 0. (18)

Recall that (T (t))t≥0 denotes the heat semigroup on L1[−1, 0]. Then it is easyto prove that the system (16) is equivalent to W (t) = BW (t) + ΦWt, t ≥ 0

W (0) = y ∈ X,

W0 = f ∈ Lp([−1, 0], X),(19)

wheref = ( f )1 g1f2g2, (20)

W (t) is as in (13) and the modified history function Wt : [−1, 0] → X is definedby

Wt(τ) :=

U(τ, 0)W (t + τ) for 0 ≤ t + τ,

U(τ, t + τ)f(t + τ) for t + τ ≤ 0.

Here U := (U(t, s))−1≤t≤s≤0 is the evolution family defined in (18). Thanks tothe equivalence between (16) and (19), to prove the existence of a solution of(16) it is sufficient to study the well-posedness of (NDE).

Definition 4.1. 1. We call a function W : R → X a classical solution of(NDE) if

(i) W ∈ C(R, X) ∩ C1(R+, X),

(ii) W (t) ∈ D(B), Wt ∈ D(Φ), t ≥ 0

(iii) W satisfies (NDE) for all t ≥ 0.

2. We call (NDE) well-posed if

(i) for every(

yf

)in a dense subspace S ⊆ X × Lp(R−, X) there is a

unique (classical) solution u(y, f, ·) of (NDE) and

(ii) the solutions depend continuously on the initial values, i.e., if a se-quence

(yn

fn

)in S converges to

(yf

)∈ S, then u(yn, fn, t) converges

to u(y, f, t) uniformly for t in compact intervals of R−.

298 FRAGNELLI

The next two results are concerned with the well-posedness of (19) and as aconsequence the existence of the solutions of (16).

Take

C :=(B Φ0 G

), (21)

with domainD(C) :=

(yf

)∈ D(B)×D(G) : f(0) = y (22)

on the product space E := X×Lp([−1, 0], X). Here the operator G is the matrix

G :=

d

dσ 0 0 00 d

dσ 0 00 0 G 00 0 0 d

dσ

,

with domain

D(G) := (W 1,p[−1, 0])2 ×D(G)×W 1,p([−1, 0], L1[0, 1]),

ddσ is the weak derivative and (G, D(G)) is the closure of

Af = f ′ + f ′′, (23)

for f in an appropriate subspace of D(G) (for details see [11, Proposition 3.1]).The following theorem follows by [9, Theorem 4.3].

Theorem 4.2. For the function h given by (15), the operator (C, D(C)) gener-ates a strongly continuous semigroup (T (t))t≥0 .

As an immediate consequence of [9, Theorem 4.5], we obtain the next theo-rem.

Theorem 4.3. If the function h is given by (15), then the delay equation (19)is well-posed. Thus the classical solutions u of (16) are given by

u(t) =

π1

(T (t)

(yf

)), t ≥ 0,

f(t), t ≤ 0,

for every(

yf

)∈ D(C). Here π1 is the projection onto the first component X of

E.

5 Stability

In this section our goal is to study the stability of the solutions of the system(16), in particular we want to find conditions such that the solutions decayexponentially. In this context positivity will be very helpful. First of all we have

299SEMIGROUPS AND GENETIC...

to observe that the function h in (15) is negative, which implies that the delayoperator Φ is negative. Then we consider the system

du1(t)dt

= −cv1(t− 1)− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t− 1, x), t ≥ 0, x ∈ (0, 1],

(24)where c is the constant in (16). Let Φ+ be the delay operator associated to thismodified system, i.e.

Φ+ :=

0 −cδ−1 0 00 0 0 00 0 0 00 0 c0δ−1 0

. (25)

Hence Φ+ is positive, thus also the semigroup T + :=(T +(t))t≥0 generated by

C+ :=(B Φ+

0 G

),

with D(C+) = D(C) is positive (see [7, Theorem 3.2]).Observe that

|T (t)| ≤ T +(t)

for all t ≥ 0 and hence‖T (t)‖ ≤ ‖T +(t)‖

for all t ≥ 0 (see [18, Proposition II.4.1]). Thus, if we want to find conditionsimplying the exponential decay of the semigroup (T (t))t≥0 , it is sufficient to findthese conditions for the semigroup (T +(t))t≥0 . To this aim we first estimatethe spectral bound of C.

For each λ ∈ C with <λ > ω0(U), where U denotes the evolution familyassociated to the heat semigroup (see (4)), we consider the bounded operatorEλ : X → Lp([−1, 0], X) defined as

Eλ :=

ελ 0 0 00 ελ 0 00 0 ελ 00 0 0 ελ

.

Here the bounded operators ελ : R → W 1,p([−1, 0], L1[0, 1]), ελ : L1[0, 1] →Lp([−1, 0], L1[0, 1]) and ελ : L1[0, 1] → Lp([−1, 0], L1[0, 1]) are defined as

(ελx)(s) := eλsx, for s ∈ [−1, 0], x ∈ R,

300 FRAGNELLI

(ελf)(s) := eλsf, for s ∈ [−1, 0], f ∈ L1[0, 1],

and

(ελf)(s) := eλsU(s, 0)f = eλsT (−s)f, for s ∈ [−1, 0], f ∈ L1[0, 1],

respectively, for (T (t))t≥0 as in (4).The following corollary follows by the positivity of (T +(t))t≥0 and by [7,

Theorem 4.1].

Corollary 5.1. For the spectral bounds of G, B+Φ+E0 and C+ the next relationholds

if s(G) and s(B + Φ+E0) < 0, then s(C+) < 0. (26)

For the proof of this corollary it is sufficient to observe (see [7, Theorem 4.1])that we can rewrite C+ as

C+ :=(B 00 G0

)(Id 0−E0 Id

)+(

0 Φ+

0 0

),

where the operator G0 is the following matrix

G0 :=

d

dσ 0 0 00 d

dσ 0 00 0 G0 00 0 0 d

dσ

,

with domain

D(G0) := (W 1,p[−1, 0])2 ×D(G0)×W 1,p([−1, 0], L1[0, 1]).

Here (G0, D(G0)) is the operator defined by

G0f := Gf and D(G0) := D(G) ∩ f(0) = 0,

where (G, D(G)) is given by (23).

Remark 5.2. The negativity of the spectral bound of G, s(G), follows by (5)and by the fact that d

dσ is the generator of the nilpotent translation semigroup(Tl(t))t≥0 on L1[0, 1] (see, e.g., [3, Lemma 1.1.12]).

Thus the problem reduces to find conditions such that s(B+Φ+E0) < 0. Tothis purpose, we firse calculate

B + Φ+E0 =

−b1 − a1 −c a1δ0 0

0 −b2 − a2 0 a2δ0

0 0 D1∆− b1 00 0 c0δ−1ε0 D2∆− b2

,

D(B + Φ+E0) = D(B),

301SEMIGROUPS AND GENETIC...

where we recall

D(B) :=

(xyfg

)∈ R2 × (W 2,1[0, 1])2 : L

(fg

)=(

xy

)and f ′(1) = g′(1) = 0

.

Hence the matrix B + Φ+E0 is of the form(A B0 Dm

)on X. The operator B is a diagonal matrix and A and Dm are triangularmatrices, i.e.

B :=(

a1δ0 00 a2δ0

),

A :=(−b1 − a1 −c

0 −b2 − a2

),

and

Dm :=(

D1∆− b1 0c0δ−1ε0 D2∆− b2

)with D(B) := (W 2,1[0, 1])2, D(A) := R2 and D(Dm) :=

(fg

)∈ (W 2,1[0, 1])2 :

f ′(1) = g′(1) = 0, respectively.Now let D ⊂ Dm with D(D) = KerL, where, as above, L : (W 2,1[0, 1])2 →

R2 is defined by

L(

fg

):=

(f ′(0)β1

+ f(0)g′(0)β∗1

+ g(0)

).

Then for the matrix

Dm :=(

A B0 Dm

),

with domain D(Dm) = R2 ×D(Dm) we have

B + Φ+E0 ⊆ Dm,

and the operator B + Φ+E0 is one-sided coupled (see [7, Definition 1.1]). Forthe spectral buond of B + Φ+E0 the following proposition holds.

Proposition 5.3. The spectral bounds of the operators B + Φ+E0, D and A +BL0 satisfy

s(B + Φ+E0) < 0 ⇔ s(D) < 0 and s(A + BL0) < 0,

where L0 := (L|ker Dm)−1 : R2 → ker(Dm) ⊆ (L1[0, 1])2.

302 FRAGNELLI

The proof of this proposition follows again by [7, Theorem 4.1], rewritingB + Φ+E0 as

B + Φ+E0 :=(

A 00 D

)(Id 0−L0 0

)+(

0 B0 0

).

Now, let C1 be the operator C1 ⊆ D1∆− b1 with domain

D(C1) := f ∈ (W 2,1[0, 1], R) : f ′(1) = 0 and f ′(0) = −β1f(0)

and C2 the operator C2 ⊆ D2∆− b2 with domain

D(C2) := f ∈ (W 2,1[0, 1], R) : f ′(1) = 0 and f ′(0) = −β∗1f(0).

Proposition 5.4. The spectral bound of the operator D satisfies the followingproperty

s(D) < 0 ⇔ s(C1) < 0 and s(C2) < 0.

The next theorem gives a stability result for (T +(t))t≥0 . It follows by [10,Theorem 4.1] and by Corollary 5.1.

Theorem 5.5. Assume that s(C1), s(C2), and s(A + BL0) are negative. Then(T +(t))t≥0 and hence (T (t))t≥0 as well are uniformly exponentially stable.

5.1 A Non Constant Diffusion in the Past

Assume now that the diffusion in the past of the mRNA presented in the cyto-plasm is not constant, but is governed by the operators A(t) := a(t)∆D, where0 < a(·) ∈ C(R−) and ∆D is the Dirichlet Laplacian on L1[0, 1] as before. Inthis case the evolution family associated to these operators is given by

U(t, s) = e(R s

ta(σ)dσ)∆D . (27)

Since the norm of the evolution family is

‖U(t, s)‖ = e(R s

ta(σ)dσ)λ0 , (28)

where λ0 denotes the largest eigenvalue of ∆D, we can compute directly thegrowth bound of (U(t, s))t≤s≤0.

Proposition 5.6 (see [4], Example 5). The growth bound of the evolution family(U(t, s))−1≤t≤s≤0 is given by

ω0(U) = infh≥0

sups+h≤0

(1h

∫ s+h

s

a(σ)dσ

)λ0,

By Theorem 4.2 one has again that there exists a solution of (NDE), butin this case the operator G defined in (23) is

Gf := f ′ + a(·)f ′′, a.e. (29)

for appropriate f . For the stability, proceeding as before, we can find sufficientconditions in order to obtain that the solutions of (4.2) decay exponentially.

303SEMIGROUPS AND GENETIC...

6 Acknowledgements

The author is supported by the Italian Programma di Ricerca di Rilevanteinteresse Nazionale ”Analisi e controllo di equazioni di evoluzione nonlineari”(cofin 2000).Part of this work has been written while the author was a Ph.D. student at theDepartment of Mathematics at the University of Tubingen and she takes theopportunity to thank Elena Babini, Klaus J. Engel, Rainer Nagel and SusannaPiazzera for many helpful discussions.

References

[1] D.J. Aidley, P.E. Stanfield, ION Channels. Molecules in Action, CambridgeUniversity Press, 2000.

[2] H.T. Banks, J.M. Mahaffy, Global asymptotic stability of certain models forprotein synthesis and repression, Quart. Appl. Math. 36, 209–221 (1978).

[3] A. Batkai, S. Piazzera, Semigroups for delay equations, Research Notes inMathematics 10, 2005.

[4] S.Brendle, R.Nagel, Partial functional differential equations with nonau-tonomous past, Disc. Cont. Dyn. Syst. 8, 953-966 (2002).

[5] S. Busenberg, J.M. Mahaffy, Interaction of spatial diffusion and delays inmodels of genetic control by repression, J. Math. Biol. 22, 313–333 (1985).

[6] V. Casarino, K.J. Engel, R. Nagel, G. Nickel, A semigroup approach toboundary feedback systems, Discrete Contin. Dyn. Syst. 12, no. 4, 761–772(2005).

[7] K.J. Engel, Positivity and stability for one-sided coupled operator matrices,Positivity 1, 103–124 (1997).

[8] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear EvolutionEquations, Springer-Verlag, 2000.

[9] G. Fragnelli, G. Nickel, Partial functional differential equations with nonau-tonomous past in Lp−phase spaces, Diff. Int. Equ. 16, 327-348 (2003).

[10] G. Fragnelli, A spectral mapping theorem for semigroups solving of PDEswith nonautonomous past, Abstract and Applied Analysis, no. 16, 933–951(2003).

[11] G. Fragnelli, Classical solutions for PDEs with nonautonomous past in Lp−spaces, Bulletin of the Belgian Mathematical Society 11, no. 1, 133–148(2004).

[12] B.C. Goodwin, Temporal organization in cells, Academic Press, New York,1963.

304 FRAGNELLI

[13] B.C. Goodwin, Oscillatory behavior of enzymatic control processes, Adv.Enzyme Reg. 13, 425–439 (1965).

[14] F. Jacob, J. Monod, On the regulation of gene activity, Cold Spring harborSymp. Quant. Biol. 26, 389–401 (1961).

[15] J.M. Mahaffy, Periodic solutions for certain protein synthesis, J. Math.Anal. Appl. 74, 72–105 (1980).

[16] J.M. Mahaffy, C.V. Pao, Models of genetic control by repression with timedelays and spatial effects, J. Math. Biol. 20, 39–58 (1984).

[17] J.M. Mahaffy, C.V. Pao, Qualitative analysis of a coupled reaction-diffusionmodel in biology with time delays, J. Math. Anal. Appl. 109, 355–371(1985).

[18] R. Nagel (ed.), One-parameter Semigroups of Positive Operators, Lect.Notes in Math 1184, Springer Verlag, 1986.

[19] V. Taglietti, C. Casella, Elementi di Fisiologia e Biofisica della Cellula, LaGoliardica Pavese, 1991.

[20] J.J. Tyson, Periodic enzyme synthesis and oscillatory repression: Why isthe period of oscillation close to the cell cycle time?, J. Theor. Biol. 103,313–328 (1983).

[21] J. Wu, Theory and Applications of Partial Functional Differential Equa-tions, Springer-Verlag, 1996.

305SEMIGROUPS AND GENETIC...

306

The irregular nesting probleminvolving triangles and rectangles

Loris FainaDipartimento di Matematica

Via L. Vanvitelli, 106123 Perugia (ITALY)Fax # 39 75 585 5024e-mail: faina@unipg.it

Abstract

This paper introduces a new geometrical method for a general two-dimensionalirregular nesting problem; an efficient algorithm is derived. The validity of this algo-rithm is testified by several statistical tests, whose results include a numerical estimateof the asymptotic performance bound and the percentage of wastage of area utilization.

AMS Subject Classification: 90b06,90c15,90c42,90c90

Key words and Phrases: Irregular Nesting, Asymptotic Performance, Method ofCones, Statistic Algorithm

1 Introduction

Cutting and Packing problems have applications accross the whole business activity, fromdesign to manifacturing and distribution. These problems have a common logical structurethat can be synthetized as follows. There are two groups of basic data, whose elementsdefine geometric bodies in one or more dimensions: the stock of large objects, and thelist of small items; the cutting and packing processes realize patterns being geometriccombinations of small items assigned to large objects without overlaps. The residualpieces, that means figures occurring in patterns not belonging to small items, are usuallytreated as trim loss. The objective of most solution techniques is to minimize the wastedmaterial.

The great majority of researchers have focused their attention on the case where boththe objects and the items are rectangular and the orientation of the items is restricted(orthogonally oriented). Many of these papers are surveyed in Dyckhoff and Finke [10]and Dowsland and Dowsland [8]; for a wide discussion about the state of the research seealso Faina [12] for the two-dimensional case and Faina [13] for the three-dimensional case.

However, there are many situations where either the pieces or the containing region areirregular in shape (see, for instance, textile, metal, or footwear industry). Problems involv-ing irregular pieces comprise the most difficult class of packing problems (see Dowsland

307JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,307-324,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

and Dowsland [7] for a survey). Human intelligence is very able to face these problems.Indeed, it has been proved by several studies comparing layouts obtained by automaticprocesses with those produced by a human expert that the computer generated solutionsare inferior in terms of minimizing the waste. However, in some industry the expertiserequired to beat the best packages often takes several years to acquire and in others thetime required to produce a manual solution may be many times that of the automaticprocess.

Although the primary objective is to minimize the waste, often the cutting time orother constraints like minimizing the movements of the cutter between the cutting opera-tions have a great relevance. Therefore good automatic computerised solutions are alwayswelcome.

1.1 State of the research

There are two main approaches for facing the irregular nesting problem. The first producesone or a series of different layouts based on piece orderings and placement polices wherethe algorithm optimizes locally by means of shifts and rotations (see, for instance, Ternoet al. [32]). The second approach is to produce an initial layout, which may be feasiblebut non-optimal or infeasible, and then to use small changes in order to improve it. Thissecond approach may incorporate a metaheuristic technique such as simulated annealingor tabu search in order to allow non-improving moves (see, for instance, Reeves [29]).

In the spirit of first approach, the method developped by Amaral et al. [2] is notewor-thy. Their method is based on a sophisticated process in order to find a suitable positionfor the next piece. Pieces are ordered in decreasing order of their areas and two differentplacement policies are used for large and small pieces, the rationale being that small piecesare used to fill holes left between the larger pieces in the layout. This reflects the strategyused by experts working on the problem manually.

Stoyan and Pankratov [31] find a way of packing identical copies of a polygon in arectangular sheet using regular patterns allowing just two orientations.

Some researchers have considered an interesting alternative in which the irregularpieces are nested inside other more regular shapes, and these simpler shapes are thenpacked into the available area. The most popular shape for nesting is the rectangle.Freeman and Shapira [15] studied the problem of enclosing a piece, whose orientation isnot restricted, in a rectangle of minimal area.

If many of the pieces are far from rectangular in shape a more effective option is tonest more than one piece together, as was developped by Adamowics and Albano [1].

An alternative to the use of rectangles is to nest all the pieces into identical polygonswhich can be used to tile the plane. Dori and Ben-Bassat [6] and Karoupi and Loftus [21]based their packing algorithms on tessellations using hexagons.

In the spirit of second approach, Ismail and Hon [18] studied the blanks nesting prob-lem, but using only two pieces and allowing 180 degree of rotation only, using a geneticalgorithm. Jain et al. [20], using simulated annealing, studied the same problem butwhere more than two pieces are involved, and these may take on any orientation. In both

308

FAINA

these methods, the fitness value of each solution is a combination of the area utilisationand an overlap penalty.

Many other authors used simulated annealing, genetic algorithm or tabu search forfacing the irregular nesting problem. We quote Lutfiyya et al. [23], Marques et al. [24],Oliveira and Ferreira [26], Dowsland and Dowsland [9], Blazewicz et al. [4], George et al.[17], Jacobs [19].

Milenkovic et al. [25] and Li and Milenkovic [22] considered the problem of planningmotions of the pieces in an already existing layout to improve the layout in some fashion:increase efficiency by shortening the length, open space for new pieces without increasingthe length, or eliminate overlap among pieces in an overlapping layout. This type ofimprovement strategy is also used by Stoyan et al. [30].

The method used to determine the geometry of the situation is essential in order toimplement any of these algorithms. Depending on the strategy used, it is necessary tofind the distance between two pieces along a given vector, the set of feasible positionsfor one piece with respect to another, or the existence of an area of overlap between anytwo pieces. The complexity of the calculations involved depends on the types of piecesconsidered and on the accuracy required.

For Qu and Sanders [28] pieces are approximated by unit squares on a rectangular grid.Preparata and Shamos [27] used basic trigonometric formulae to do these calculations, likethe intersection of two lines or intersection of two polygons.

Things are even worse when non-convex shapes are involved. One possibility is to cutthe shape into a number of convex pieces, as suggested by Cuninghame-Green [5].

One of the most used concepts in determining the interaction between two pieces isthe concept of the no-fit polygon. This was first introduced by [1] as an efficient way ofdetermining the minimum enclosure for a cluster of two or more pieces.

An important practical problem linked to the irregular cutting problem is to cut out agiven shape or design from a given piece of parent material. Bhadury and Chandrasekaran[3] studied the problem to find a sequence of guillottine cuts to cut out a convex polygonPin from a convex polygon Pout ⊃ Pin, such that the total length of the cutting sequenceis minimized.

1.2 The aim of this paper

We study the problem of nesting triangles and rectangles on a strip with a fixed widthand virtually infinite length, minimizing the length required. This is the first stage of astudy whose goals is to nest convex polygons of any number of edges.

The limited number of sides permitted here is motivated by the exceptional accuracyof the calculations which allows us, in some cases, to get an optimal solution. Indeed, apeculiar characteristic of our algorithm cone2d is the capability to solve puzzles and thisgives many assurances on the validity of the algorithm.

We use a strategy which is an evolution of the geometrical method introduced in[12, 13, 14] for rectangular pieces, and that has been theoretically proved to lead to anoptimal solution. This new geometrical procedure generates a particular finite class ofplacings to which a statistical global optimization algorithm, based on simulated annealing,

309

THE IRREGULAR NESTING...

is applied. In some relevant cases, this reduction scheme is proved to lead to an optimalsolution (see Section 3.1).

The pieces are considered one at a time and no overlap is tolerated in any stage of theprocess. In each stage of the allocation process, there are some free zones of conical shapewhere the pieces can be nested in such a way that no check for overlapping is necessary(with great saving of time!).

We consider any number of different pieces, and these may take any orientation; no apriori ranking rule is necessary.

After generating a starting feasible layout, the algorithm cone2d uses small changes inorder to improve it, incorporating a fully statistical technique based on simulated anneal-ing.

We do not have a mathematical proof for the optimality of this algorithm except forthe capability to solve puzzles. However, we believe that there is a substantial difference,on a principle level, between developping an algorithm whose layout is as close as possibleto an optimal solution (and the capability to solve puzzles is the minimum request in thissense) and the description of a mere heuristic procedure which, at most, incorporates somemetaheuristic technique, but having no care at all of the theoretical validity of the usedalgorithms. For instance, simulated annealing is an algorithm which has deep theoreticalfoundations, but in most of the papers quoted in the reference section which assert touse simulated annealing we see no traces of a proof or, at least, a discussion about theconvergence of the algorithm presented and/or any relations with an optimal solution.

For sure it is difficult to prove the asymptotic convergence of simulated annealing toan optimal solution of an irregular nesting problem. Indeed we did not get it; but, atleast, we prove the asymptotic convergence of the algorithm cone2d to an optimal solutioneach time the items involved form a perfect puzzle.

In experimental sciences, the results, for being valid, should be at least reproducible;in mathematics, the results should be at least theoretically proved; operational researchis in between these two philosophiae, and we believe that it is our duty to prove as muchas possible, giving motivations for the unproved and reasons for the conjectures.

We will show the validity of the algorithm cone2d by reporting the results of many testsbased on randomly generated initial set of conditions, valuated with several parameters likethe worst-case performance ratio, the asymptotic performance bound and the percentageof wastage of area utilisation. In some cases, a graphic representation of the solution isalso presented, in order to show the difficult link between the parameters and the validityof the algorithm cone2d (see figure 9).

2 Statement of the Problem

Let S be a rectangular strip, with a fixed width w and unbounded length; consider Sembedded into a two dimensional Cartesian frame, in such a way that the lower left sidecorner of S coincides with the origin and each side of S is parallel to a reference frameaxis, i.e. S = (x, y) : 0 ≤ y ≤ w, x ≥ 0.

Let Q be the set of all closed triangular and rectangular items contained in S. Given

310

FAINA

Figure 1: A bit of notations about the items.

an item q ∈ Q, we denote by l0(q) and l1(q) its width and length respectively, and byl2(q) its diagonal, if q is a rectangle; otherwise, we denote by l0(q), l1(q), l2(q) the threeedge lengths of q and by a0(q), a1(q), a2(q) the angles opposed to l0(q), l1(q), and l2(q)respectively. For the sake of shortness, when there will be no risk of confusion, we willwrite li, ai instead of li(q), ai(q) (see figure 1).

Let L = q1, · · · , qn ⊂ Q be a list of n items. A geometrical combination of the itemsof L is called a feasible placing P if the interior of the items do not overlap.

Let Π(L, S) be the set of all the feasible placings P of L contained in S. If P ∈ Π(L, S),then the length of the strip S containing P is called the length of the placing P , and it isdenoted by len(P ).

The two-dimensional irregular nesting problem consists in finding a feasible placingPLopt ∈ Π(L, S) which has the minimal length, i.e.

len(PLopt) = minP∈Π(L,S)

len(P ).(2.1)

Clearly, this is a natural generalization of the regular two dimensional nesting probleminvolving only rectangles, and therefore it is NP-hard (see [16]); this means that, in general,optimal solution s are not known . From the statement of the problem (2.1) we derivethat no restrictions are made neither on the orientation of the items nor on the numberof different items.

2.1 The Method of the Cones

For placing an item, any nesting algorithm, sooner or later, should face the followingquestions: Which are the possible positions where the item could be located? Does theitem fit into the strip? Is the item overlapped to the already placed items?

Mainly, there are two leading strategies: to store a lot of informations about somespecial positions on the strip, which permit to reduce the problem of overlapping; or tolocate approximately an initial position of the item and then to apply some strategy foreliminating the possible overlaps. Both the strategies have their strenghts and weaknesses:the first strategy implies the storage of a lot of positions to avoid the penalty of a poorplacement, but every position implies a lot of computations to do; the second strategyhas the heavy duty to check and eliminate the possible overlaps; but even a simple goodroutine for checking the overlapping between two triangles is everything except easy.

311

THE IRREGULAR NESTING...

Figure 2: left: Description of a conical zone; right: Placement of an item into a conicalzone.

Figure 3: The conical zone used for the placing of an item is replaced with two new conicalzones: case of a triangular item.

We cannot say that one strategy is better than the other. In the papers [12, 13], wedevelopped algorithms for the regular cutting and packing problems in the spirit of thefirst strategy. We operate this choice also in the present two dimensional irregular nestingproblem.

The strategy of our method is to determine a certain number of zones where it ispossible to place an item without troubling about overlaps. Since these zones will have aconical shape, we will call this geometrical method the method of the cones. The methodof the cones develops through three stages: the generation of the conical zones; the de-scription of the strategy for placing an item inside such conical zones; and, each time anew item is placed, a reset policy for revising all the conical zones in order to preservetheir properties.

Assume to have stored some conical zones; we describe now what happens when anitem is placed in a particular conical zone. Suppose first to have a triangular item. Let Cbe the vertex and a be the amplitude of the conical zone (see figure 2-left), and supposethat the angle a2 of the triangle q is smaller than a. Then, it is possible to place thetriangle in the conical zone; we put the corner relative to the angle a2 on the point C andlean the side l0 on the lower side (with respect to the clockwise orientation) of the cone(see figure 2-right). If the triangle remains into the strip, we find an admissible placingfor the triangular item q. Now two new conical zones are naturally determined: the firstconical zone has the vertex in C and amplitude equal to a′ = a− a2 (see figure 3-left); thesecond conical zone has the vertex in D and amplitude equal to a′′ (see figure 3-centre).It is to underline that the amplitude of the second conical zone is, in general, smaller thanπ− a1, because we should guarantee that no other items occupy that conical zone. In thecase a = a2, the first conical zone has vertex in E and amplitude equal to a′ (see figure3-right). In the same spirit, figure 4 indicate how to operate with a rectangular item.

Since the conical zones have to be truncated for fitting into the strip S, some tuning

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Figure 4: Generation of two new conical zones: case of a rectangular item.

Figure 5: The amplitude of the new conical zones has to be tuned when their vertices lieon the border of the strip.

is necessary when the vertices of the new conical zones lie on the border of the strip (seefigure 5).

Clearly each time we introduce a new item, all the conical zones already stored shouldbe revised; first we check if the new item intersects the interior of a conical zone and, inthe positive case, we reduce the amplitude of that conical zone in the spirit of figure 6.

Since for the choice of the sequence of conical zones where to check for the placing ofthe current item the algorithm cone2d uses a metaheuristic technique, we postpone thecomplete description of the method of the cones to the subsection 3.1.

3 The base of the nesting algorithm: the simulated anneal-

ing

Simulated annealing algorithms are based on the analogy between the simulation of theannealing of solids and the problem of solving large combinatorial optimization problems.

These algorithms continuously transform the current configuration into another oneby a small perturbation. The new proposed configuration, called transition, is acceptedeither if the cost functional is lowered or on a probabilistic basis depending on a non-zeroprobability which decreases exponentially as the cost functional increases. This acceptance

Figure 6: Reduction of the amplitude of the other conical zones after the placing of a newitem.

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criterion is strongly guided by a control parameter which is lowered step by step, and whichreduces the number of transitions accepted as the algorithm proceeds.

It can be proved (see for example Faina [12] for an essential bibliography) that if anynew transition from a starting configuration has the same probability to be generated thenthis process leads to an optimal solution with probability 1 as the control parameter goesto zero. It is important to underline that the solution obtained by simulated annealingdoes not depend on the initial configuration.

This framework solves the problem from a theoretical point of view; but asymptoticconvergence is attained possibly in infinite time. Therefore, in any implementation ofsimulated annealing, asymptotic convergence can be only approximated, and the simulatedannealing turns into a heuristic algorithm.

Indeed, the number of transitions for each value ck of the control parameter, for in-stance, must be finite and limk→+∞ ck can only be approximated in a finite number ofvalues ck. Due to these approximations the algorithm is not guaranteed to find a globalminimum with probability 1.

3.1 The nesting algorithm cone2d

In the notation introduced in this section, a feasible placing P ∈ Π(L, S) is considered aconfiguration and the length of the placing P , len(P ), is the cost functional.

Given a list L, the items are placed one by one following a certain order that could beas simple as that in which the items appear in the list (the initial ordering of the list isnot important, since simulated annealing does not depend on the initial configuration).

At the very beginning, we have only one conical zone with vertex in the origin andamplitude equal to π

2 . In general, we have to consider six different positions for valuatingthe possibility to locate a triangular item into a conical zone, and two different positionsin the case of a rectangular item. Note that an item having at least a side not greater thatw can be always placed into a conical zone with vertex lying on the x-axis and amplitudeequal to π

2 .The first item is located, choosing randomly one among all the feasible positions of

the item inside the initial conical zone. Then, two new conical zones are generated; weintroduce always an additional conical zone with vertex in (hmax, 0) and amplitude equalto π

2 , where hmax is the length reached by the items already placed. This is a safety zone,which prevents the halt of the algorithm (see, for example, figure 7).

By induction, suppose to have located the i-th item; the geometrical method of thecones locates at most (n+ 1) conical zones where the (i+ 1)-th item could be located anda conical zone is choosen at random, until the (i + 1)-th item is placed; then we reducethe already stored conical zones in order to eliminate overlaps with the (i + 1)-th item;and so on.

In this way we obtain the initial configuration. Then, we operate a small perturbationto this initial configuration by modifying a little the order of the items, for instance bychanging the position of any two items -at random- and by constructing a new configura-tion as shown above. Let Π∗(L, S) ⊂ Π(L, S) be the set of all the finite feasible placingsthat can be obtained from the method of the cones. Since the transition mechanism de-pends only on the order of the items, and this order is perturbed at random, any new

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Figure 7: This is a simple case showing the necessity of the safety zone. The only twoavailable conical zones (the shadowed areas) have the amplitude smaller than π

2 , andtherefore no rectangle can be placed at this stage.

configuration has the same probability of being generated. This property guarantees thatsimulated annealing will converge to a global optimal solution of the problem

minP∈Π∗(L,S)

len(P ).(3.2)

In general, an optimal solution of (3.2) has a length bigger than an optimal solution of(2.1). It has proven difficult to find a relation between problem (3.2) and (2.1). However,if the items of L form a perfect puzzle, then it is always possible to find a suitable orderfor the items in such a way that the items’ puzzle is represented by a placing of Π∗(L, S).This property holds because, in the case of a puzzle, the items are located in the positionsand with the orientations which are peculiar of the method of the cones.

Thus, in these particular but relevant cases, the two problems are equivalent, andtherefore our algorithm converges asymptotically to an optimal solution of problem (2.1).

3.2 The implementation of the algorithm cone2d

For the implementation of the algorithm we should specify the following parameters:

• the initial value of the control parameter, c0;

• the final value of the control parameter, cf (stop criterion);

• the number of new transitions generated for each value of the control parameter;

• a rule for changing the current value of the control parameter, ck, into the next one,ck+1.

A choice for these parameters is referred to as a ‘ cooling schedule ’.The initial value c0 is determined in such a way that virtually all transitions are

accepted. The stop criterion which determines the final value of the control parameter,consists in fixing the number of values of ck, say it Fk, for which the algorithm can beexecuted, or by terminating if consecutive decrements of the control parameter are identicalfor a given number of times, (a ‘ stalemate ’ condition), whichever occur first.

For controlling the number of transitions generated, we have to face two conflictingrequirements: the first is that for each value of the control parameter a minimal numberof transitions should be accepted; the second is that, since transitions are accepted with

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decreasing probability as ck → 0, we should aim to prevent the generation of large numbersof transitions for low values of ck. A good compromise is to keep fixed the number oftransitions generated, say it Lk, choosing a quite big number depending on the size of theproblem, and to fix a maximum number of accepted transitions for each value of ck, say itMk. In this way, for values of ck quite far from 0, the number of transitions generated willprobably be much smaller than Lk (since a lot of transitions are accepted), while for smallvalue of ck this number will more likely be closer to Lk, assuring that a sufficient numberof transitions are accepted. Thus, the number of accepted transitions is substantial foreach value of ck, and there is little waste of computational time for the initial steps.

For decrementing the control parameter, we use the rule

ck+1 = αck for k ≥ 0,

where α is a constant smaller than but close to 1.The cooling schedules depends on the number of items to place and on the objective

to achieve. Indeed, heavier cooling schedules (this means that c0, Fk,Lk,Mk are big) givesbetter and almost optimal solutions of problem (3.2) but with a larger computationaltime, and vice versa. The aim of the following tests was to present cooling scheduleswhich guarantee a good quality of the solutions in a reasonable computational time.

4 Numerical Tests

The computer codes for the algorithm cone2d is written in the programming languageC, under the LINUX operative system. The computational time is computed through avirtual timer counter, which increments both when the routine executes and when thesystem is executing on behalf of the routine. This choice means that the computationaltime measures the time spent by the routine both in user and kernel space. The platformused is a PENTIUM III 450 MHz.

The evaluation of the algorithm cone2d was performed by means of a self-comparisontest on the basis of randomly generated initial set of conditions. Let cone2d(L) be thefinal placing obtained by the algorithm cone2d for a given list of items L. The mainperformance measure of the nesting algorithm is the asymptotic performance bound, whichcharacterizes the behaviour of the ratio of len(cone2d(L)) over len(PLopt); if there areconstants α and β such that for any list of items L,

len(cone2d(L)) ≤ α · len(PLopt) + βDLmin,(4.3)

where DLmin = maxbi∈Llen(P

biopt ), then α is called an asymptotic performance bound

for the algorithm cone2d.The number DL

min represents the minimal length necessary for placing each single item;it represents a necessary correction to the formula (4.3) in the case of relatively small listsof items, for avoiding an unfair big value of α even in the case of quasi-optimal solutions(see, for example, figure 8).

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Figure 8: Optimal solution with a big percentage of wastage of area utilization.

Actually, since we do not have an estimate of len(P Lopt), we use a different form of

formula (4.3) in the numerical estimations

len(cone2d(L)) ≤ α∗ · len(A(L)) +DLmin,

where A(L) is the lower bound of len(P Lopt), equal to the length such that A(L) · w =∑

bi∈L area of bi (β = 1); therefore α∗ is called an approximate asymptotic performancebound.

We introduce another two parameters for a more complete evaluation: the worst-caseperformance ratio rw

rw =len(cone2d(L))

len(A(L)),

and the percentage of wastage of area utilization A%,

A% =len(cone2d(L))− len(A(L))

len(cone2d(L))· 100.

A comparison between rw and α∗ explains better the role of DLmin. Clearly, the two value

get closer as soon as the list of items becomes larger and larger.Furthermore, it is to underline that A% does not coincide with the true wastage of area

in comparison with the optimal solution, but it is, in general, much bigger. This meansthat a small value of A% implies a good performance of the algorithm; but a relativelybig value of A% (or rw) should not be necessarily interpreted as a failure of the algorithm.Some example reported in figure 9 help understanding better this situation.

The numerical tests have been performed as follows:

• each test has been executed on the basis of 100 runs;

• in each test, we have:

– a fixed number of items (between 8 and 1000) whose parameters are uniformlygenerated random numbers: in the case of a rectangle, the two sides are betweenw10 and w; in the case of a triangle, one side is between w

10 and w, another sideis between w

10 and 2w and the angle between them has an amplitude between0.2 and π

2 ;

– a fixed percentage of triangles and rectangles in the list of items;

– each run has as output the numbers α∗, rw, A%, and the computational timeneeded t;

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items=16, % triang=100, α∗ = 0.16, rw = 1.43, A% = 30

items=16, % triang=100, α∗ = 0.55, rw = 1.43, A% = 30

items=16, % triang=100, α∗ = 0.59, rw = 1.52, A% = 34

items=16, % triang=100, α∗ = 0.65, rw = 1.42, A% = 29

Figure 9: Some outputs of the algorithm cone2d; the big values of A% and rw do notcorrectly express the quality of the solution obtained.

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Table 1: Codes for cooling schedules used in numerical tests.

Test Number c0 α Max n. conf. Max n. conf. Max n. decr. Stalemate

type of runs generated (Lk) accepted (Mk) control param. (Fk)

A 100 50 0.7 16000 1000 30 20B 100 10 0.9 12000 1000 20 10C 100 10 0.9 16000 1000 30 20D 100 10 0.9 12000 1000 30 10E 100 10 0.9 12000 1000 20 10F 100 50 0.7 12000 800 10 10G 100 1 0.9 100 10 1 1P 1 10 0.8 20000 1000 20 5

• the output of each test returns the average value of α∗, rw, A% and t, plus theirbest/worst performances.

The Table 1 contains the cooling schedules of all the test performed.

5 Remarks on the Numerical Tests

The tests have the purpose to valuate the algorithm cone2d with respect to very generalconditions, where the lists of items contain every kind of pieces, from that very big,through that very thin and long, to that very small. These conditions are very difficult;the algorithm cone2d has no shortcuts of any kind.

The complexity of these problems, mainly measured by the number of stored positionsand feasible placings for each item, start to become uncontrollable as soon as the numberof items exceeds 16; notwithstanding the results up to 128 items are still reasonable fromall points of view (see Table 2).

Concerning numerical results for instances with many triangles, it should be said thatthe number of triangles influences the size of feasible region, since for each triangle and foreach cone there are six possible placements. Therefore such instances are more difficult tobe solved compared to instances with the same number of items but more rectangles, andthis fact may explain the reduced performance of the algorithm. In the case of assenceof triangles, the algorithm cone2d is compared with the algorithm zone2d developped bythe author in [14]; zone2d treats only rectangles, therefore it is an important comparisonfor cone2d. These results are shown in Table 4 (see [14] for the values related to zone2d);clearly, zone2d is better than cone2d, but cone2d is not that bad. The results for 64 and128 items show better the penalization of cone2d that ”does not know it has no triangles!”.

Figure 10 shows an example of perfect puzzle solved by cone2d; from the dimensionsof the items involved, one can extrapolate the complexity of the puzzle, even in this caseof 9 items only.

Finally, Table 3 reports the results of cone2d stopped after very few iterations; thisseparate test shows the quality of the initial placings of the algorithm and the quality’slimitations of the final placings of cone2d in the case of very large list of items (within acomputational time of few seconds).

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Table 2: Tests for the algorithm cone2d.

Number % of Test α∗ Perf. Ratio A% Comp. Time in sec.of items Triangles type mean min/max mean min/max mean min/max mean min/max

8 100 C 0.65 0.10/1.08 1.51 1.26/2.60 33.05 21.09/61.59 9.18 6.32/11.8050 A 0.78 0.43/1.03 1.26 1.06/1.59 20.69 6.26/37.09 46.17 8.59/72.0450 B 0.81 0.45/1.12 1.30 1.10/1.59 22.74 9.16/37.30 5.23 3.06/6.800 C 0.79 0.64/1.02 1.10 1.06/1.25 9.82 6.15/20.32 7.26 5.26/8.67

16 100 C 1.00 0.59/1.56 1.52 1.27/2.10 33.76 21.61/52.46 22.29 18.26/29.25100 D 1.03 0.71/1.67 1.55 1.32/2.10 35.39 24.62/52.42 11.35 8.39/14.2350 D 1.01 0.79/1.26 1.31 1.12/1.55 23.49 11.07/35.86 11.9 8.28/17.550 D 0.94 0.84/1.06 1.11 1.06/1.21 10.64 6.45/17.47 10.32 8.19/12.86

32 100 C 1.17 0.98/1.30 1.44 1.30/1.56 30.81 23.08/36.14 37.42 30.42/47.37100 F 1.20 1.05/1.37 1.46 1.34/1.68 31.87 25.56/41.56 20.94 16.10/24.8950 E 1.10 0.99/1.24 1.27 1.13/1.42 21.57 11.82/29.57 42.21 31.21/54.830 E 1.03 0.98/1.08 1.12 1.07/1.17 10.87 7.29/15.08 38.30 29.84/49.41

64 100 E 1.30 1.17/1.45 1.44 1.32/1.58 30.72 24.28/36.99 116.6 91.9/144.350 E 1.17 1.07/1.28 1.27 1.18/1.39 21.31 5.25/28.47 148.1 107.2/190.750 F 1.19 1.11/1.30 1.28 1.20/1.40 22.39 16.96/28.63 99.8 77.98/127.00 E 1.08 1.05/1.12 1.13 1.10/1.17 11.84 9.25/14.80 137.2 108.0/159.8

128 100 E 1.36 1.29/1.46 1.44 1.36/1.54 30.66 26.83/35.11 415.6 356.2/490.450 E 1.22 1.17/1.28 1.27 1.22/1.34 21.50 18.25/25.56 568.6 466.4/698.650 F 1.23 1.17/1.32 1.28 1.22/1.36 22.38 18.33/26.94 332.9 285.0/392.00 E 1.12 1.09/1.16 1.44 1.12/1.18 12.99 10.61/15.81 533.7 452.9/612.9

Table 3: Tests for the algorithm cone2d stopped after few iterations.

Number % of Test α∗ Perf. Ratio A% Comp. Time in sec.of items Triangles type mean min/max mean min/max mean min/max mean min/max

32 100 G 1.71 1.38/2.24 1.99 1.66/2.59 49.52 39.93/61.46 0.02 0.02/0.0450 G 1.42 1.14/1.79 1.59 1.30/1.98 36.95 23.21/49.63 0.03 0.02/0.040 G 1.18 1.08/1.29 1.26 1.16/1.39 21.13 14.25/28.35 0.02 0.02/0.04

64 100 G 1.80 1.58/2.39 1.94 1.71/2.50 48.29 41.63/60.78 0.08 0.07/0.1150 G 1.48 1.30/1.68 1.57 1.40/1.77 36.46 28.72/43.57 0.11 0.09/0.140 G 1.24 1.16/1.42 1.29 1.20/1.48 22.78 17.08/32.59 0.09 0.08/0.11

128 100 G 1.86 1.67/2.20 1.93 1.75/2.28 48.24 43.00/56.21 0.29 0.24/0.3650 G 1.51 1.38/1.69 1.57 1.43/1.75 36.32 30.11/43.05 0.39 0.35/0.460 G 1.28 1.21/1.35 1.30 1.24/1.38 23.45 19.36/27.62 0.34 0.31/0.39

1000 100 G 1.85 1.77/1.92 1.86 1.78/1.93 46.50 43.89/48.28 15.91 14.85/17.2150 G 1.50 1.44/1.57 1.51 1.45/1.58 34.08 31.08/36.91 21.30 20.02/22.760 G 1.32 1.29/1.37 1.33 1.30/1.37 24.95 23.12/27.25 19.46 18.63/20.66

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Figure 10: Example of perfect puzzle solved by cone2d in 30 seconds. We use the coolingschedule P .

Table 4: Comparison between zone2d and cone2d (only rectangular items!). We reportjust the mean values of the parameter measured; the first value is referred to zone2d andthe second to cone2d.

Number α∗ A% Comp. Time in sec.of items mean mean mean

8 0.74/0.79 9.69/9.82 5.34/7.26

16 0.87/0.94 6.74/10.64 9.13/10.32

32 0.95/1.03 5.32/10.87 18.75/38.30

64 1.00/1.08 5.96/11.84 40.10/137.2

128 1.03/1.12 5.95/12.99 91.28/533.7

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6 Conclusions

In this paper we propose a new geometrical method which reduces, in some cases, thetwo dimensional irregular nesting problem (involving triangles and rectangles) to a finitereduction scheme. The algorithm derived, cone2d, is able to solve puzzles. In the case ofabsence of triangles, cone2d is compared with a fully regular algorithm (only for rectangularitems), zone2d, developped by the author in [14]. This comparison gives a ”lower bound”for the algorithm cone2d; the reasonable gap between each couple of parameter in Table 4denotes that the increased complexity of the strategy for the algorithm cone2d due to theaddition of triangles does not affect too much the quality of its final solutions, except forthe computational time.

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Weak convergence and Prokhorov’s Theorem

for measures in a Banach space

Maria Cristina Isidori

Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia

Via Vanvitelli 1, 06123 - PERUGIA, ITALY

Fax: +39-75-5855024

e-mail: isidori@dipmat.unipg.it

Abstract

We prove the sufficient part of Prokhorov’s Theorem in the vector setting.

AMS Subject Classification : 46G10, 28B05, 46B09

Key words and Phrases: Weak convergence, semivariation, Rybakov control, prokhorov’s Theorem.

1 Introduction

As it is well known the Prokhorov’s Theorem has a very important role in the study of cylin-

drical measures and in general in probability theory ([14]).

Vector versions of the Prokhorov’s Theorem have been studied in the literature ([5], [7], [13],

[14], [18]) in relation to several problems and applications (marginal problem, existence of

the projective limit). However the authors either consider vector measures ranging in the

positive cone of a Banach lattice or, even in the case of more general target spaces (as in

[13], [14]), deal with special families of Radon measures.

325JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,325-347,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

2 M.C. Isidori

In this paper we shall achieve one implication of Prokhorov’s Theorem for a sequence of

abstract measures in bv(Ω,Σ, X), where X is a general Banach space, along the ideas of

Billingsley [2]. Indeed, we consider weak convergence in terms of the integral of continuous

functions. The integral that is usually considered in the classic case ([2]), can be lifted to

the vector case, since only continuous bounded integrands are taken into consideration (see

for instance [18]).

In [4] Brooks and Martellotti show that in this particular class of integrands, the integral

can be equivalently defined in the monotone sense, according to the scalar definition due to

De Giorgi and Letta ([9]): this equivalence easily yields the validity of the vector version of

Alexandroff Theorem ([12] IV.9.15).

The proof is rather long and elaborated, and goes along the scalar one as given in [2]: how-

ever one has to replace upper and lower limits by Cauchy nets, and also needs suitable

extension results in the vector setting as those given by [17] [20].

The author whishes to thank Professor Anna Martellotti for the interesting and useful sug-

gestions and the referees for their valuable suggestions.

2 Notations and Preliminaries

Let Ω be a separable metric space, Σ the Borel σ-algebra of Ω. Let d be the metric on Ω.

We denote by Cb(Ω) the set of all continuous and bounded functions f : Ω → IR.

If Pn and P are finite real measures, we say that Pn weakly converges to P , and we write

Pn P ,if for every f ∈ Cb(Ω)

limn

∫ΩfdPn =

∫ΩfdP.

326

Weak convergence 3

A set A ∈ Σ whose boundary ∂A satisfies P (∂A) = 0 is called a P -continuity set.

Let X be a Banach space, X∗ its dual and X∗1 the unit ball of X∗. Let m : Σ → X be a

finitely additive measure, or simply a vector measure.

Definition 2.1 A vector measure m is said to be strongly bounded whenever given a se-

quence (An)n of pairwise disjoint members of Σ the limit limnm(An) is zero.

Theorem 2.2 ([16]) A strongly bounded vector measure m is bounded.

From Theorem 2.2 it follows that all countably additive measures, defined on σ-algebras,

are strongly bounded.

Definition 2.3 The variation of m is the set function | m |: Σ → [0,+∞] defined by

| m | (E) = supπ

∑A∈π

‖m(A)‖

where the supremum is taken over all partitions π of E into a finite number of pairwise

disjoint elements of Σ.

If | m | (Ω) < +∞, then m is of bounded variation (b.v.).

We denote by ca(Ω,Σ, X) the set of all countably additive measures m : Σ → X and by

bvca(Ω,Σ, X) the subset of all countably additive measures of bounded variation.

Definition 2.4 The semivariation of m is the function ‖m‖ : Σ → [0,+∞] defined by

‖m‖(E) = sup| x∗m | (E) : x∗ ∈ X∗, ‖x∗‖ ≤ 1.

If ‖m‖(Ω) < +∞, then m is of bounded semivariation.

327

4 M.C. Isidori

One can easily prove the following relationship:

‖m(E)‖ ≤ ‖m‖(E) ≤| m | (E), (1)

for every E ∈ Σ.

Definition 2.5 A set A ∈ Σ will be called an m-continuity set if ‖m‖(∂A) = 0.

Definition 2.6 A finitely additive bounded measure m is regular if for every E ∈ Σ and

for every ε > 0 there exist an open set G and a closed set F with F ⊆ E ⊆ G such that

‖m‖(G \ F ) < ε.

Lemma 2.7 ([8]) Since Ω is a metrizable space, every countably additive measure is regular.

Definition 2.8 Let m and ν : Σ → [0,+∞] be positive subadditive functions. We say that

m is absolutely continuous with respect to ν, and we write m ν, if for every ε > 0 there

exists δ(ε) > 0 such that for every E ∈ Σ with ν(E) < δ it is m(E) < ε.

Definition 2.9 Let m and ν : Σ → [0,+∞] be positive subadditive functions. We say that

m is equivalent to ν if they are absolutely continuous with respect to each other.

Definition 2.10 A positive subadditive function λ : Σ → [0,+∞] is a control measure for

m if λ is equivalent to ‖m‖.

Theorem 2.11 ([3]) If m is a strongly bounded vector measure, then m admits a control

measure, which is finitely additive.

Theorem 2.12 ([11]) If m is a strongly bounded vector measure then there exists x∗ ∈ X∗

with ‖x∗‖ = 1 such that | x∗m | is a control measure for m.

328

Weak convergence 5

In this case we say that λ =| x∗m | is a Rybakov control for m.

Definition 2.13 ([4]) Let f : Ω → [0,+∞] be a measurable function.

f is ( )-integrable with respect to m if ‖m‖(ω ∈ Ω : f(ω) > t) is Lebesgue integrable. In

this case, as it is shown in [4], the X-valued vector function

t 7→ m(ω ∈ Ω : f(ω) > t) turns out to be Bochner integrable on [0,+∞]. In this case we set

∫Ωfd‖m‖ =

∫ +∞

0‖m‖(ω ∈ Ω : f(ω) > t)dt,

and ∫Ωfdm =

∫ +∞

0m(ω ∈ Ω : f(ω) > t)dt.

Also in [4] it is proved that if f is ( )-integrable then, for every A ∈ Σ, f1A is ( )-integrable.

Therefore we can set ∫Afdm =

∫Ωf1Adm,

and ∫Afd‖m‖ =

∫Ωf1Ad‖m‖.

These integrals satisfy the following properties:

(2.13.1)∫A∪B

fdm =∫

Afdm+

∫Bfdm, where A,B ∈ Σ, A ∩B = ∅;

(2.13.2) If f = k in A then∫

Afdm = k ·m(A);

(2.13.3) If f ≤ g then∫Ωfd‖m‖ ≤

∫Ωgd‖m‖;

(2.13.4) If A ⊂ B then∫

Afd‖m‖ ≤

∫Bfd‖m‖.

If f takes values in IR we say that f is ( )-integrable if f+ and f− are ( )-integrable and we

set ∫Ωfdm =

∫Ωf+dm−

∫Ωf−dm,

329

6 M.C. Isidori

and ∫Ωfd‖m‖ =

∫Ωf+d‖m‖ −

∫Ωf−d‖m‖.

Theorem 3.2 in [4] shows that every bounded function is ( )-integrable.

Definition 2.14 Let mk and m be countably additive vector measures. We say that mk

weakly converges to m, and we write mk m, if for every f ∈ Cb(Ω) and for every x∗ ∈ X∗

it is

limkx∗(

∫Ωfdmk) = x∗(

∫Ωfdm).

If A is an algebra, we denote by σ(A) the σ-algebra generated by A.

Theorem 2.15 ([20]) Let µ : A → X be a countably additive set function, where A is an

algebra, such that for every monotone sequence (An)n ∈ A there exists limn µ(An). Then µ

can be extended to σ(A) in a countably additive way.

Definition 2.16 A family of sets L is a lattice if L is closed under finite joints and meets,

and ∅ ∈ L.

Let L be a lattice and λ : L → X be a function. We say that λ is strongly additive if the

following hold:

1) λ(A) + λ(B) = λ(A ∪B) + λ(A ∩B) with A,B ∈ L;

2) λ(∅) = 0.

Theorem 2.17 ([17]) Every strongly additive set function λ on L has a unique extension

to an additive set function defined on the ring generated by L.

330

Weak convergence 7

We recall that a subnet of a net g : Λ → X is the composition g ϕ, where M is directed

and ϕ : M → Λ is increasing and cofinal in Λ (i.e. for each λ ∈ Λ, there is some µ ∈M such

that λ ≤ ϕ(µ)).

3 The Prokhorov’s Theorem for tight families in bvca(Ω,Σ,X)

According to [2] we set the following definition:

Definition 3.1 A family Π ⊆ bvca(Ω,Σ, X) is weakly sequentially compact if for every

sequence (mk)k ⊆ Π there exists a subsequence (mkj)j and a countably additive vector

measure β such that mkj β.

Definition 3.2 A family Π ⊆ bvca(Ω,Σ, X) is uniformly tight if for every ε > 0 there exists

a compact set Kε ⊂ Ω such that

| m | (Kε) >| m | (Ω)− ε,

for every m ∈ Π.

Definition 3.3 A family Π ⊆ bvca(Ω,Σ, X) is uniformly sequentially tight if for every

(mk)k ⊆ Π there exists a subsequence (mkj)j uniformly tight.

Theorem 3.4 (Prokhorov) Let Π ⊆ bvca(Ω,Σ, X) be a family of measures such that

1) Π is equibounded in the sense of the variations;

2) Π is uniformly sequentially tight;

3) for every compact subset S of Ω there exists a weakly sequentially compact set WS in X

such that m(S) : m ∈ Π ⊂WS.

331

8 M.C. Isidori

Then Π is weakly sequentially compact.

Proof: Step 1 Fix (mk)k in Π. We shall construct the weak limit β on Σ.

In order to do this, first we shall define a subsequence of indexes for which the measures and

their variations simultaneously converge. This construction will coincide with that in [2] for

the sequence (| mk |)k.

Since Π is uniformly sequentially tight, there exists a sequence of compact sets Ku with

K1 ⊂ K2 ⊂ · · · and a subsequence (mkj)j such that

| mkj| (Ω \Ku) <

1u,

for every u ∈ IN and j ∈ IN. We will continue to denote (mkj) by (mk).

Let D = xj : j ∈ IN be dense in Ω and let S be the family B(xj , ρ) : j ∈ IN, ρ ∈ Q+.

Let H consist of ∅ and of the finite unions of sets of the form A ∩Ku, for A ∈ S and u ≥ 1.

H is at most countable and closed with respect to the finite unions and every set of H is

compact. We set

H′= H1 ∩H2,Hi ∈ H, i = 1, 2.

We define the function ψ : H×H → H′by ψ(H1,H2) = H1∩H2. Then ψ is onto and so H′

is

countable. Let H ∈ H′. Then H is compact and by assumption mk(H), k ∈ IN ⊂WH , for

some relatively compact set WH . Therefore there exists a subsequence of (mk(H))k which

weakly converges to an element of X. By a diagonal argument we can determine indexes kj

such that the following two limits exist:

α(H) = (w)− limjmkj

(H); (2)

σ(H) = limj| mkj

| (H). (3)

332

Weak convergence 9

We assume that the subsequence (mkj)j is weakly convergent.

We have

‖α(H)‖ ≤ limj| mkj

| (H) = σ(H) (4)

Claim 1) Let G ⊂ Ω be an open set. Then α(H) : H ∈ H,H ⊂ G is a strongly Cauchy

net.

Proof: Fix G open in Ω. We set

ΛG = H ∈ H : H ⊆ G.

The set ΛG is non empty by definition of H and it is directed with respect to the inclusion.

Define

P (G) = supH∈ΛG

σ(H) (5)

for every open set G, and for M ∈ Σ set

γ(M) = infP (G), G open and M ⊂ G. (6)

As in the proof of Prokhorov’s Theorem ([2]), one proves that P ≡ γ on the open sets and

that it is countably additive on Σ.

Let ε > 0 be fixed; for every H ∈ H, there exists an open set G0 ⊃ H such that

P (G0) < γ(H) + ε,

and so from (6) we have

γ(H) > P (G0)− ε = supT⊂G0

σ(T )− ε > σ(H)− ε.

By the arbitrariness of ε, γ ≥ σ on H. Moreover, for every open set G, there exists H0 ⊂

G, H0 ∈ H, such that

0 ≤ P (G)− σ(H0) <ε

2.

333

10 M.C. Isidori

Let Hi ∈ ΛG with Hi ⊃ H0, i = 1, 2. Then we have

0 ≤ σ(H1) + σ(H2)− 2σ(H1 ∩H2) ≤

≤ γ(H1) + γ(H2)− 2σ(H0) ≤ 2[P (G)− σ(H0)] < ε.

Since H1,H2 and H1 ∩H2 ∈ H′, from (3) it follows that there exists j∗ such that for j > j∗

σ(H1) + σ(H2)− 2σ(H1 ∩H2)− | mkj| (H1 4H2) < ε.

Therefore | mkj| (H1 4H2) < 2ε, for j > j∗.

So, for every x∗ ∈ X∗1 , it is

| x∗mkj| (H1 4H2) < 2ε,

for every j > j∗.

Then we have

| x∗α(H1)− x∗α(H2) |= limj| x∗mkj

(H1)− x∗mkj(H2) |≤

≤ lim supj

| x∗mkj| (H1 4H2) ≤ 2ε.

Hence the net α(H),H ∈ H,H ⊂ G is strongly Cauchy. 2

Now for every open set G we can define

β(G) = limH∈ΛG

α(H) (7)

Claim 2) The set function defined in (7) is finitely additive.

Proof: We observe first that α is finitely additive on H. In fact if H1,H2 ∈ H, for every

x∗ ∈ X∗1 it is

x∗α(H1 ∪H2) = limjx∗mkj

(H1 ∪H2) =

334

Weak convergence 11

= limjx∗mkj

(H1) + limjx∗mkj

(H2)− limjx∗mkj

(H1 ∩H2) =

= x∗[α(H1) + α(H2)− α(H1 ∩H2)].

By the arbitrariness of x∗ we obtain

‖α(H1 ∪H2)− [α(H1) + α(H2)− α(H1 ∩H2)]‖ = 0,

namely

α(H1 ∪H2) = α(H1) + α(H2)− α(H1 ∩H2). (8)

Note that, if H1 ∩H2 = ∅, then from (8)

α(H1 ∪H2) = α(H1) + α(H2).

Now we prove that β is finitely additive.

By definition of β we have β(∅) = 0.

Let G1, G2 be two disjoint open sets. We define the family

HG1∪G2 = H1 ∪H2 : Hi ∈ H,Hi ⊂ Gi, i = 1, 2.

We want to prove that α(H1 ∪ H2) : Hi ∈ H,Hi ⊂ Gi, i = 1, 2 is a subnet of the net

α(H) : H ∈ H,H ⊂ G1 ∪G2. Obviously HG1∪G2 is directed with respect to the inclusion.

Let ϕ : HG1∪G2 → H ∈ H : H ⊂ G1 ∪G2 be the identity function. ϕ is clearly monotone

with respect to the inclusion. It remains to prove that ϕ is cofinal. Let H ∈ ΛG1∪G2 . Then

H can be written as

H =p⋃

j=1

(B(xj , ρj) ∩Kuj ) ⊂ G1 ∪G2,

where B(xj , ρj) ∈ S, the countable basis fixed in the beginning.

Define

F1 = ω ∈ H : d(ω,Gc1) ≥ d(ω,Gc

2)

335

12 M.C. Isidori

F2 = ω ∈ H : d(ω,Gc2) ≥ d(ω,Gc

1)

As in the proof of Prokhorov’s Theorem [2], it follows that F1 ⊂ G1 and F2 ⊂ G2. Moreover

H = F1 ∪ F2. Let j ∈ 1, · · · , p be fixed. The set F1 ∩ B(xj , ρj) ∩Kuj is compact in G1,

and so there exist y1, · · · yn ∈ D and r1, · · · rn ∈ Q+ such that

F1 ∩B(xj , ρj) ∩Kuj ⊂n⋃

i=1

B(yi, ri) ∩Kuj ⊂ G1.

Set H1,j =⋃n

i=1B(xi, ρi) ∩ Kuj . Obviously H1,j ∈ ΛG1 . Iterating this construction for

j ∈ 1, · · · p, we have

F1 = F1 ∩H ⊂p⋃

j=1

H1,j .

Since H is closed under finite unions and every H1,j ⊂ G1, setting H1 =⋃p

j=1H1,j , we obtain

F1 ⊂ H1 ⊂ G1, where H1 ∈ H. So we have proved that for every H ∈ ΛG1∪G2 there exist

H1 ∈ ΛG1 ,H2 ∈ ΛG2 such that H ⊂ H1 ∪H2, namely ϕ is cofinal. (Note that ϕ is cofinal

even if G1, G2 are not disjoint sets). Then, by e) pag. 75 of [21], it is

limH1∪H2∈ΛG1∪G2

α(H1 ∪H2) = limH⊂G1∪G2,H∈H

α(H),

and so

β(G1 ∪G2) = limH1∪H2∈ΛG1∪G2

(α(H1) + α(H2)).

By definition of β it follows that for every ε > o there exist H1 and H2 such that for every

H′i ⊃ Hi, with H

′i ∈ ΛGi , i = 1, 2,

‖β(G1 ∪G2)− α(H′1 ∪H

′2)‖ ≤

ε

3;

‖β(G1)− α(H′1)‖ ≤

ε

3;

‖β(G2)− α(H′2)‖ ≤

ε

3.

336

Weak convergence 13

Thus β is finitely additive on open sets. 2

By (7) it follows that for every open set A in Ω there exists HεA ∈ H such that ‖β(A)−

α(H)‖ < ε4 , for every H ∈ H with Hε

A ⊂ H ⊂ A.

Define Λ′G = H ′ ∈ H′ : H ′ ⊂ G. We shall show that

Claim 3) β can be equivalently defined as

β(G) = limH′∈Λ′

G

α(H ′).

Proof: Let the open set G and ε > 0 be fixed. It is enough to show that there exists

HεG ∈ ΛG such that for every H ∈ H and H

′ ∈ H′with Hε

G ⊂ H ⊂ G and HεG ⊂ H

′ ⊂ G it

is ‖α(H′)− α(H)‖ < ε. Let G be an open set of Ω, and let ε > 0 be fixed. From (5) there

exists HεG ⊂ G, Hε

G ∈ H, such that

0 ≤ P (G)− σ(HεG) <

ε

4.

Let H′ ∈ H′

be a subset of G. We will prove that there exists H0 ∈ H such that H′ ⊂ H0 ⊂

G.

Since H′ ∈ H′

, there exist H1,H2 ∈ H such that H′= H1 ∩H2. So H

′is the finite union of

sets of the form Al ∩Bl ∩Kul∩Kvl

, where Al, Bl ∈ S, ul, vl ≥ 1 and 1 ≤ l ≤ t, for some

t ∈ IN . Since H′ ⊂ G and G is open, for every compact set Al ∩Bl ∩Kul

∩Kvlthere exists

a finite family Dli1≤i≤nl

⊂ S such that

Al ∩Bl ∩Kul∩Kvl

⊂⋃

1≤i≤nl

Dli ⊂

⋃1≤i≤nl

Dli ⊂ G,

337

14 M.C. Isidori

and therefore

Al ∩Bl ∩Kul∩Kvl

⊂⋃

1≤i≤nl

Dli ∩Kvl

⊂ G.

It follows that

H′=

⋃1≤l≤t

Al ∩Bl ∩Kul∩Kvl

⊂⋃

1≤l≤t

(⋃

1≤i≤nl

Dli ∩Kvl

).

Then, setting H0 =⋃

1≤l≤t,1≤i≤nl(Dl

i ∩Kvl), we find that

H0 ∈ H, H′ ⊂ H0 ⊂ G. (9)

If H ∈ H and H′ ∈ H′

are such that HεG ⊂ H ∩H ′

, and H0 ∈ H is obtained from (9), we

have that

σ(HεG) ≤ σ(H

′) ≤ σ(H0) ≤ P (G);

σ(HεG) ≤ σ(H) ≤ P (G);

σ(HεG) ≤ σ(H ∩H ′

) ≤ σ(H) ≤ P (G);

and for every x∗ ∈ X∗1 ,

| x∗α(H)− x∗α(H′) |=| lim

jx∗mj(H)− lim

jx∗mj(H

′) |=

=| limj

(x∗mj(H)− x∗mj(H′)) |≤ lim sup

j[| x∗mj | (H −H ∩H ′

) +

+ | x∗mj | (H′ −H ∩H ′

)] = lim supj

| x∗mj | (H 4H′) =

= lim supj

[| x∗mj | (H)+ | x∗mj | (H′)− 2 | x∗mj | (H ∩H ′

)] =

= limj

[| x∗mj | (H)+ | x∗mj | (H′)− 2 | x∗mj | (H ∩H ′

)] =

= σ(H) + σ(H′)− 2σ(H ∩H ′

) ≤

≤| σ(H)− P (G) | + | σ(H′)− P (G) | +2 | P (G)− σ(H ∩H ′

) |=

= (P (G)− σ(H)) + (P (G)− σ(H′)) + 2(P (G)− σ(H ∩H ′

)) ≤

≤ 4 | P (G)− σ(HεG) |< ε.

338

Weak convergence 15

It follows that

‖α(H)− α(H′)‖ < ε.

2

Claim 4) Let G be the family of the open sets of Ω. Then the set function β defined on

G is strongly additive and it has a unique countably additive extension to Σ.

Proof: The family G is a lattice of sets because it is closed under finite joints and meets

and ∅ ∈ G. Obviously β(∅) = 0.

We will prove that

β(A) + β(B) = β(A ∪B) + β(A ∩B), (10)

for every A,B ∈ G.

Let A,B ∈ G be fixed. We consider the collection

HA ⊕HB = H1 ∪H2 : H1 ⊂ A,H2 ⊂ B,Hi ∈ H, i = 1, 2.

Obviously HA ⊕HB is directed by inclusion.

Let ϕ : HA⊕HB → H ∈ H : H ⊂ A∪B be the identity. Clearly ϕ is monotone. Moreover

as in the proof of Claim 2) ϕ is cofinal.

Let ε > 0 be fixed. From the definition of β and from Claim 3) there exist HεA∪B ∈

ΛA∪B, HεA∩B ∈ ΛA∩B, Hε

A ∈ ΛA, HεB ∈ ΛB such that

‖β(A ∪B)− α(H)‖ < ε,

if H ∈ H and HεA∪B ⊂ H ⊂ A ∪B;

‖β(A ∩B)− α(H)‖ < ε,

339

16 M.C. Isidori

if H ∈ H and HεA∩B ⊂ H ⊂ A ∩B;

‖β(A)− α(H)‖ < ε,

if H ∈ H and HεA ⊂ H ⊂ A;

‖β(B)− α(HεB)‖ < ε,

and

‖α(H′)− α(Hε

B)‖ < ε,

if H′ ∈ H′

and HεB ⊂ H

′ ⊂ B.

Since ϕ is cofinal there exist H1,H2 ∈ H with H1 ⊂ A,H2 ⊂ B such that HεA∪B ⊂ H1 ∪

H2 ⊂ A ∪ B. We set H′

= HεA ∪ H1 ⊂ A and H

′′= Hε

B ∪ H2 ⊂ B; so it follows that

HεA∪B ⊂ H

′ ∪H ′′ ⊂ A ∪ B. Set H∗1 = Hε

A∩B ∪H′and H∗

2 = HεA∩B ∪H

′′. Then it follows

that H∗1 ⊃ Hε

A, H∗2 ⊃ Hε

B, H∗1 ∪H∗

2 ⊃ HεA∪B, A ∩ B ⊃ H∗

1 ∩H∗2 ⊃ Hε

A∩B. Remind that,

since Hi ∈ H, i = 1, 2, H∗1 ∩H∗

2 ∈ H′. Therefore we have

‖α(HεA∩B)− α(H∗

1 ∩H∗2 )‖ < ε,

hence, clearly

‖β(A ∩B)− α(H∗1 ∩H∗

2 )‖ < 2ε.

Moreover, observing that H ⊂ H′, from (8)

‖α(H∗1 ∪H∗

2 ) + α(H∗1 ∩H∗

2 )− α(H∗1 )− α(H∗

2 )‖ = 0.

Therefore we easily find

‖β(A ∪B) + β(A ∩B)− β(A)− β(B)‖ < 5ε.

By the arbitrariness of ε, (10) is proved.

By Theorem 2.17 β has a unique finitely additive extension defined on the algebra R

340

Weak convergence 17

generated by G.

Moreover, by (4) we have, for every G ∈ G

‖β(G)‖ = limH∈ΛG

‖α(H)‖ ≤ limH∈ΛG

σ(H) = supH∈ΛG

σ(H) ≤ γ(G); (11)

since Ω is a metrizable space, the result is true also for every G ∈ R (see [2]), and so β

is countably additive and strongly bounded on R, since γ is (see [2]). By Theorem 2.15,

since strong boundedness implies that for every monotone sequence (An)n limn β(An) ex-

ists, there exists a unique countably additive extension of β to Σ. Without loss of generality

we will continue to denote with β this extension. 2

Step 2) We will prove that mkjconverges weakly to β.

Claim 1) For every open set G of γ-continuity it is

limH∈ΛG

mkj(H) = mkj

(G) (12)

uniformly in kj.

Proof: Let (mkj)jbe the subsequence chosen in the proof of Step 1 (at the beginning).

Fix kj ; since mkjis regular (by Lemma 2.7), for every ε > 0 there exists a compact set

Fε,kj⊂ G such that

‖mkj‖(G \ Fε,kj

) < ε,

being Fε,kj= F ∩Ku, where F ⊂ G is closed, u ∈ IN and ‖mkj

‖(Ω\Ku) <12ε

, ‖mkj‖(G\

F ) <12ε

, and so there are B(x(ε,kj)1 , r

(ε,kj)1 ), · · · , B(x(ε,kj)

n , r(ε,kj)n ) ∈ S such that

Fε,kj⊂

n⋃i=1

B(x(ε,kj)i , r

(ε,kj)i ) ⊂ G.

341

18 M.C. Isidori

Let H =⋃n

i=1B(x(ε,kj)i , r

(ε,kj)i )∩Ku ∈ ΛG. By inclusion, ‖mkj

‖(G\H) < ε. Hence for every

H ∈ ΛG,H ⊃ H it is ‖mkj‖(G \H) < ε, and a fortiori ‖mkj

(G \H)‖ < ε.

This proves that

limH∈ΛG

mkj(H) = mkj

(G). (13)

We want to show now that the limit in (13) is uniform with respect to j.

Let γ be the measure defined in (6) (see [2]) and let G be an open set such that γ(∂G) = 0.

Then by Portmanteau’s Theorem [2] it is

P (G) = γ(G) ≥ lim supj

| mkj| (G).

By definition of P , for every ε > 0 there exists H0 ∈ ΛG such that

P (G)− σ(H0) < ε.

So it follows that

lim supj

| mkj| (G)− | mkj

| (H0) ≤

≤ lim supj

| mkj| (G)− lim

j| mkj

| (H0) ≤

≤ P (G)− σ(H0) < ε.

Thus there exists j∗ such that for H ⊃ H0,H ⊂ G,

| mkj| (G)− | mkj

| (H) < ε,

for every j > j∗. Thus

‖mkj(G \H)‖ ≤| mkj

| (G \H) =| mkj| (G)− | mkj

| (H) < ε,

for every j > j∗. Since limH∈ΛGmkj

(H) = mkj(G) for each kj , we can determineHε

1 , · · · ,Hεj∗ ∈ ΛG

such that

‖mkp(G)−mkp(H)‖ < ε,

342

Weak convergence 19

for every H ⊃ Hεp and for every p = 1, · · · j∗.

We set H = H0 ∪Hε1 ∪ · · · ∪Hε

j∗ with H ∈ ΛG. Let H ⊃ H. Then we have

‖mkj(G)−mkj

(H)‖ < ε,

for every j ∈ IN . 2

Now we can apply Lemma I.7.6 of [12] and we obtain

β(G) = limH∈ΛG

α(H) = limH∈ΛG

[(w)− limjmkj

(H)] =

= (w)− limj

[ limH∈ΛG

mkj(H)] = (w)− lim

jmkj

(G)

Therefore, for every open set G of γ-continuity, we have

β(G) = (w)− limjmkj

(G) (14)

By complementation (14) is true also for every closed set of γ-continuity.

Let A be the algebra of γ-continuity sets: we want to prove that the equality (14) is verified

for every M ∈ A.

Let x∗ ∈ X∗1 and ε > 0 be fixed. Then we want to find a j ∈ IN such that for each j > j

| x∗[β(M)−mkj(M)] |< ε. (15)

Observe that

| x∗mkj(M)− x∗β(M) |≤

≤| x∗mkj(M)− x∗mkj

(M0) | + | x∗mkj(M0)− x∗β(M0) | +

+ | x∗β(M0)− x∗β(M) |≤| x∗mkj(M \M0) | +

+ | x∗[mkj(M0)− β(M0)] | +‖β(M \M0)‖.

343

20 M.C. Isidori

By (11) it is ‖β(M \M0)‖ ≤ γ(M \M0) = 0.

Moreover, since M0 is a γ-continuity open set, from (14) there exists j∗ ∈ IN such that for

every j > j∗ it is

| x∗mkj(M0)− x∗β(M0) |≤ ε

2.

Since | mkj| γ (see [2]), from the Portmanteau’s Theorem it follows that there exists

j∗∗ ∈ IN such that for every j > j∗∗

| γ(M)− | mkj| (M) |< ε

4,

| P (M0)− | mkj| (M0) |< ε

4.

Since M ∈ A, we have γ(M) = P (M0). Therefore it is

| x∗mkj(M \M0) |≤ ‖mkj

‖(M \M0) ≤

≤| mkj| (M \M0) ≤| mkj

| (M \M0) <ε

2.

Hence (15) is proved.

The next step will be finally to prove weak convergence.

Let A ∈ A and let f ∈ Cb(Ω). We set H = t : γ(f = t) > 0. Since γ is bounded, H is at

most countable. We observe that for every t 6∈ H it is (f = t) ∈ A. Since ‖β(•)‖ ≤ γ(•), if

t 6∈ H then the set (f > t) is a β-continuity set. Let x∗ ∈ X∗ be fixed and assume f to be

positive. Since f is bounded, then∣∣∣∣∣x∗∫Ωfdmkj

− x∗∫Ωfdβ

∣∣∣∣∣ =∣∣∣∣∫

Ωfd(x∗mkj

)−∫Ωfd(x∗β)

∣∣∣∣ =

=∣∣∣∣∫ +∞

0x∗mkj

(f > t)dt−∫ +∞

0x∗β(f > t)dt

∣∣∣∣ =

=

∣∣∣∣∣∫[0,sup f ]\H

[x∗mkj(f > t)− x∗β(f > t)]dt

∣∣∣∣∣ ≤≤

∫[0,sup f ]\H

| x∗mkj(f > t)− x∗β(f > t) | dt.

344

Weak convergence 21

By (15)

| x∗β(f > t)− x∗mkj(f > t) |→ 0

in [0, sup f ] \H. Moreover it is

| x∗mkj(f > t)− x∗β(f > t) |≤| x∗β(f > t) | + | x∗mkj

(f > t) |≤

≤ ‖x∗‖ · [‖mkj‖(Ω) + ‖β‖(Ω)] < +∞.

By the Dominated Convergence Theorem the assertion follows.

Remark 3.5 For an m ∈ ca(Ω,Σ, X), and A ∈ Σ we denote by mA the induced measure

mA(E) = m(A ∩ E). Then from Prokhorov’s Theorem it follows that for every A ∈ A,

mAkj βA. It suffices to observe that, for A ∈ A and B ∈ Σ fixed, B is a βA-continuity

set if B ∩A is a β-continuity set.

References

[1] P. BILLINGSLEY ”Convergence of Probability Measures”, John Wiley and Sons, Inc.,

New York, (1968).

[2] P. BILLINGSLEY ”Weak Convergence of Measures: Applications in Probability”, Re-

gional conference series in applied mathematics, Siam, (1977).

[3] J. K. BROOKS ”On the existence of a control measure for strongly bounded vector

measures”, Bull. Amer. Math. Soc., 77, (1971), 999-1001.

[4] J. K. BROOKS–A. MARTELLOTTI ”On the De Giorgi-Letta integral in infinite di-

mensions”, Atti Sem. Mat. Fis. Univ. Modena, 4, (1992), 285-302.

345

22 M.C. Isidori

[5] M. J. MUNOZ BOUZO ”A Prokhorov’s theorem for Banach lattice valued measures”,

R. Acad. Ci. Madrid, 89, (I-II), (1995), 111-127.

[6] D. CANDELORO ”Uniforme esaustivita e assoluta continuita”, Boll. Umi., 4-B (6),

(1985), 709-724.

[7] E. D’ANIELLO–R. M. SHORTT ”Vector-valued capacities”, Rend. Circ. Mat. Palermo,

46, (1997).

[8] M. DEKIERT ”Kompaktheit, Fortsetzbarkeit und Konvergenz von Vectormassen”, Dis-

sertation, University of Essen, (1991).

[9] E. DE GIORGI–G. LETTA ”Une notion generale de convergence faible des fonctions

croissantes d’ensemble”, Ann. Scuola Norm. Sup. Pisa, 33, (1977), 61-99 .

[10] N. DINCULEANU ”Vector Measures”, Pergamon Press, (1967).

[11] J. DIESTEL–J.J. UHL ”Vector measures”, Amer. Math. Soc. Surveyes, 15, Providence,

(1968).

[12] N. DUNFORD–J.T. SCHWARTZ ”Linear Operators I; General Theory ”, Interscienze,

New York, (1958).

[13] F. J. FERNANDEZ–P. JIMENEZ GUERRA ”On the Prokhorov’s Theorem for Vector

Measures”, Comm. Math. Univ. S. Pauli, 39, (1990), 129-141.

[14] F. J. FERNANDEZ–P. JIMENEZ GUERRA ”Projective Limits of Vector Measures”,

Revista Matematica de la Univ. Complutense de Madrid, 3, (1990), 125-142.

[15] G. H. GRECO ”Integrale monotono”, Rend. Sem. Mat. Univ. Padova, 57, (1977), 149-

166.

346

Weak convergence 23

[16] M. P. KATS ”On the extension of vector measures”, Sib. Math. Zhurn., 13 (5), (1972),

1158-1168.

[17] Z. LIPECKI ”On strongly additive set functions”, Coll. Math., 22, (1971), 255-256.

[18] M. MARZ–R. M. SHORTT ”Weak convergence of vector measures” Publ. Math. De-

brecen, 45, (1994), 71-92.

[19] W. RUDIN ”Analisi reale e complessa”, Boringhieri, (1974).

[20] M. SION ”A theory of semigroup valued measures”, Lecture Notes in Math., Springer-

Verlag, New York, (1973).

[21] S. WILLARD ”General Topology”, Addison-Wesley Publishing Company (1970).

347

348

Fitted mesh B-spline collocation method for solvingsingularly perturbed reaction-diffusion problems

Mohan K. Kadalbajoo 1 and Vivek K. Aggarwal2

1,2Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

email: kadal@iitk.ac.in, vivekkumar.ag@gmail.com

Abstract.

In this paper we develop B-spline collocation method for solving a class of singularlyperturbed reaction diffusion equations given by

−ǫu′′ + a(x)u(x) = f(x), a(x) ≥ a

∗> 0

u(0) = α, u(1) = β(0.1)

Here we use the fitted mesh technique to generate piecewise uniform mesh. Our schemeleads to a tridiagonal linear system. The convergence analysis is given and the methodis shown to have uniform convergence of second order. Numerical illustrations aregiven in the end to demonstrate the efficiency of our method.

Key words: Singularly perturbed boundary value problems, B-spline collocationmethod, fitted mesh method, boundary layers, uniform convergence.

1 Introduction

We consider the following class of singularly perturbed two-point boundaryvalue problems

(1.1) Lu ≡ −ǫu′′(x) + a(x)u(x) = f(x) where 0 ≤ x ≤ 1

subject to

(1.2) u(0) = α u(1) = β, α, β ∈ R

where ǫ is a small positive parameter and a(x) and f(x) are bounded con-tinuous functions. For a(x) > 0, Most the equations of type, (1.1) possessesboundary layers [1], i.e regions of rapid change in the solution near the end-points, at the both of the end points. Due to this unique characteristic, thesetype of problems are very important in application point of view. These classof problems arises in various fields of science and engineering, for instance, fluidmechanics, quantum mechanics, optimal control, chemical-reactor theory, aero-dynamics, reaction-diffusion process, geophysics etc.

There are three principal approaches to solve numerically these type of prob-lems, namely, the Finite Difference methods [2] and [3], the Finite Elementmethods [4] and the Spline Approximation methods . First two methods havebeen used by numerous researchers. Enright [5] reported on an ongoing investi-gation into the performance of numerical methods for two-point boundary valueproblems. He outlined how methods based on multiple shooting, collocation

349JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,349-365,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

2

and other discretizations can share a common structure. Surla and Jerkovic [6]considered the singularly perturbed boundary value problem using spline collo-cation method. Sakai and Usmani [7] gave a new concept of B-spline in termof hyperbolic and trigonometric splines which are different from earlier ones. Itis proved that the hyperbolic and trigonometric B-spline are characterized by aconvolution of some special exponential functions and a characteristic functionon the interval [0,1].

In this paper we use the third approach, namely, B-Spline collocation method[8], [9] and [10], to solve the problems of above type. In section 2 we have giventhe derivation for the B-spline method and mesh strategy . Uniform convergencefor the method have been discussed in section 3. In sections 4 and 5 we havesolved some problems using the method and the results and graphs have alsobeen shown.

2 B-spline collocation method

We subdivide the interval [0, 1], and we choose piecewise uniform mesh points

represented by π = x0, x1, x2, ........xN, such that x0 = 0 and xN = 1 and h(Discussed in the next section) is the piecewise uniform spacing. We defineL2[0, 1] is a vector space of all the square integrable function on [0,1], and X bethe linear subspace of L2[0, 1]. Now we define for i = 0, 1, 2, ....N.(2.1)

Bi(x) =1

h3

(x − xi−2)3, if x ∈ [xi−2, xi−1]

h3 + 3h2(x − xi−1) + 3h(x − xi−1)2 − 3(x − xi−1)

3, if x ∈ [xi−1, xi]

h3 + 3h2(xi+1 − x) + 3h(xi+1 − x)2 − 3(xi+1 − x)3, if x ∈ [xi, xi+1]

(xi+2 − x)3, if x ∈ [xi+1, xi+2]

0, otherwise.

We introduce four additional knots as x−2 < x−1 < x0 and xN+2 > xN+1 > xN .From the above equation (2.1) we can simply check that each of the functionsBi(x) is twice continuously differentiable on the entire real line, Also

(2.2) Bi(xj) =

4, if i = j

1, if i − j = ±1

0, if i − j = ±2.

and that Bi(x) = 0 for x ≥ xi+2 and x ≤ xi−2.Similarly we can show that

(2.3) B′i(xj) =

0, if i = j

± 3

h, if i − j = ±1

0, if i − j = ±2.

and

(2.4) B′′i (xj) =

−12

h2, if i = j

6

h2, if i − j = ±1

0, if i − j = ±2.

350 KADALBAJOO,AGGARWAL

3

Each Bi(x) is also a piece-wise cubic with knots at π, and Bi(x) ∈ X . LetΩ = B−1, B0, B1, ......BN+1 and let Φ3(π) = span Ω. The functions Ω arelinearly independent on [0, 1], thus Φ3(π) is (N +3) -dimensional. Even one canshow that Φ3(π) ⊆sp X . Let L be a linear operator whose domain is X andwhose range is also in X . Now we define

(2.5) S(x) = c−1B−1(x)+c0B0(x)+c1B1(x)+........+cNBN (x)+cN+1BN+1(x)

Then force S(x) to satisfy the collocation equations plus the boundary conditions.We have

(2.6) LS(xi) = f(xi) 0 ≤ xi ≤ N

and

(2.7) S(0) = α, S(1) = β

On solving the equation (2.6) we get

ci−1(−ǫ B′′i−1(xi) + aiBi−1(xi)) + ci(−ǫ B′′

i (xi) + aiBi(xi))

+ ci+1(−ǫ B′′i+1(xi) + aiBi+1(xi)) = fi ∀i = 0, 1, 2....N

(2.8)

where a(xi) = ai and f(xi) = fi. Solving equation (2.8) we get

(2.9) (−6ǫ+aih2)ci−1+(12ǫ+4aih

2)ci+(−6ǫ+aih2)ci+1 = h2fi, ∀i = 0, 1, ...N

The given boundary conditions (2.7) becomes

(2.10) c−1 + 4c0 + c1 = α

and

(2.11) cN−1 + 4cN + cN+1 = β

Equations (2.9),(2.10) and (2.11) lead to a (N +3)× (N +3) tridiagonal systemwith (N+3) unknowns CN = (c−1, c0, .....cN+1)

t ( where t stands for transpose).Now eliminating c−1 from first equation of (2.9) and (2.10):we find

(2.12) 36ǫ c0 = f0h2 − α(−6ǫ + a0h

2)

Similarly, eliminating cN+1 from the last equation of (2.9) and from (2.11), wefind

(2.13) 36ǫ cN = fN h2 − β(−6ǫ + aN h2)

Coupling equations (2.12) and (2.13) with the second through (N-1)st equationsof (2.9), we are lead to the system of (N+1) linear equations TxN = dN in the

(N+1) unknowns xN = (c0, .....cN )t with right-hand side dN = (f0h2 −α(−6ǫ +

351FITTED MESH B-SPLINE COLLOCATION...

4

a0h2), h2f1, h

2f2............h2fN−1, fN h2 −β(−6ǫ+ aN h2)) and the coefficient ma-

trix is(2.14)

36ǫ 0 0 0 · · · 0

−6ǫ + a1h2 12ǫ + 4a1h

2 −6ǫ + a1h2 0 · · · 0

0...

...... 0 · · ·

0 0 −6ǫ + aih2 12ǫ + 4aih

2 −6ǫ + aih2 · · ·

......

......

......

0 · · · 0 −6ǫ + aN−1h2 12ǫ + 4aN−1h

2 −6ǫ + aN−1h2

0 · · · 0 0 0 36ǫ

Since a(x) > 0, it is easily seen that the matrix T is strictly diagonally dominantand hence nonsingular. Since T is nonsingular, we can solve the system TxN =dN for c0, c1, ........cN and substitute into the boundary equations (2.10) and(2.11) to obtain c−1 and cN+1. Hence this method of collocation applied to(1.1) using a basis of cubic B-spline has a unique solution S(x) .

2.1 Mesh selection strategy

We form the piecewise -uniform grid in such a way that more pts are generatedin the boundary layers region than outside of it.

we divide the interval [0,1] into three sub-intervals (0, κ), (κ, 1−κ) and (1−κ, 1),where

(2.15) κ = min1/4, c√

ǫ lnN; c is a constant.

Assuming N = 2r with r ≥ 3 be the total no. of subintervals in the partitionsof [0,1].we divide the intervals (0, κ) and (1 − κ, 1) each into N/4 equal mesh elements,while the interval (κ, 1 − κ) is divided into N/2 equal mesh elements. Theresulting piecewise mesh depends upon just one parameter κ. Obviously Nowwe consider

h = h1 in the interval [0, κ]

h1 =4κ

Nxi = xi−1 + h1 for i = 1, 2, 3....N/4.

(2.16)

where x0 = 0. Also

h = h2 in the interval [κ, 1 − κ]

h2 =2(1 − 2κ)

Nxi = xi−1 + h2 for i = N/4 + 1, .....3N/4

(2.17)

Similarly

352 KADALBAJOO,AGGARWAL

5

h = h3 in the interval [1 − κ, 1]

h3 =4κ

Nxi = xi−1 + h3 for i = 3N/4 + 1, ........N.

(2.18)

3 Derivation for Uniform Convergence

First we prove the following lemmaLemma 3.1. The B-splines B−1, B0, .......BN+1 defined in equation (2.1),

satisfy the inequality

N+1∑

i=−1

|Bi(x)| ≤ 10, 0 ≤ x ≤ 1

Proof. We know that

|N+1∑

i=−1

Bi(x)| ≤N+1∑

i=−1

|Bi(x)|

At any ith nodal point xi, we have

N+1∑

i=−1

|Bi| = |Bi−1| + |Bi| + |Bi+1| = 6 < 10

Also we have

|Bi(x)| ≤ 4 and |Bi−1(x)| ≤ 4 for x ∈ [xi−1, xi]

similarly|Bi−2(x)| ≤ 1 and |Bi+1(x)| ≤ 1 for x ∈ [xi−1, xi]

Now for any point x ∈ [xi−1, xi] we have

N+1∑

i=−1

|Bi(x)| = |Bi−2| + |Bi−1| + |Bi| + |Bi+1| ≤ 10

Hence this proves the lemma.Now to estimate the error ‖u(x)−S(x)‖∞, let Yn be the unique spline interpo-

late from Φ3(π) to the solution u(x) of our boundary value problem (1.1)-(1.2).If f(x) ∈ C2[0, 1], and u(x) ∈ C4[0, 1], and it follows from the De Boor-Hallerror estimates that

(3.1) ‖Dj(u(x) − Yn)‖∞ ≤ γjh4−jc , j = 0, 1, 2.

353FITTED MESH B-SPLINE COLLOCATION...

6

where hc = maxh1, h2, h3 and γj ’s are independent of hc and N . Let

(3.2) Yn(x) =N+1∑

i=−1

biBi(x)

Where

S(x) =

N+1∑

i=−1

ciBi(x)

is our collocation solution. It is clear that from equation (3.1) that

(3.3) |LS(xi) − LYn(xi)| = |f(xi) − LYn(xi) + Lu(xi) − Lu(xi)| ≤ λh2c

where λ = [ǫ γ2 + γ0‖a(x)‖∞h2c ]. Also LS(xi) = Lu(xi) = f(xi). Let LYn(xi) =

fn(xi) ∀i, and fn = (fn(x0), fn(x1), ............fn(xN ))t. Now from the systemTxN = dN and equation (3.3) that the ith coordinate [T (xN − yn)]i of T (xN −yn), yn = (b0, b1, ........bN )t, satisfy the inequality

(3.4) |[T (xN − yn)]i| = h2c |fi − fi| ≤ λh4

c

since (TxN )i = h2cfi and (Tyn)i = h2

c fn(xi) for i = 1, 2, 3...N − 1. Also

(3.5) (TxN )0 = h2cf0−α(−6ǫ+a0hc

2), and (Tyn)0 = h2

c fn(x0)−α(−6ǫ+a0hc

2)

similarly(3.6)

(TxN)N = h2cfN −β(−6ǫ+ aN hc

2), and (Tyn)N = h2

c fn(xN )−β(−6ǫ+ aN hc

2)

But the ith coordinate of [T (xN − yn)] is the ith equation(3.7)(−6ǫ + aih

2c)δi−1 + (12ǫ + 4aih

2c)δi + (−6ǫ + aih

2c)δi+1 = ξi ∀i = 1, 2, ....N − 1.

where δi = bi − ci, −1 ≤ i ≤ N + 1,and

(3.8) ξi = h2c [f(xi) − fn(xi)] ∀i = 1, 2, 3, .....N − 1.

Obviously from equation (3.3)

(3.9) |ξi| ≤ λh4c

Let ξ = max1≤x≤N−1 |ξi|. Also consider δ = (δ−1, δ0, ........δN+1)t, then we define

ei = |δi|, and e = max1≤i≤N−1 |ei|. Now equation (3.7) becomes

(3.10) (12ǫ + 4aih2c)δi = ξi + (6ǫ − aih

2c)(δi−1 + δi+1) ∀i = 1, 2, ....N − 1.

Taking absolute values with sufficiently small hc. We have

(3.11) (12ǫ + 4aih2c) ei ≤ ξ + 2e(6ǫ − aih

2c)

354 KADALBAJOO,AGGARWAL

7

Also 0 < a∗ ≤ a(x) for all x. We get

(3.12) (12ǫ + 4a∗h2c) ei ≤ ξ + 2e(6ǫ − a∗h2

c) ≤ ξ + 2e(6ǫ + a∗h2c)

In particularly

(3.13) (12ǫ + 4a∗h2c) e ≤ ξ + 2e(6ǫ + a∗h2

c)

which gives

(3.14) 2a∗h2c e ≤ ξ ≤ λh4

c implies e ≤ λh2c

2a∗

To estimate e−1, e0, eN and eN+1, first we observe that the first equation of the

system T (xN − yn) = h2c [FN − fn] (where FN = (f0, f1, .....fN )) gives

(3.15) 36ǫ δ0 = h2c [f0 − f0]

which gives

(3.16) e0 ≤ λh4c

36ǫ

similarly we can calculate

(3.17) eN ≤ λh4c

36ǫ

Now e−1 and eN+1 can be calculated using equations (2.10) and (2.11) as δ−1 =(−4δ0 − δ1) and δN + 1 = (−4δN − δN−1)

(3.18) e−1 ≤ λh4c

9ǫ+

λh2c

2a∗

also

(3.19) eN+1 ≤ λh4c

9ǫ+

λh2c

2a∗

using value λ = [ǫ γ2 + γ0‖a(x)‖∞h2c ], we get

(3.20) e = max−1≤i≤N+1

ei ⇒ e ≤ ωh2c

where ω =γ2h2

c

9+ λ

2a∗where (

h6c

ǫ∼ 0).

The above inequality enables us to estimate ‖S(x)−Yn(x)‖∞, and hence ‖u(x)−S(x)‖∞. In particular

(3.21) S(x) − Yn(x) =

N+1∑

i=−1

(ci − bi)Bi(x).

355FITTED MESH B-SPLINE COLLOCATION...

8

thus

(3.22) |S(x) − Yn(x)| ≤ max |ci − bi|N+1∑

i=−1

|Bi(x)|.

But

(3.23)

N+1∑

i=−1

|Bi(x)| ≤ 10, 0 ≤ x ≤ 1 (using lemma 1).

Combining equations (3.20) (3.22) and (3.23), we see that

(3.24) ‖S − Yn‖∞ ≤ 10ωh2c.

But ‖u−Yn‖∞ ≤ γ0h4c . Since ‖u−S‖∞ ≤ ‖u−Yn‖∞ + ‖Yn −S‖∞, we see that

(3.25) ‖u − S‖∞ ≤ Mh2c

Where M = 10ω + γ0h2c . Combining the results we have proved.

Theorem 3.2. The collocation approximation S(x) from the space Φ3(π)to the solution u(x) of the boundary value problem (1.1) with (1.2) exists. Iff ∈ C2[0, 1], then

‖u(x) − S(x)‖∞ ≤ Mh2c

for h2c sufficiently small and M is a positive constant (independent of ǫ).

4 Numerical Results

In this section the numerical results of some model problems are presented.

Problem 1.: We take

(4.1) −ǫu′′ + u = 1 + 2√

ǫ[exp(−x/√

ǫ) + exp((x − 1)/√

ǫ)],

subject to

(4.2) u(0) = 0 and u(1) = 0

which has the exact solution

u(x) = 1 − (1 − x) exp(−x/√

ǫ) − x exp[(x − 1)/√

ǫ].

Problem 2. : We solve [11]

(4.3) −ǫu′′ + u = − cos2(πx) − 2ǫπ2 cos(2πx),

subject to

(4.4) u(0) = 0 and u(1) = 0

356 KADALBAJOO,AGGARWAL

9

which has the exact solution

u(x) = [exp(−1(1 − x)/√

ǫ) + exp(−x/√

ǫ)]/[1 + exp(−1/√

ǫ)] − cos2(πx).

Problem 3.: We consider

(4.5) −ǫu′′ + u = x,

subject to

(4.6) u(0) = 1 and u(1) = 1 + exp(−1/√

ǫ).

which has the exact solution

u(x) = exp(−x/√

ǫ) + x.

Problem 4.: We solve [12]

(4.7) −ǫu′′ + (1 + x)u = −40[x(x2 − 1) − 2ǫ],

subject to

(4.8) u(0) = 0 and u(1) = 0

which has the exact solution

u(x) = 40x(1 − x).

The maximum error for problem 1 for different values of ǫ(uniform mesh) andgrid points N is presented in Table 1.

TABLE 1Maximum error for Problem 1

Uniform Mesh

ǫ = 2−k N=16 N=32 N=64 N=128 N=256 N=512 N=1024k =2 9.424E-04 2.352E-04 5.879E-05 1.469E-05 3.674E-06 2.175E-06 2.296E-07

4 2.240E-03 5.576E-04 1.392E-04 3.480E-05 8.700E-06 5.203E-06 5.437E-076 5.462E-03 1.339E-03 3.333E-04 8.325E-05 2.081E-05 1.778E-05 1.300E-068 2.015E-02 4.663E-03 1.146E-03 2.848E-04 7.113E-05 6.551E-05 4.445E-0610 6.921E-02 1.851E-02 4.292E-03 1.054E-03 2.623E-04 2.510E-04 1.637E-0512 1.582E-01 6.637E-02 1.769E-02 4.107E-03 1.008E-03 9.861E-04 6.270E-0514 2.318E-01 1.552E-01 6.495E-02 1.728E-02 4.014E-03 3.968E-03 2.454E-0416 2.593E-01 2.299E-01 1.537E-01 6.424E-02 1.708E-02 1.698E-02 9.748E-0418 2.662E-01 2.583E-01 2.289E-01 1.530E-01 6.389E-02 6.371E-02 3.941E-0320 2.677E-01 2.657E-01 2.578E-01 2.284E-01 1.526E-01 2.471E-01 1.693E-0225 2.680E-01 2.679E-01 2.677E-01 2.666E-01 2.626E-01 2.471E-01 1.961E-0130 2.679E-01 2.679E-01 2.679E-01 2.679E-01 2.677E-01 2.672E-01 2.652E-01

357FITTED MESH B-SPLINE COLLOCATION...

10

The maximum error for problem 1 for different values of ǫ(piecewise uniformmesh) and grid points N is presented in Table 2.

TABLE 2Maximum error for Problem 1

Using Fitting Mesh

ǫ = 2−k N=16 N=32 N=64 N=128 N=256 N=512 N=1024k =2 9.424E-04 2.352E-04 5.879E-05 1.469E-05 3.674E-06 9.043E-07 2.296E-07

4 2.240E-03 5.576E-04 1.392E-04 3.480E-05 8.700E-06 2.141E-06 5.437E-076 5.462E-03 1.339E-03 3.333E-04 8.325E-05 2.081E-05 5.123E-06 1.300E-068 2.866E-02 5.755E-03 1.144E-03 2.848E-04 7.113E-05 1.750E-05 4.445E-0610 5.226E-02 2.152E-02 8.340E-03 3.114E-03 1.122E-03 3.737E-04 1.094E-0412 5.962E-02 2.811E-02 1.283E-02 5.743E-03 2.559E-03 1.136E-03 5.159E-0414 6.162E-02 3.028E-02 1.469E-02 7.019E-03 3.322E-03 1.560E-03 7.484E-0416 6.219E-02 3.092E-02 1.531E-02 7.540E-03 3.681E-03 1.775E-03 8.706E-0418 6.237E-02 3.112E-02 1.551E-02 7.720E-03 3.828E-03 1.875E-03 9.312E-0420 6.244E-02 3.119E-02 1.558E-02 7.778E-03 3.879E-03 1.915E-03 9.590E-0425 6.249E-02 3.124E-02 1.561E-02 7.808E-03 3.903E-03 1.936E-03 9.752E-0430 6.249E-02 3.124E-02 1.562E-02 7.811E-03 3.903E-03 1.937E-03 9.763E-04

The maximum error for problem 2 for different values of ǫ and grid points Nis presented in Table 3.

TABLE 3Maximum error for Problem 2

Using Fitting Mesh

ǫ = 2−k N=16 N=32 N=64 N=128 N=256 N=512 N=1024k=4 7.098E-03 1.779E-03 4.450E-04 1.112E-04 2.782E-05 6.955E-06 1.738E-066 4.073E-03 1.016E-03 2.541E-04 6.353E-05 1.588E-05 3.970E-06 9.926E-078 6.305E-02 2.441E-03 9.336E-04 2.323E-04 5.803E-05 1.450E-05 3.626E-0610 7.752E-02 5.022E-02 3.928E-03 9.612E-04 2.392E-04 5.028E-05 1.276E-0512 6.341E-02 3.066E-02 2.021E-02 3.803E-03 9.633E-04 2.768E-04 5.988E-0514 6.237E-02 3.174E-02 1.576E-02 6.303E-03 5.367E-03 9.917E-04 2.398E-0416 6.251E-02 3.119E-02 1.581E-02 7.909E-03 3.468E-03 9.731E-04 9.635E-0418 6.250E-02 3.124E-02 1.560E-02 7.871E-03 3.940E-03 1.826E-03 6.840E-0420 6.250E-02 3.125E-02 1.562E-02 7.804E-03 3.921E-03 1.963E-03 9.404E-0425 6.250E-02 3.125E-02 1.562E-02 7.812E-03 3.906E-03 1.952E-03 9.759E-0430 6.250E-02 3.125E-02 1.562E-02 7.812E-03 3.906E-03 1.953E-03 9.765E-04

The maximum error for problem 3 for different values of ǫ and grid points Nis presented in Table 4.

TABLE 4Maximum error for Problem 3

Using Fitting Mesh

358 KADALBAJOO,AGGARWAL

11

ǫ = 2−k N=16 N=32 N=64 N=128 N=256 N=512 N=1024k =2 1.910E-04 4.771E-05 1.193E-05 2.983E-06 7.458E-07 1.864E-07 4.661E-08

4 9.553E-04 2.337E-04 5.938E-05 1.484E-05 3.710E-06 9.276E-07 2.319E-076 3.921E-03 9.635E-04 2.398E-04 5.989E-05 1.497E-05 3.742E-06 9.355E-078 1.465E-02 7.796E-03 9.635E-04 2.398E-04 5.989E-05 1.497E-05 3.742E-0610 2.208E-02 1.507E-02 3.921E-03 9.635E-04 2.398E-04 2.989E-05 1.256E-0512 2.761E-02 1.581E-02 4.842E-03 7.921E-03 9.635E-04 7.118E-04 5.269E-0514 3.133E-02 1.670E-02 9.525E-03 6.947E-03 6.252E-03 9.074E-04 5.977E-0416 3.353E-02 1.774E-02 9.125E-03 5.266E-03 3.898E-03 3.522E-03 3.432E-0318 3.472E-02 1.848E-02 9.279E-03 4.747E-03 2.783E-03 2.080E-03 1.879E-0320 3.534E-02 1.890E-02 9.476E-03 4.683E-03 2.416E-03 1.435E-03 1.078E-0325 3.586E-02 1.928E-02 9.703E-03 4.763E-03 2.334E-03 1.156E-03 1.374E-0330 3.595E-02 1.935E-02 9.749E-03 4.790E-03 2.346E-03 1.154E-03 1.381E-03

The comparison Table for problem 4 with the existing methods for differentvalues of ǫ and grid points N is presented in Tables 5 and 6.

TABLE 5Maximum error for Problem 4

Fitted Mesh: N=16

ǫ Miller [13] Niijima [14] Niijima [12] Our method(Unif. Mesh)

0.1E-03 0.25E-01 0.26E-01 0.65E-04 1.776E-150.1E-04 0.21E-01 0.24E-01 0.36E-04 1.776E-150.1E-05 0.70E-02 0.17E-01 0.33E-04 1.776E-150.1E-06 0.75E-03 0.69E-02 0.26E-04 1.776E-150.1E-07 0.74E-04 0.23E-02 0.20E-04 1.783E-150.1E-08 0.67E-05 0.76E-03 0.20E-04 1.783E-150.1E-09 0.00E+00 0.24E-03 0.11E-04 1.784E-15

TABLE 6Maximum error for Problem 4

Fitted Mesh : N=32

ǫ Miller [13] Niijima [14] Niijima [12] Our method(Unif. Mesh)

0.1E-03 0.64E-02 0.65E-02 0.59E-04 1.776E-150.1E-04 0.61E-02 0.64E-02 0.21E-04 1.776E-150.1E-05 0.41E-02 0.56E-02 0.35E-04 1.776E-150.1E-06 0.77E-03 0.31E-02 0.39E-04 1.776E-150.1E-07 0.76E-04 0.12E-02 0.21E-04 1.776E-150.1E-08 0.67E-05 0.38E-03 0.21E-04 1.788E-150.1E-09 0.00E+00 0.13E-03 0.14E-04 1.789E-15

359FITTED MESH B-SPLINE COLLOCATION...

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=128

Figure 4.1: Problem 1. with uniform mesh for ǫ = 10−3

360 KADALBAJOO,AGGARWAL

13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

X−axis

Y−

axis

Su

For N=64

S(x)=Appro. sol.u(x)=Exact sol.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=128

Figure 4.2: Problem 1. with Fitted mesh for ǫ = 10−3

361FITTED MESH B-SPLINE COLLOCATION...

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=128

Figure 4.3: Problem 2. with Fitted mesh for ǫ = 10−3

362 KADALBAJOO,AGGARWAL

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

X−axis

Y−

axis

SuS(x)=Appro. sol.

u(x)=Exact sol.

N=64

Figure 4.4: Problem 3. with Fitted mesh for ǫ = 10−3

363FITTED MESH B-SPLINE COLLOCATION...

16

5 Discussion

We have described a B-spline collocation method for solving singularly per-turbed problems using fitted mesh technique. We have applied this method forproblems having second derivative term and function term. These type of prob-lems are very important physically and difficult to solve because of two boundarylayers at both of the end points. In Table 1, we have shown the maximum errorfor the problem 1 using B-spline collocation method with uniform mesh. It isvery clear from the Table that for large value of ǫ we have better results butas we decrease the value of ǫ, the accuracy decreases rapidly. So we concludethat uniform mesh is not a good technique for such kind of problems, becausenumber of mesh points in the boundary layer region should be much more thanthat from the outer region. In Tables 2,3 and 4 we have shown that maximumerror for problems 2,3 and 4 using fitted mesh technique (piecewise -uniformmesh) respectively. For

N ≥ exp1

4√

ǫ

our mesh behaves as uniform mesh. It can be seen from the respective Tables thatif we take more points in the boundary layer region then we obtain better resultswhich implies that the use of piecewise-uniform mesh is quite advantageous. InTables 5 and 6, we have given the comparison with the different methods appliedto the same kind of problems.

To further corroborate the applicability of the proposed method, graphs havebeen plotted for problems for x ∈ [0, 1] versus the computed solution obtainedat different values of x for a fixed ǫ.

6 Conclusion

As is evident from the numerical results, this method gives O(h2) accuracy.The results obtained using this method are better than using the stated existingmethods with the same no. of knots. Also this method produce a spline functionwhich may be used to obtain the solution at any point in the range, whereas thefinite-difference methods [12]-[14], gives the solution only at the chosen knots.

REFERENCES

1. P.W.Hemker, J.J.H.Miller(Eds.), ”Numerical analysis of singular perturba-tion problems”, Academic Press, New York, 1979.

2. P.A.Markowich, ”A finite difference method for the basic stationary semi-conductor device equations,:in progress in Science and Computations”, Vol.5, Numerical boundary value ODEs, Birkhauser, Boston, MA, 1985, pp 285-301.

3. D.Herceg, ”Uniform fourth order difference scheme for a singular perturba-tion problem”, Numer.Math., Vol. 56, (7) pp 675-693, 1990.

4. E.O’Riordan, M.Stynas, ”A uniformly accurate finite element method fora singularly perturbed one-dimensional reaction-diffusion problem”, Math.Comput., 47 (176), pp 555-570, 1986.

5. W.H.Enright, ”Improving the performance of numerical methods for two-point boundary value problems,:in progress in Science and Computations”,

364 KADALBAJOO,AGGARWAL

17

Vol. 5, Numerical boundary value ODEs, Birkhauser, Boston, MA, 1985, pp107-119.”,

6. K.Surla, V.Jerkovic, ”Some possibilites of applying spline collocations tosingular perturbation problems”, Numerical methods and ApproximationTheory, Vol. II, Novisad, pp 19-25, 1985.

7. M.Sakai, R.A.Usmani, ”On exponential splines”, J.Appro. Theory, Vol. 47,pp 122-131, 1986.

8. L.L.Schumaker, ”Spline functions: Basic Theory”, Krieger Publishing Com-pany, Florida , 1981.

9. J.M.Ahlberg, E.N.Nilson and J.L.Walsh, ”The Theory of splines and theirapplications”, Academic Press, New York, 1967.

10. G.Micula, ”Handbook of Splines”, Kluwer Academic Publishers, Dor-drecht,The Netherlands 1999.

11. E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, ”Uniform Numerical methodsfor problems with initial and boundary layers”, Boole Press, Dublin , 1980.

12. K. Niijima, ”On a three-point difference scheme for a singular perturbationproblem without a first derivative term II”, Mem. Numer. Math., Vol. 7, pp11-27, 1980.

13. J.J.H. Miller, ”On the convergence, uniformly in ǫ, of difference schemes fora two-point boundary singular perturbation problem, Numerical Analysis ofSingular Perturbation Problems”, Academic Press, New York , pp 467-474,1979.

14. K. Niijima, ”On a three-point difference scheme for a singular perturbationproblem without a first derivative term I”, Mem. Numer. Math., Vol. 7, pp1-10, 1980.

365FITTED MESH B-SPLINE COLLOCATION...

366

A Procedure for Improving ConvexityPreserving Quasi-Interpolation

Costanza Conti and Rossana Morandi

Dip. di Energetica ”Sergio Stecco”, Universita di FirenzeVia Lombroso 6/17, 50134 Firenze, Italy.

costanza.conti@unifi.it, rossana.morandi@unifi.it

Abstract

In this paper a procedure is proposed to improve the shape pre-serving quasi-interpolant scheme defined by positive quadratic splineswith local support proposed in [3]. In fact, starting from that quasi-interpolant, a new shape preserving quasi-interpolant is defined byquasi-interpolating the weighted error obtained at the previous step.This strategy provides an error reduction at the data points and guar-antees the IP1-reproduction property as well as the end point interpo-lation condition.

Keywords: Quasi-interpolation, Convexity preservation.

1 Introduction

The problem of constructing a quasi-interpolant shape preserving function fit-ting a data set (xi, fi)n

i=1 is known to be of great interest in the applicationto graphical problems and in the construction of functions and curves follow-ing the shape suggested by the data (see for instance [2], [4] and referencestherein). In particular, a shape preserving quasi-interpolant in its simplestform, i.e. expressed as σ(x) =

∑n+1i=0 fiCi(x) for x ∈ [x1, xn], was defined

in [1], [5] and in [3] by using multiquadric functions and natural quadraticsplines with local support, respectively. Here, dealing with convexity preser-vation, an improving procedure based on a more general quasi-interpolationscheme is proposed. The procedure defines a new quasi-interpolant function,namely σ, of the more general type

σ(x) :=n+1∑

i=0

Λi(f)Ci(x), x ∈ [x1, xn]

367JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,367-378,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

where Λn+1i=0 are suitable functionals and Cin+1

i=0 are positive quadraticsplines with local support. The quasi-interpolant is obtained by quasi- inter-polating the error function, that is

σ(x) := σ(x) +n+1∑

i=0

wie(xi)Ci(x), x ∈ [x1, xn]

with win+1i=0 suitably chosen weights, x0, xn+1, f0, fn+1 additional data and

e(xi) := fi − σ(xi), i = 0 . . . , n + 1.

This strategy provides the convexity preservation, the error reduction at thedata points, the IP1-reproduction and the end point interpolation.

The outline of the paper is as follows. In section 2 some background informa-tion is given and in section 3 the strategy for improving the quasi-interpolantis presented. Also the reduction of the error at the data points is investigatedtogether with the shape preservation properties. Finally, in section 4 somepictures are given to illustrate the good performance of the method.

2 Preliminaries

As we refer to the quasi-interpolant scheme given in [3], we start by recallingit shortly. We also include in this section two new theorems useful for furtherdiscussions. We begin by recalling the following

Definition 2.1 A set of real values (xi, fi)i=1,...,n with x1 < . . . < xn,is called monotone increasing (decreasing) if fi ≤ fi+1, i = 1, . . . , n − 1(fi > fi+1, i = 1, . . . , n− 1) and convex (concave) if ∆i−1 ≤ ∆i ≤ ∆i+1, i =1, . . . , n − 2 (∆i−1 ≥ ∆i ≥ ∆i+1, i = 1, . . . , n − 2), where ∆i = fi+1−fi

xi+1−xi, i =

1, . . . , n− 1.

Given the data set fi, xini=1 we define hi = xi+1−xi, i = 1, . . . , n−1 and the

extra knots x1−i = x1− i ·h1, i = 1, 2, 3 and xn+i = xn−1 + i ·hn−1, i = 1, 2, 3.Then, we approximate the first derivatives of any function g at the point xi

by D12,ig(x)|x=xi

=∑1

k=−1 γikg(xi+k), where the coefficients γi

kk=−1,0,1 are

γi−1 = −hi

hi−1(hi−1+hi), γi

0 = hi−hi−1

hi−1hi, γi

1 = hi−1

hi(hi−1+hi). It is easy to show that

D12,ig(x)|x=xi

is a IP2-exact approximation of the first derivative of g at xi,

368 CONTI,MORANDI

i.e. it is exact for every polynomial of degree two. Next, we define thefunctions

Ci(x) = 1hi−1

D12,i−1v(x− y)|y=xi−1

− (hi+hi−1)hihi−1

D12,iv(x− y)|y=xi

+ 1hi

D12,i+1v(x− y)|y=xi+1

, i = 0, . . . , n + 1,(1)

where the discrete derivatives of v(x− y) = −14|x− y|(x− y) are done with

respect to the y variable at the specified points. To be more specific, thefunctions Ci, i = 0, . . . , n + 1 are

Ci(x) = −1hi−2(hi−2+hi−1)

v(x− xi−2) + 1hi−2hi−1

v(x− xi−1)+

( 1hi−1(hi−2+hi−1)

+ 1hi(hi−1+hi)

)v(x− xi)+−1

hihi+1v(x− xi+1) + 1

hi+1(hi+hi+1)v(x− xi+2),

(2)

that is quadratic splines. These functions are used to define the quasi-interpolant σ(x) :=

∑n+1i=0 fiCi(x) where the extra values are f0 := 2f1 − f2

and fn+1 := 2fn− fn−1. The properties of σ are summarized in the followingtheorem whose proof is given in [3].

Theorem 2.1 The functions Cii∈ZZ defined in (2) are piecewise quadraticpositive functions with local support on [xi−2, xi+2], for i ∈ ZZ. The quasi-interpolant σ(x) =

∑n+1i=0 fiCi(x) is IP1-reproducing and interpolating the end

points (x1, f1), (xn, fn). Furthermore, if the data set is monotone and/orconvex (concave), so is the quasi-interpolant σ.

The second theorem is about derivative interpolation at the end points. Thiscan be useful when dealing with locally monotone and locally convex/concavedata. The proof easily follows from the Ci definition by differentiation.

Theorem 2.2 The quasi-interpolant σ(x) =∑n+1

i=0 fiCi(x) is such that σ′(x1) =∆1 and σ′(xn) = ∆n−1.

We conclude this section with a theorem discussing the error behaviour atthe data points i.e. e(xi) = fi − σ(xi), 1 ≤ i ≤ n.

Theorem 2.3 The sign of the error at the data points depends on the con-vexity/concavity of the data.

369...QUASI-INTERPOLATION

Proof. For 1 ≤ i ≤ n, we have Ci−1(xi) = hi

2(hi−1+hi), Ci(xi) = 1

2and

Ci+1(xi) = hi−1

2(hi−1+hi). Thus, the error at the knots can be written as

e(xi) := fi − fi−1Ci−1(xi)− fiCi(xi)− fi+1Ci+1(xi)

= 12fi − fi−1

hi

2(hi−1+hi)− fi+1

hi−1

2(hi−1+hi)

= hi−1hi

2(hi−1+hi)(∆i−1 −∆i) , 1 ≤ i ≤ n.

(3)

3 Improving the convexity preserving quasi-

interpolation

In this section we present a strategy to define a new convexity preservingquasi-interpolant method. Without loss of generality, convex data is as-sumed. The concave case can be treated similarly.

Based on the properties of σ, the idea is to define a new quasi-interpolant σby quasi-interpolating the error at the data points, i.e. by considering

σ(x) := σ(x) +n+1∑

i=0

wie(xi)Ci(x) , (4)

with e(xi) := fi−σ(xi), 0 ≤ i ≤ n and the weights wi, 0 ≤ i ≤ n positive realvalues. The extra values are defined as e(x0) := −e(x2), e(xn+1) := −e(xn−1).Obviously, the quasi-interpolant σ can be also written as

∑n+1i=0 fiCi(x), hav-

ing set fi := fi + wie(xi)n+1i=0 . A lemma states the properties of σ.

Lemma 3.1 The quasi-interpolant σ defined in (4) is IP1-reproducing. Fur-thermore, setting w0 := w2 and wn+1 := wn−1 the end point interpola-tion is kept. Last, defining the new error at the data points as e(xi) :=f(xi)− σ(xi), i = 1, . . . , n, then the following holds for i = 1, . . . , n,

e(xi) = e(xi)− wi−1hie(xi−1)

2(hi−1 + hi)− wi

e(xi)

2− wi+1

e(xi+1)hi−1

2(hi−1 + hi).

Proof. The first statement obviously follows from the IP1-reproduction of σ,while the second one is just a matter of computation. The last one is the

370 CONTI,MORANDI

consequence of the local support of Ci and their values at the knots beingCi−1(xi) = hi

2(hi−1+hi), Ci(xi) = 1

2and Ci+1(xi) = hi−1

2(hi−1+hi).

3.1 Error reduction

Let us now investigate the error behaviour of the proposed procedure and theinfluence of the weight choice. First, we give the following theorem whoseproof is trivial.

Theorem 3.1 Let win+1i=0 be equal to a given positive constant less or equal

to 2. Then |e(xi)| ≤ maxj=1,...,n |e(xj)|, for 1 ≤ i ≤ n.

Figure 3.1. shows the behaviour of the quasi-interpolants and of piecewiselinear functions interpolating the error at the knots. The graphs are obtainedby iterating three times the procedure (4) assuming wi = 1, i = 0, . . . , n+1.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

Fig. 3.1. Quasi-interpolants (left) and errors (right)

Next, we discuss the case of non-uniform weights. In particular, Theorem3.2. analyzes the influence of the weights determined by the algorithm below.

371...QUASI-INTERPOLATION

Algorithm 3.1

1. Set wi := 0 for i = 0, . . . , n + 1

2. If e(x2) 6= 0&e(x3) 6= 0, set w2 := C ·min1, (h2+h3)e(x3)h3e(x2)

3. For i = 3, . . . , n− 23.1. If e(xi−1) 6= 0 &e(xi) 6= 0 &e(xi+1) 6= 0set wi := minai, bi, ciwith ai := C − wi−1hie(xi−1)

(hi−1+hi)e(xi), bi := C(hi+hi+1)e(xi+1)

hi+1e(xi)and

ci := −(hi−2+hi)hi−2e(xi)

[(wi−1 − C)e(xi−1) + wi−2hi−1e(xi−2)(hi−2+hi−1)

]

4. If e(xn−2) 6= 0 &e(xn−1) 6= 0set wn−1 := minan−1, bn−1with an−1 := C − wn−2hn−1e(xn−2)

(hn−2+hn−1)e(xn−2),

and bn−1 := −(hn−3+hn−2)hn−3e(xn−1)

[(wn−2 − C)e(xn−2) + wn−3hn−2e(xn−3)(hn−3+hn−2)

]

5. set w0 := w2

6. set wn+1 := wn−1

Theorem 3.2 Let win+1i=0 non negative real values defined by means of the

Algorithm 3.1, assuming C ≤ 4. If fini=1 is a convex data set, then |e(xi)| ≤

|e(xi)| for 1 ≤ i ≤ n. Furthermore, if C ≤ 2 it also follows that e(xi) ≤ 0 for1 ≤ i ≤ n.

Proof. Assuming |e(xi)| ≤ |e(xi)| for i = 2, . . . , n− 2 leads to

2e(xi) ≤ wi−1hie(xi−1)

2(hi−1 + hi)+ wi

e(xi)

2+ wi+1

e(xi+1)hi−1

2(hi−1 + hi)≤ 0 .

The inequality on the right hand side is always satisfied. For the inequalityon the left hand side the following six cases are possible:

If e(xi) = 0, then wi−1 := 0, wi+1 := 0.

If e(xi) 6= 0, then

if e(xi−1) = 0 & e(xi+1) = 0, then wi ≤ C,

372 CONTI,MORANDI

if e(xi−1) = 0 & e(xi+1) 6= 0, then wi ≤ C, wi+1 ≤ (hi−1+hi)[(wi−C)e(xi)]−hi−1e(xi+1)

,

if e(xi−1) 6= 0 & e(xi+1) = 0, then

wi−1 ≤ C(hi−1 + hi)e(xi)

hie(xi−1), wi ≤ C − wi−1hie(xi−1)

(hi−1 + hi)e(xi),

if e(xi−1) 6= 0 & e(xi+1) 6= 0, then

wi−1 ≤ C(hi−1+hi)e(xi)hie(xi−1)

, wi ≤ C − wi−1hie(xi−1)(hi−1+hi)e(xi)

wi+1 ≤ − (hi−1+hi)[(wi−C)e(xi)+wi−1hie(xi−1)

hi−1+hi]

hi−1e(xi+1).

Combining all the previous conditions on wi, we conclude that the choice

C ≤ 4 implies |e(xi)| ≤ |e(xi)|, i = 1, . . . , n.

Moreover, if we also require the non-positivity of e(xi), then

wi−1hie(xi−1)

2(hi−1 + hi)+ wi

e(xi)

2+ wi+1

e(xi+1)hi−1

2(hi−1 + hi)≥ e(xi) .

This is true whenever win+1i=0 are set as in Algorithm 3.1 with C ≤ 2.

3.2 Convexity preservation

To keep convexity, it is convenient to set the weights wi in such a way thatthe new data set fi = fi + wie(xi)n+1

i=0 is convex as well. Thus, let us

consider the quantities ∆i = fi+1−fi

hi, i = 0, . . . , n. It is easy to see that

∆1 − ∆0 = ∆1 −∆0, ∆n − ∆n−1 = ∆n −∆n−1 and

∆2 − ∆1 = (∆2 −∆1) + w3e(x3)−w2e(x2)h2

− w2e(x2)h1

,

∆i+1 − ∆i = (∆i+1 −∆i) + wi+2e(xi+2)−wi+1e(xi+1)hi+1

−wi+1e(xi+1)−wie(xi)hi

, i = 2, . . . , n− 3 ,

∆n−1 − ∆n−2 = (∆n−1 −∆n−2)− wn−1e(xn−1)hn−1

−wn−1e(xn−1)−wn−2e(xn−2)hn−2

.

(5)

The following algorithm can be used for choosing the weights in order topreserve convexity.

373...QUASI-INTERPOLATION

Algorithm 3.2

1. Set wi := 0, i = 0, . . . n + 1

2. If e(x2) 6= 0, set

w2 := h2(∆2−∆3)e(x2)

3. For i = 3, . . . , n− 23.1. If e(xi) 6= 0 set

wi := minhi(∆i−∆i+1)e(xi)

,hi−1(∆i−2−∆i−1+wi−1e(xi−1)

hi−1+hihi−1hi

−wi−2e(xi−2)

hi−2

e(xi)

4. If e(xn−1) 6= 0, set

wn−1 :=hn−2(∆n−3−∆n−2+wn−2e(xn−2)

hn−2+hn−3hn−2hn−3

−wn−3e(xn−3)

hn−3)

e(xn−1)

5. set w0 := w2

6. set wn+1 := wn−1

Theorem 3.3 Let win+1i=0 non negative real values defined via Algorithm

3.2 and fini=1 be a convex data set. Then the new data set fin

i=1 is alsoconvex.

The proof of this result based on (5) is along the same lines as the proof ofTheorem 3.2.

3.3 Monotonicity preservation

Now, assuming the data set is convex and monotone, the combination of thefollowing algorithms with the previous one guarantees the convexity and themonotonicity preservation. The first algorithm is related to the monotoneincreasing case, while the second one to the monotone decreasing case.

374 CONTI,MORANDI

Algorithm 3.3

1. For i = 2, . . . , n− 1, set wi := C2. For i = 2, . . . , n− 1

if e(xi+1) 6= 0 set wi+1 := −hi∆i+wie(xi)e(xi+1)

3. set w0 := w2

4. set wn+1 := wn−1

Algorithm 3.4

1. For i = 2, . . . , n− 1, set wi := C2. For i = n− 1, . . . , 2

if e(xi) 6= 0, set wi := hi∆i+wi+1e(xi+1)e(xi)

3. set w0 := w2

4. set wn+1 := wn−1

Theorem 3.4 Let win+1i=0 non negative real values defined via the previous

algorithms where C is any non-negative real value. Let fini=1 be convex and

monotone data. Then, the new data set fin+1i=0 is monotone as well.

Now, as the reduction of the error is obviously desirable, in case convexdata or convex and monotone data are given, the combination of the abovealgorithms (3.1, 3.2, 3.3 or 3.4) yields an improved shape preserving quasi-interpolant. Here, combination means that the final set of weights is definedby taking, elementwise, the minumun among the set produced by the algo-rithms.

Remarks. It is important to note that, due to the non symmetry of theproposed algorithms, even in case of symmetric data the improved quasi-interpolant σ given in (4) turns out to be non symmetric. A trivial solutionof this drawback is the weight “symmetrization” by taking the minimum be-tween the corresponding weights. In addition, we could performe algorithmsanalogous to the one discussed above also in a converse way ( i.e. checkingthe inequality from right to left). After symmetrization of this second sweepthe highest symmetric weights (between the two sweeps) can be used in (4)to reduce the error as much as possible.

According to the results given in Theorem 3.2, the choice C ≤ 2 in Algorithm3.1 allows us to iterate the procedure until an index i, such that both wi 6= 0and e(xi) 6= 0, is found. Furthermore, locally convex and monotone data canbe handled taking into consideration the results of Theorem 2.2.

375...QUASI-INTERPOLATION

4 Examples

We conclude this paper by showing some applications of the described pro-cedure. Three examples are given to put in evidence its effectiveness. Foreach of them two pictures are presented. The ones on the left display theshape preserving quasi-interpolants σ (−·) and σ (−) and the ones on theright the piecewise linear functions interpolating the values of the error at theknots, i.e. e(xi), ı = 1 . . . , n (−·) and e(xi), i = 1 . . . , n (−), respectively.In the pictures on the left the data sets fin

i=1 and fini=1 are denoted by

the symbols ‘∗′ and ‘+′, respectively. In all the examples the constant is setas C = 2.

The first example is related to data coming from the function f(x) = − log(x)when evaluated at the abscissae 0.01, 0.7, 3.6, 8, 13.4, 17. This set ofdata is convex and monotone decreasing. Thus, the weights are obtainedcombining algorithms 3.1, 3.2, 3.4.

The second one concerns the function f(x) = x −√

x2/2− x− 4 and the

points 4.1, 4.8, 1 + 3√

2, 5.9, 6.4, 7, 7.5. Here we combine algorithms3.1, 3.2 as we deal with convex data.

The last example deals with concave data taken by evaluating the functionf(x) =

√1− x2 at the points −1, −0.75, −0.35, 0, .5, .75, 1. The

concave quasi-interpolant is performed via algorithm 3.1 and the analogousto algorithm 3.2 for concave data.

0 2 4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Fig. 4.1. Quasi-interpolants (left) and errors (right)

376 CONTI,MORANDI

4 4.5 5 5.5 6 6.5 7 7.53.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

4 4.5 5 5.5 6 6.5 7 7.5−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Fig. 4.2. Quasi-interpolants (left) and errors (right)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Fig. 4.3. Quasi-interpolants (left) and errors (right)

References

[1] R. K. Beatson and M. J. D. Powell, Univariate multiquadric approxi-mation, Constructive Approximation, 8, 275–288 (1992).

[2] C. Conti and R. Morandi, Piecewise C1 Shape-Preserving Hermite In-terpolation, Computing, 56, 323–341 (1996).

[3] C. Conti, R. Morandi and C. Rabut, IP1-reproducing, Shape-preservingQuasi-Interpolant Using Positive Quadratic Splines with Local Sup-port, Journal of Computational and Applied Mathematics, 104, 111–122(1999).

377...QUASI-INTERPOLATION

[4] P. Costantini, An Algorithm for Computing Shape-Preserving Interpo-lating Spline of Arbitrary Degree, Journal of Computational and AppliedMathematics,22, 89–136 (1988).

[5] Z. Wu and R. Schaback, Shape preserving properties and convergence ofunivariate multiquadric quasi-interpolation, Acta Mathematicae Sinica,Study Bulletin, 10, No. 4, 441–446 (1994).

378 CONTI,MORANDI

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381

TABLE OF CONTENTS,JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,2006 ON THE UNIFORM EXPONENTIAL STABILITY OF EVOLUTION FAMILIES IN TERMS OF THE ADMISSIBILITY OF AN ORLICZ SEQUENCE SPACE, C.PREDA.S.DRAGOMIR,C.CHILARESCU,………………………………………253 GENERALIZED QUASILINEARIZATION TECHNIQUE FOR THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH GENERAL MIXED BOUNDARY CONDITIONS,B.AHMAD,R.KHAN,……………………………………………....267 EXISTENCE AND CONSTRUCTION OF FINITE TIGHT FRAMES, P.CASAZZA,M.LEON,……………………………………………………………...277 SEMIGROUPS AND GENETIC REPRESSION,G.FRAGNELLI,………………....291 THE IRREGULAR NESTING PROBLEM INVOLVING TRIANGLES AND RECTANGLES,L.FAINA,…………………………………………………………...307 WEAK CONVERGENCE AND PROKHOROV’S THEOREM FOR MEASURES IN A BANACH SPACE,M.ISIDORI,……………………………………………………...325 FITTED MESH B-SPLINE COLLOCATION METHOD FOR SOLVING SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS, M.KADALBAJOO,V.AGGARWAL,………………………………………………..349 A PROCEDURE FOR IMPROVING CONVEXITY PRESERVING QUASI- INTERPOLATION,C.CONTI,R.MORANDI,………………………………………..367

382

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Editor in -Chief: George Anastassiou Department of Mathematical Sciences The University Of Memphis Memphis,TN 38152,USA tel.901-678-3144,fax 901-678-2480 e-mail ganastss@memphis.edu www.msci.memphis.edu/~ganastss/jcaam Areas:Approximation Theory, Probability,Moments,Wavelet, Neural Networks,Inequalities,Fuzzyness. Associate Editors: 1) Ravi Agarwal Florida Institute of Technology Applied Mathematics Program 150 W.University Blvd. Melbourne,FL 32901,USA agarwal@fit.edu Differential Equations,Difference Equations,inequalities 2) Shair Ahmad University of Texas at San Antonio Division of Math.& Stat. San Antonio,TX 78249-0664,USA SAHMAD@utsa.edu Differential Equations,Mathematical Biology 3) Drumi D.Bainov Medical University of Sofia P.O.Box 45,1504 Sofia,Bulgaria drumibainov@yahoo.com Differential Equations,Optimal Control, Numerical Analysis,Approximation Theory 4) Carlo Bardaro Dipartimento di Matematica & Informatica Universita' di Perugia Via Vanvitelli 1 06123 Perugia,ITALY tel.+390755855034, +390755853822, fax +390755855024 bardaro@unipg.it , bardaro@dipmat.unipg.it Functional Analysis and Approximation Th.,

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Summability,Signal Analysis,Integral Equations, Measure Th.,Real Analysis 5) Francoise Bastin Institute of Mathematics University of Liege 4000 Liege BELGIUM f.bastin@ulg.ac.be Functional Analysis,Wavelets 6) Paul L.Butzer RWTH Aachen Lehrstuhl A fur Mathematik D-52056 Aachen Germany tel.0049/241/80-94627 office, 0049/241/72833 home, fax 0049/241/80-92212 Butzer@rwth-aachen.de Approximation Th.,Sampling Th.,Signals, Semigroups of Operators,Fourier Analysis 7) Yeol Je Cho Department of Mathematics Education College of Education Gyeongsang National University Chinju 660-701 KOREA tel.055-751-5673 Office, 055-755-3644 home, fax 055-751-6117 yjcho@nongae.gsnu.ac.kr Nonlinear operator Th.,Inequalities, Geometry of Banach Spaces 8) Sever S.Dragomir School of Communications and Informatics Victoria University of Technology PO Box 14428 Melbourne City M.C Victoria 8001,Australia tel 61 3 9688 4437,fax 61 3 9688 4050 sever.dragomir@vu.edu.au, sever@sci.vu.edu.au Math.Analysis,Inequalities,Approximation Th., Numerical Analysis, Geometry of Banach Spaces, Information Th. and Coding 9) A.M.Fink Department of Mathematics Iowa State University Ames,IA 50011-0001,USA

tel.352-392-9011 pardalos@ufl.edu Optimization,Operations Research 24) Svetlozar T.Rachev Dept.of Statistics and Applied Probability Program University of California,Santa Barbara CA 93106-3110,USA tel.805-893-4869 rachev@pstat.ucsb.edu AND Chair of Econometrics and Statistics School of Economics and Business Engineering University of Karlsruhe Kollegium am Schloss,Bau II,20.12,R210 Postfach 6980,D-76128,Karlsruhe,Germany tel.011-49-721-608-7535 rachev@lsoe.uni-karlsruhe.de Mathematical and Empirical Finance, Applied Probability, Statistics and Econometrics 25) Paolo Emilio Ricci Universita' degli Studi di Roma "La Sapienza" Dipartimento di Matematica-Istituto "G.Castelnuovo" P.le A.Moro,2-00185 Roma,ITALY tel.++39 0649913201,fax ++39 0644701007 riccip@uniroma1.it,Paoloemilio.Ricci@uniroma1.itOrthogonal Polynomials and Special functions, Numerical Analysis, Transforms,Operational Calculus, Differential and Difference equations 26) Cecil C.Rousseau Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA tel.901-678-2490,fax 901-678-2480 ccrousse@memphis.edu Combinatorics,Graph Th., Asymptotic Approximations, Applications to Physics 27) Tomasz Rychlik Institute of Mathematics Polish Academy of Sciences Chopina 12,87100 Torun, Poland T.Rychlik@impan.gov.pl Mathematical Statistics,Probabilistic Inequalities 28) Bl. Sendov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria

tel.515-294-8150 fink@math.iastate.edu Inequalities,Ordinary Differential Equations 10) Sorin Gal Department of Mathematics University of Oradea Str.Armatei Romane 5 3700 Oradea,Romania galso@uoradea.ro Approximation Th.,Fuzzyness,Complex Analysis 11) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis, Memphis,TN 38152,USA tel.901-678-2484 jgoldste@memphis.edu Partial Differential Equations, Semigroups of Operators 12) Heiner H.Gonska Department of Mathematics University of Duisburg Duisburg,D-47048 Germany tel.0049-203-379-3542 office gonska@informatik.uni-duisburg.de Approximation Th.,Computer Aided Geometric Design 13) Dmitry Khavinson Department of Mathematical Sciences University of Arkansas Fayetteville,AR 72701,USA tel.(479)575-6331,fax(479)575-8630 dmitry@uark.edu Potential Th.,Complex Analysis,Holomorphic PDE,Approximation Th.,Function Th. 14) Virginia S.Kiryakova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria virginia@diogenes.bg Special Functions,Integral Transforms, Fractional Calculus 15) Hans-Bernd Knoop Institute of Mathematics Gerhard Mercator University D-47048 Duisburg Germany tel.0049-203-379-2676

bsendov@bas.bg Approximation Th.,Geometry of Polynomials, Image Compression 29) Igor Shevchuk Faculty of Mathematics and Mechanics National Taras Shevchenko University of Kyiv 252017 Kyiv UKRAINE shevchuk@univ.kiev.ua Approximation Theory 30) H.M.Srivastava Department of Mathematics and Statistics University of Victoria Victoria,British Columbia V8W 3P4 Canada tel.250-721-7455 office,250-477-6960 home, fax 250-721-8962 harimsri@math.uvic.ca Real and Complex Analysis,Fractional Calculus and Appl., Integral Equations and Transforms,Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th. 31) Ferenc Szidarovszky Dept.Systems and Industrial Engineering The University of Arizona Engineering Building,111 PO.Box 210020 Tucson,AZ 85721-0020,USA szidar@sie.arizona.edu Numerical Methods,Game Th.,Dynamic Systems, Multicriteria Decision making, Conflict Resolution,Applications in Economics and Natural Resources Management 32) Gancho Tachev Dept.of Mathematics Univ.of Architecture,Civil Eng. and Geodesy 1 Hr.Smirnenski blvd BG-1421 Sofia,Bulgaria Approximation Theory 33) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock Germany manfred.tasche@mathematik.uni-rostock.de Approximation Th.,Wavelet,Fourier Analysis, Numerical Methods,Signal Processing,

knoop@math.uni-duisburg.de Approximation Theory,Interpolation 16) Jerry Koliha Dept. of Mathematics & Statistics University of Melbourne VIC 3010,Melbourne Australia koliha@unimelb.edu.au Inequalities,Operator Theory, Matrix Analysis,Generalized Inverses 17) Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston,RI 02881,USA kulenm@math.uri.edu Differential and Difference Equations 18) Gerassimos Ladas Department of Mathematics University of Rhode Island Kingston,RI 02881,USA gladas@math.uri.edu Differential and Difference Equations

Image Processing,Harmonic Analysis 34) Chris P.Tsokos Department of Mathematics University of South Florida 4202 E.Fowler Ave.,PHY 114 Tampa,FL 33620-5700,USA profcpt@math.usf.edu,profcpt@chuma1.cas.usf.edu Stochastic Systems,Biomathematics, Environmental Systems,Reliability Th. 35) Lutz Volkmann Lehrstuhl II fuer Mathematik RWTH-Aachen Templergraben 55 D-52062 Aachen Germany volkm@math2.rwth-aachen.de Complex Analysis,Combinatorics,Graph Theory

Preface

This special issue contains invited research papers on “Wavelets and Applications.” Results in wavelets,

frames, subdivision schemes, spline wavelets, and wavelet applications in statistics/bio-statistics were among

the areas discussed.

The paper by Bruce Kessler, following a macroelement approach, deals with construction of an orthogonal

scaling vector of differentiable fractal interpolation functions, with symmetry properties, that generates a

space including C1 cubic splines.

In his paper, Qun Mo shows that normalized tight multiwavelet frames with highest possible order of van-

ishing moments can be derived from refinable functions under the multiwavelet settings.

The paper by Bin Han and Thomas P.-Y. Yu, entitled Face-Based Hermite Subdivision Schemes , studies

the symmetry and structure of the vectors for the sum rule orders of the refinement mask for a type of vector

Hermite subdivision scheme, called “face-based Hermite subdivision scheme by the authors. Examples of

the refinement masks with the quincunx dilation matrix and 2I2 are constructed.

In the paper by Don Hong and Qingbo Xue, piecewise linear prewavelets over a bounded domain with regular

triangulation are constructed by investigating the orthogonal conditions directly.

The contribution by Lutai Guan and Tan Yang gives results on spline-wavelets on refined grids and appli-

cations. So-called addition spline-wavelets are constructed using natural spline interpolation skills. Applica-

tions in image compression are also included.

The following two papers are on wavelet applications in statistics/bio-statistics. In the paper by Jianzhong

Wang, the author first gives a brief review of wavelet shrinkage and then applies the variational method for

wavelet regression and shows the relation between the variational method and the wavelet shrinkage.

In the paper by Don Hong and Yu Shyr, many applications of wavelets in cancer study are reviewed,

in particular, in medical imaging such as in detection of microcalcifications in digital mammography and

functional magnetic resonance imaging, and molecular biology area such as in genome sequence analysis,

protein structure investigation, and gene expression data analysis. Wavelet-based method for MALDI-TOF

mass spectrometry data analysis is also reviewed.

We are grateful to all the authors who contributed to the special issue. We appreciate all the referees who

reviewed the submitted papers. Their efforts and suggestions made it possible to publish scientific papers

of high quality. We would like to express our special thanks to George Anastassiou for his patience and

support and to Benedict Bobga, Shuo Chen, Yuannan Diao, Brad Dyer, Qingtang Jiang, David Roach,

Bashar Shakhtour, Wasin So, and Qiyu Sun for their assistance in the editing of this special issue. Last

but not least we also thank the support of the Department of Biostatistics and Ingram Cancer Center of

Vanderbilt University, and the Department of Mathematics, Middle Tennessee State University.

Don HongMiddle Tennessee State UniversityMurfreesboro, TN, USA

Yu ShyrVanderbilt UniversityNashville, TN, USA

391

JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.3,391,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,391,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

HONG.SHYR

392

An Orthogonal Scaling Vector Generating a Spaceof C1 Cubic Splines Using Macroelements

Bruce KesslerDepartment of MathematicsWestern Kentucky University

Bowling Green, KY, USAbruce.kessler@wku.edu

Abstract

The main result of this paper is the creation of an orthogonal scaling vector of fourdifferentiable functions, two supported on [−1, 1] and two supported on [0, 1], that gen-erates a space containing the classical spline space S1

3 (Z) of piecewise cubic polynomialson integer knots with one derivative at each knot. The author uses a macroelementapproach to the construction, using differentiable fractal function elements defined on[0, 1] to construct the scaling vector. An application of this new basis in an imagecompression example is provided.

AMS Subject Classification Numbers: 42C40, 65D15Keywords: orthogonal bases, scaling vectors, spline spaces, macroelements, wavelets, multi-wavelets

1 Introduction

Orthogonal scaling vectors have numerous applications in signal and image processing, in-cluding image compression, denoising, and edge detection. Geronimo, Hardin, and Masso-pust were among the first to use more than one scaling function, in their construction ofthe GHM orthogonal scaling vector in [7]. The space generated by the GHM functions andtheir integer translates had approximation order 2, like the space generated by the singlecompactly-supported D4 orthogonal scaling function constructed by Daubechies in [3], butalso contained the space of continuous piecewise linear polynomials on integer knots. Also,by using more than one function, they were able to build scaling functions and waveletswith symmetry properties. The only single compactly-supported scaling function with sym-metry properties is the characteristic function on [0, 1), known as the Haar basis. Othertypes of orthogonal scaling vectors have been constructed in [4], [8], [11], and [13] that havecompactly-supported elements that lack continuous derivatives. Han and Jiang constructed alength-4 scaling vector in [9] that has approximation order 4 and two continuous derivatives.

Hardin and Kessler put the constructions of Geronimo and Hardin from [7] (with Mas-sopust) and [4] (with Donovan) into a macroelement framework in [10]. The construction ofscaling vectors from a single macroelement supported on [0, 1] is important, since these typesof scaling functions will be orthogonal interval-by-interval, and the intervals over which thebasis is defined need not be of uniform length. This adaptability to an arbitrary partition ofR promises to give greater flexibility to the use of these types of bases in applications. Whilebuilding bases on variable length intervals is outside the scope of this paper, we should keepin mind that most of the results here can be easily adapted to intervals of arbitrary length.

393JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,393-413,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

The main result in this paper is the creation of a orthogonal scaling vector of length 4 ofcompactly-supported, differentiable piecewise fractal interpolation functions, with symmetryproperties, that generates a space including the classical spline space S1

3 (Z) of differentiablecubic polynomials on integer knots. This scaling vector has a symmetric/antisymmetricpair supported on [−1, 1] and a symmetric/antisymmetric pair supported on [0, 1]. Theconstruction will follow a macroelement approach; that is, we construct a macroelementwith the desirable properties using fractal functions and then construct a scaling vectorfrom the macroelement. Fractal functions have been used previously in [4], [6], [7], and [11]to construct orthogonal scaling vectors. We show that a shorter-length scaling vector cannot be found with the same properties, and then show this new basis in use in an imagecompression example. We believe this to be the first scaling vector to be constructed withall of the above-mentioned properties.

1.1 Scaling Vectors

A vector Φ = (φ1, φ2, . . . , φr)T of functions defined on Rk is said to be refinable if

Φ = Nk2

∑giΦ(N · −i) (1)

for some integer dilation N > 1, i ∈ Zk, and for some sequence of r × r matrices gi. (The

normalization factor Nk2 can be dropped, but is convenient for applications.) A scaling vector

is a refinable vector Φ of square-integrable functions where the set of the components of Φand their integer tranlates are linearly independent. An orthogonal scaling vector Φ is ascaling vector where the functions φ1, . . . , φr are compactly supported and satisfy

〈φi, φj(· − n)〉 = δi,jδ0,n, i, j ∈ 1, . . . , r, n ∈ Zk,

where the inner product is the standard L2(Rk) integral inner product

〈f, g〉 =

∫Rk

f(x)g(x)dx

and δ is Kronecker’s delta (1 if indices are equal, 0 otherwise.) A scaling vector Φ is said togenerate a closed linear space denoted by

S(Φ) = closL2 span φi(· − j) : i = 1, . . . , r, j ∈ Z .

Two scaling vectors Φ and Φ are equivalent if S(Φ) = S(Φ). The scaling vector Φ is said toextend Φ, or be an extension of Φ, if S(Φ) ⊂ S(Φ).

Scaling vectors are important because they provide a framework for analyzing functionsin L2(Rk). A multiresolution analysis (MRA) of L2(Rk) of multiplicity r is a set of closedlinear spaces (Vp) such that

1. · · · ⊃ V−2 ⊃ V−1 ⊃ V0 ⊃ V1 ⊃ V2 · · · ,

2.⋃

p∈Z Vp = L2(Rk),

3.⋂

p∈Z Vp = 0,

394 KESSLER

4. f ∈ V0 iff f(N−j·) ∈ Vj, and

5. there exists a set of functions φ1, . . . , φr whose integer translates form a Riesz basis ofV0.

From the above definitions, it is clear that scaling vectors can be used to generate MRA’s,with V0 = S(Φ). Jia and Shen proved in [15] that if the components of a scaling vectorΦ are compactly-supported, then Φ will always generate an MRA. All the scaling vectorsdiscussed in this paper will consist of compactly-supported functions, and therefore, willgenerate MRA’s. A function vector Ψ = (ψ1, . . . , ψr(Nk−1))

T , such that ψi ∈ V−1 for i =1, . . . , r(Nk − 1) (see [14]) and such that S(Ψ) = V−1 V0, is called a multiwavelet, and theindividual ψi are called wavelets.

1.2 Macroelements on [0, 1]

We will use the notation f (j)(x) to denote the jth derivative of f(x), with the conventionf (0)(x) = f(x). As a convenience, we will use the notation f (j)(0) and f (j)(1) to denotelim

x→0+f (j)(x) and lim

x→1−f (j)(x), respectively, although the notation may not always be mathe-

matically rigorous.A Cp macroelement defined on [0, 1] is a vector of the form (l1, . . . , lk, r1, . . . , rk,m1, . . . ,mn)T

where the set of elements are linearly independent functions supported on [0, 1] with p con-tinuous derivatives such that

1. l(j)i (0) = r

(j)i (1) = 0 for i = 1, . . . , k,

2. m(j)i (0) = m

(j)i (1) = 0 for i = 1, . . . , n, and

3. l(j)i (1) = r

(j)i (0) for i = 1, . . . , k

for j = 0, . . . , p. A macroelement is orthonormal if 〈li, rj〉 = 0 for i, j ∈ 1, . . . , k, 〈li,mj〉 =〈ri,mj〉 = 0 for i ∈ 1, . . . , k and j ∈ 1, . . . , n, and each element is normalized.

A macroelement Λ is refinable if there are (2k+n)× (2k+n) matrices p0, . . . , pN−1 suchthat

Λ(x) =√NpiΛ(Nx− i) for x ∈

[i

N,i+ 1

N

], i = 0, . . . , N − 1. (2)

Because of the linear independence of the components of Λ, the matrix coefficients will beunique if they exist. Note that a refinable C0 macroelement by necessity has li(1) = ri(0) 6= 0for some i, since if all elements were 0 at x = 0, 1, then each element would be 0 at N j-adicpoints on [0, 1] as j →∞. Hence, each element would be 0, a contradiction. Likewise, notethat a refinable C1 macroelement by necessity has l′i(1) = r′i(0) 6= 0 for some i, since if allelements had a zero derivative at x = 0, 1, then each element would be a constant function,also a contradiction.

Lemma 1. A refinable Cp macroelement Λ = (l1, . . . , lk, r1, . . . , rk,m1, . . . ,mn)T definedon [0, 1] has an associated scaling vector Φ of length k + n and support [−1, 1]. If themacroelement Λ is orthonormal, then the scaling vector Φ is equivalent to an orthogonalscaling vector.

395AN ORTHOGONAL SCALING...

Proof. Let l = (l1, . . . , lk)T , r = (r1, . . . , rk)

T , and m = (m1, . . . ,mn)T . Let ali, a

ri , and am

i

be the k × k, k × k, and k × n matrices, respectively, such that[al

i ari am

i

]Λ(2x− i) = l(x) if x ∈

[i

N,i+ 1

N

], i = 0, . . . , N − 1.

Likewise, let bli, bri , and bmi be the k × k, k × k, and k × n matrices, respectively, such that[

bli bri bmi]Λ(2x− i) = r(x) if x ∈

[i

N,i+ 1

N

], i = 0, . . . , N − 1

and let cli, cri , and cmi be the n× k, n× k, and n× n matrices, respectively, such that[cli cri cmi

]Λ(2x− i) = m(x) if x ∈

[i

N,i+ 1

N

], i = 0, . . . , N − 1.

Then the matrix coefficients in (2) can be written in block-matrix form

pi =

ali ar

i ami

bli bri bmicli cri cmi

.Note that many of the block matrices are redundant: al

i−1 = ari , b

li−1 = bri , and cli−1 = cri

for i = 0, . . . , N − 1 since the macroelement components are continuous, and alN−1 = br0 due

to the endpoint conditions of the macroelement. Also, many of the block matrices are zeromatrices: ar

0 = blN−1 = 0k×k and cr0 = clN−1 = 0n×k due to the endpoint conditions of themacroelement.

Define

φi(x) =1√2

li(x+ 1) for x ∈ [−1, 0]ri(x) for x ∈ [0, 1]

, i = 1, . . . , k, and (3)

φk+i(x) = mi(x) for x ∈ [0, 1], i = 1, . . . , n. (4)

Then the function vector Φ = (φ1, . . . , φk+n)T satisfies (1), with

gi =

[ar

N+i amN+i

0n×k 0n×n

]i = −N, . . . ,−1 and gi =

[bri bmicri cmi

]i = 0, . . . , N − 1.

Hence, Φ is refinable, and supported completely in [−1, 1].If Λ is orthonormal, then by definition, Φ meets the criteria of an orthogonal scal-

ing vector, except that possibly 〈φi, φj〉 6= 0 for i, j ∈ 1, . . . , k, i 6= j, and for i, j ∈k + 1, . . . , k + n, i 6= j. However, we may replace φ1, . . . , φk with an orthonormalset φ1, . . . , φk and φk+1, . . . , φk+n with an orthonormal set φk+1, . . . , φk+n, so thatφ1, . . . , φk, φk+1, . . . , φk+n is an orthogonal scaling vector.

Let span Λ refer to the span of the elements of Λ. Two macroelements Λ and Λ areequivalent if span Λ = span Λ. The macroelement Λ is said to extend Λ, or be an extensionof Λ, if span Λ ⊂ span Λ. In this paper, we will extend macroelements for the purpose ofextending scaling vectors, using the following lemma. We use the notation χ[a,b] to be thecharacteristic function defined by

χ[a,b] =

1 for x ∈ [a, b],0 otherwise.

396 KESSLER

Lemma 2. Let Λ be a refinable Cp macroelement defined on [0, 1], and let Φ be the associatedscaling vector as defined in (3) and (4), for p = 0, 1. If Λ is a Cp macroelement extensionof Λ, then the associated scaling vector Φ as defined in (3) and (4) is an extension of Φ.

Proof. Let Λ = l1, . . . , lk, r1, . . . , rk,m1, . . . ,mn and Λ = l1, . . . , lk′ , r1, . . . , rk′ , m1, . . . , mn′,where Λ is an extension of Λ, so k ≤ k′ and n ≤ n′. Consider a basis element φ ∈φi(· − j) : φi ∈ Φ, i ∈ 1, . . . , k + n, j ∈ Z. If suppφ ⊂ [j, j + 1] for some j ∈ Z,then φ(x+ j) ∈ span m1, . . . ,mn ⊂ span m1, . . . , mn′. From the definition of Φ in (4),then φ ∈ S(Φ).

Otherwise, suppφ ⊂ [j, j + 2] for some j ∈ Z. Let l = φχ[j,j+1] and r = φχ[j+1,j+2].

Then l(x + j) ∈ span l1, . . . , lk,m1, . . . ,mn ⊂ span l1, . . . , lk′ , m1, . . . , mn′ and r(x +j + 1) ∈ span r1, . . . , rk,m1, . . . ,mn ⊂ span r1, . . . , rk′ , m1, . . . , mn′. From the match-up conditions in the definition of the macroelement and the definition of Φ in (3), thenφ ∈ S(Φ). Thus, S(Φ) ⊂ S(Φ) and Φ is an extension of Φ.

In the following example, we show a simple way to extend a macroelement, and hence, ascaling vector.Example 1. Let l1 = xχ[0,1] and r1 = (1−x)χ[0,1]. Then Λ = (l1, r1)

T is a C0 macroelement.It is also refinable, since

Λ(x) =√

2

[1

2√

20

12√

21√2

]Λ(2x) if x ∈

[0, 1

2

][

1√2

12√

2

0 12√

2

]Λ(2x− 1) if x ∈

[12, 1].

Therefore, we have the scaling vector Φ = (φ1) given in (3) and (4), with

Φ(x) =√

2

[1

2√

2Φ(2x+ 1) +

1√2Φ(2x) +

1

2√

2Φ(2x− 1)

].

Then Φ is the linear B-spline, and generates S(Φ) = S01 (Z), the spline space of continuous

piecewise linear polynomials on integer knots.Both Λ and Φ can be extended by the addition of the function m1(x) = φ2 = 4x(1 −

x)χ[0,1]. Then Λ = (l1, r1,m1)T is refinable, since

Λ(x) =√

2

1

2√

20 0

12√

21√2

01√2

0 14√

2

Λ(2x) if x ∈[0, 1

2

]

1√2

12√

20

0 12√

20

0 1√2

14√

2

Λ(2x− 1) if x ∈[

12, 1],

and, by Lemma 1, we have the scaling vector Φ = (φ1, φ2)T , with

Φ(x) =√

2

([ 12√

20

0 0

]Φ(2x+ 1) +

[1√2

0

0 14√

2

]Φ(2x) +

[1

2√

20

1√2

14√

2

]Φ(2x− 1)

).

397AN ORTHOGONAL SCALING...

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1 φ1 φ2

Figure 1: The scaling functions φ1 and φ2 from Example 1.

By Lemma 2, Φ is an extension of Φ. In fact, S(Φ) = S02 (Z) ⊃ S0

1 (Z). Both functions areillustrated in Figure 1.

1.3 Fractal Interpolation Functions

Let C0([0, 1]) denote the space of continuous functions defined over [0, 1] that are 0 at x = 0, 1.Let C1([0, 1]) denote the subspace of differentiable functions in C0([0, 1]) whose derivativesare 0 at x = 0, 1. Let Λ be a refinable macroelement of length n, and let Π be a functionvector of length k defined by

Π(x) = piΛ(Nx− i) for x ∈[i

N,i+ 1

N

], i = 0, . . . , N − 1,

for some k × n matrices pi such that Π(x) ∈ C0([0, 1])k. Then a vector Γ of the form

Γ(x) = Π(x) +N−1∑i=0

siΓ(Nx− i) ∈ C0([0, 1])k, (5)

where each si is a k × k matrix and maxi‖si‖∞ < 1, is a vector of fractal interpolation

functions (FIF’s). (See [1] and [2] for a more detailed introduction to FIF’s.) By definition,the vector Λ = (ΛT ,ΓT )T is a refinable C0 macroelement that extends Λ.

Lemma 3. Let Γ be a FIF satisfying (5). If Π(x) ∈ C1([0, 1])k and maxi=1,...,N−1

‖si‖∞ < 1N

,

then Γ ∈ C1([0, 1])k.

Proof. Since Γ satisfies (5), then

Γ′(x) = Π′(x) +N−1∑i=0

NsiΓ′(Nx− i).

Since Π′(x) ∈ C0([0, 1])k and maxi=1,...,N−1

‖Nsi‖∞ = N maxi=1,...,N−1

‖si‖∞ < N 1N

= 1, then Γ′(x) is

by definition a FIF, and Γ′(x) ∈ C0([0, 1])k. Therefore, Γ(x) ∈ C1([0, 1])k.

398 KESSLER

Consider a C0 or C1 macroelement Λ = (l1, . . . , lk, r1, . . . , rk,m1, . . . ,mn)T defined on[0, 1] that is not orthogonal. We can not simply apply the Gram-Schmidt process to thecomponents of Λ to obtain an orthonormal macroelement, since the resulting functions willnot satisfy the endpoint criteria. In fact, we can not apply the process to any subset ofelements that includes a li and rj and still have the same type of macroelement. (We couldgo from a C1 macroelement to a C0 macroelement, but the associated scaling vector wouldnot be an extension of the original.) However, we can apply the Gram-Schmidt process tothe set of functions M = m1, . . . ,mn to get M = m1, . . . , mn, and then subtract PM ,the orthogonal projection onto the space spanned by M , from each of the other elements,giving the equivalent macroelement

Λ = ((I − PM)l1, . . . , (I − PM)lk, (I − PM)r1, . . . , (I − PM)rk, m1, . . . , mn)T .

If〈(I − PM)li, (I − PM)rj〉 = 0 for i, j = 1, . . . , k, (6)

(and each element is normalized), then Λ is an orthonormal macroelement. This is thefractal function approach for extending a macroelement: add FIF’s to the set M , hencethe macroelement, so that (6) is satisfied. (See [5] for a broader discussion on constructingintertwined MRA’s.)Example 2. The scaling vector shown in this example was originally constructed by Geron-imo, Hardin, and Massopust in [7], although not in the macroelement context, and is recon-structed by Hardin and Kessler in detail using macroelements in [10]. It is widely known asthe GHM scaling vector.

Consider from Example 1 the C0 macroelement Λ and the scaling vector Φ = (φ1) thatgenerates S0

1 (Z). In order to extend Λ to an orthonormal C0 macroelement, we construct aFIF satisfying

u(x) = φ1(2x− 1) + s0u(2x) + s1u(2x− 1), maxi=0,1

|si| < 1,

such that 〈(I − Pu)l1, (I − Pu)r1〉 = 0. It was shown in [7] and [10] that the orthogonalitycondition is satisfied by s0 = s1 = −1

5. By letting

l1 =(I − Pu)l1‖(I − Pu)l1‖

, r1 =(I − Pu)r1‖(I − Pu)r1‖

, and m1 =u

‖u‖,

we have the orthonormal C0 macroelement Λ = (l1, r1, m1)T , equivalent to (l1, r1, u)

T andan extension of Λ. The associated scaling vector Φ = (φ1, φ2)

T defined in (3) and (4) is theorthogonal GHM scaling vector, and is illustrated in Figure 2.Example 3. The scaling vector shown in this example was originally constructed by Dono-van, Geronimo, and Hardin in [4], although not in the macroelement context, and again byHardin and Kessler in detail in [10] using a macroelement approach.

Consider from Example 1 the C0 macroelement Λ and the scaling vector Φ = (φ1, φ2)T

that generates S02 (Z). In order to extend Λ to an orthonormal C0 macroelement, we construct

a FIF satisfying

u(x) = φ2(2x)− φ2(2x− 1) + su(2x) + su(2x− 1) for |s| < 1,

399AN ORTHOGONAL SCALING...

-1 -0.5 0.5 1

-0.5

0.5

1

1.5

2

φ1

φ2

Figure 2: The orthogonal GHM scaling vector Φ from Example 2.

such that 〈(I − PM)l1, (I − PM)r1〉 = 0, where M = φ2, u. (Note that 〈φ2, u〉 = 0, sinceφ2 and u are symmetric and antisymmetric, respectively, about x = 1

2.) It was shown in [4]

and [10] that the orthogonality condition is satisfied by s = 2−√

106

≈ −0.1937. By letting

l1 =(I − PM)l1‖(I − PM)l1‖

, r1 =(I − PM)r1‖(I − PM)r1‖

, m1 =m1

‖m1‖, and m2 =

u

‖u‖,

we have the orthonormal C0 macroelement Λ = (l1, r1, m1, m2), equivalent to (l1, r1,m1,m2)T

and an extension of Λ. The associated scaling vector Φ = (φ1, φ2, φ3)T defined in (3) and (4)

is an orthogonal scaling vector, and is illustrated in Figure 3.

-1 -0.5 0.5 1

-1

1

2 φ1

φ2

φ3

Figure 3: The orthogonal scaling vector Φ from Example 3.

2 Main Results

In this section, we will construct a refinable C1 macroelement on [0, 1] which, in turn, providesa scaling vector Φ that generates S1

3 (Z). We will show that two additional functions areneeded to extend that macroelement to an orthonormal macroelement that is refinable withdilation 2, and an explicit construction of that macroelement and the associated orthogonalscaling vector will be provided. Lastly, although general methods are available for finding a

400 KESSLER

multiwavelet associated with an orthogonal scaling vector (see [10], [16], and [17]), we willshow an equivalent macroelement approach to constructing the wavelets.

2.1 Scaling Vector Generating S13(Z)

Consider the following basis for cubic polynomials defined on [0, 1] and 0 elsewhere:

l1(x) = (−2x3 + 3x2)χ[0,1], l2(x) = (x3 − x2)χ[0,1],

r1(x) = (2x3 − 3x2 + 1)χ[0,1], and r2(x) = (x3 − 2x2 + x)χ[0,1].

One may verify that Λ = (l1, l2, r1, r2)T is a C1 macroelement. It is also refinable, since

Λ =√

2

1

2√

23

4√

20 0

− 18√

2− 1

8√

20 0

12√

2− 3

4√

21√2

01

8√

2− 1

8√

20 1

2√

2

Λ(2x) if x ∈[0, 1

2

],

1√2

0 12√

23

4√

2

0 12√

2− 1

8√

2− 1

8√

2

0 0 12√

2− 3

4√

2

0 0 18√

2− 1

8√

2

Λ(2x− 1) if x ∈[

12, 1].

(7)

From direct computation, we know that

〈l1, r1〉 =9

70, 〈l1, r2〉 =

13

420, 〈l2, r1〉 = − 13

420, and 〈l2, r2〉 = − 1

140,

so Λ is not an orthonormal macroelement. As in (3), we define

φi(x) =1√2

li(x+ 1) if x ≤ 0ri(x) if x ≥ 0

, i = 1, 2,

so that we have the scaling vector Φ = (φ1, φ2)T satisfying

Φ(x) =√

2

([1

2√

23

4√

2

− 18√

2− 1

8√

2

]Φ(2x+ 1) +

[1√2

0

0 12√

2

]Φ(2x) +

[1

2√

2− 3

4√

21

8√

2− 1

8√

2

]Φ(2x− 1)

),

where S(Φ) = S13 (Z). Both functions are illustrated in Figure 4.

2.2 An Orthogonal, Refinable Extension

We may extend Φ to a refinable vector of length 3 by adding a single m1 or by adding l3 andr3 functions to the macroelement Λ. However, as we show in the following theorem, neithermethod will produce an orthogonal extension. Finding a length-4 orthogonal extension ofΦ is nontrivial. The proof of the following theorem gives a construction of one such scalingvector using the fractal function idea on the C1 macroelement Λ defined above.

401AN ORTHOGONAL SCALING...

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1φ1

φ2

Figure 4: The scaling vector Φ generating S13 (Z).

Theorem 1. The orthogonal scaling vector of least length that extends the length-2 scalingvector which generates S1

3 (Z) has length 4.

Proof. There are two parts to the proof: we first show that a single m1 or l3-r3 pair addedto the macroelement, thereby adding one function to the scaling vector, can not be foundsuch that all of the necessary orthogonality conditions are satisfied, and then we show thata 2-vector of fractal functions (with symmetry properties, no less) can be found so that thenecessary conditions are satisfied.

Suppose that the single normalized function m1 added to the macroelement Λ gives a C1

extension of the macroelement. Since we are assuming no symmetry properties for m1, wehave four orthogonality conditions from (6), which reduce to the the four equations

〈li, rj〉 = 〈li,m1〉〈rj,m1〉, i, j ∈ 1, 2.

The system of four equations with the unknowns 〈l1,m1〉, 〈l2,m1〉, 〈r1,m1〉, and 〈r2,m1〉 isinconsistent, so no single function m1 can satisfy all of the necessary conditions.

Now suppose that two orthonormal functions l3 and r3 added to the macroelement Λ givea C1 extension of the macroelement. To be an orthogonal extension, l3 and r3 will need tosatisfy the four orthogonality conditions

〈li − 〈li, l3〉l3, rj − 〈rj, r3〉r3〉 = 0 i, j ∈ 1, 2,

while satisfying the endpoint conditions

li(1)− 〈li, l3〉l3(1) = ri(0)− 〈ri, r3〉r3(0) i = 1, 2.

The endpoint conditions force 〈l1, l3〉 = 〈r1, r3〉 and 〈l2, l3〉 = 〈r2, r3〉, the values of which wewill denote α and β, respectively. Then the four orthogonality conditions become

α(〈l1, r3〉+ 〈r1, l3〉) = 970

α〈r2, l3〉+ β〈l1, r3〉 = 13420

α〈l2, r3〉+ β〈r1, l3〉 = − 13420

β(〈l2, r3〉+ 〈r2, l3〉) = − 1140

402 KESSLER

with unknowns 〈l1, r3〉, 〈l2, r3〉, 〈r1, l3〉, and 〈r2, l3〉. Again, the system is inconsistent, so nopair of functions l3 and r3 can satisfy all of the necessary conditions.

To show that a length-4 orthogonal scaling vector is possible, we will actually constructone by adding two functions m1 and m2 to the macroelement Λ. Consider two FIF m1 andm2 and function vector Γ = (m1,m2)

T satisfying

Γ(x) =

[α 00 β

]Φ(2x− 1) +

[0 qs t

]Γ(2x) +

[0 −q−s t

]Γ(2x− 1)

where the maximum ∞-norm of the matrix coefficients of Γ(2x) and Γ(2x−1) is 1 and α, β 6=0 are chosen so that ‖m1‖ = ‖m2‖ = 1. The extended macroelement Λ = (l1, l2, r1, r2,m1,m2)

T

is refinable, with

Λ(x) =√

2

12√

23

4√

20 0 0 0

− 18√

2− 1

8√

20 0 0 0

12√

23

4√

21√2

0 0 01

8√

2− 1

8√

20 1

2√

20 0

α√2

0 0 0 0 q√2

0 β√2

0 0 s√2

t√2

Λ(2x) if x ∈

[0, 1

2

],

1√2

0 12√

23

4√

20 0

0 12√

2− 1

8√

2− 1

8√

20 0

0 0 12√

23

4√

20 0

0 0 18√

2− 1

8√

20 0

0 0 α√2

0 0 − q√2

0 0 0 β√2

− s√2

t√2

Λ(2x− 1) if x ∈

[12, 1].

(8)

Let M = m1,m2. In order to construct an orthonormal macroelement equivalent to Λ,the elements of M must satisfy the following conditions:

〈(I − PM)li, (I − PM)rj〉 = 0, i, j ∈ 1, 2.

One can verify that m1 and m2 are symmetric and antisymmetric, respectively, aboutx = 1

2, so that 〈m1,m2〉 = 0. Also, due to symmetry,

〈r1,m1〉 = 〈l1,m1〉, 〈r1,m2〉 = −〈l1,m2〉, 〈r2,m1〉 = −〈l2,m1〉, and 〈r2,m2〉 = 〈l2,m2〉.

Thus, the four orthogonality conditions reduce to the three equations970− 〈r1,m1〉2 + 〈r1,m2〉2 = 0

1140

− 〈r2,m1〉2 + 〈r2,m2〉2 = 013420

− 〈r1,m1〉〈r2,m1〉+ 〈r1,m2〉〈r2,m2〉 = 0.(9)

Using direct computation, we know that

〈r1, r1〉 = 〈l1, l1〉 =13

35, 〈r2, r2〉 = 〈l2, l2〉 =

1

105, and 〈r1, r2〉 = −〈l1, l2〉 =

11

210.

403AN ORTHOGONAL SCALING...

Using the coefficients in (8), we may expand and solve for 〈r1,m1〉, 〈r1,m2〉, 〈r2,m1〉, and〈r2,m2〉 in the following system:

2〈r1,m1〉 = α2

2〈r2,m1〉 = −14q(〈r1,m2〉 − 2〈r2,m2〉) + 13α

120

2〈r1,m2〉 = s〈r1,m1〉+ 3s2〈r2,m1〉+ t〈r1,m2〉 − 3t

2〈r2,m2〉 − 19β

420

2〈r2,m2〉 = 3s4〈r2,m1〉+ t

4〈r2,m2〉 − β

168.

(10)

The systems (9) and (10) have the solutions

〈r1,m1〉 =α

4, 〈r1,m2〉 = −1

4

√α2 − 72

35, 〈r2,m1〉 =

1

216

(13α−

√7α2 − 72

5

),

〈r2,m2〉 =1

216

(√

7α− 13

√α2 − 72

35

), q =

26α− 20√

7α2 − 725

5√

7α+ 70√α2 − 72

35

,

s =13104− 6125α2 + 35α

√7α2 − 72

5+ 51

√7αβ + 147β

√α2 − 72

35√7(504 + 140α2 + 29

√5α√

35α2 − 72), and

t =35280− 7

√5√

35α2 − 72(1465α− 9√

7β) + 5α(8575α− 9√

7β)

35(504 + 140α2 + 29√

5α√

35α2 − 72).

Again using the coefficients in (8), we may expand 〈m1,m1〉 and 〈m2,m2〉 and numericallysolve for α and β. After substituting the above results into the initial expansions

q2 − 2qα〈r1,m2〉+13

35α2 = 1 and s2 + t2 − 2sβ〈r2,m1〉+ 2tβ〈r2,m2〉+

β2

105= 1,

we find the approximate solutions α ≈ 1.63240645 and β ≈ 14.19575451, so that

〈r1,m1〉 ≈ 0.40810161, 〈r1,m2〉 ≈ −0.19487303, 〈r2,m1〉 ≈ 0.08869880,

〈r2,m2〉 ≈ −0.02691878, q ≈ 0.01570030, s ≈ 0.45783086, and t ≈ −0.03034240.

Since ∥∥∥∥[0 qs t

]∥∥∥∥∞

=

∥∥∥∥[ 0 −q−s t

]∥∥∥∥∞≈ 0.48817326 <

1

2,

then by Lemma 3, m1,m2 ∈ C1([0, 1]). By setting

l1 =(I − PM)l1‖(I − PM)l1‖

, l2 =(I − PM)l2‖(I − PM)l2‖

, r1 =(I − PM)r1‖(I − PM)r1‖

, and r2 =(I − PM)r2‖(I − PM)r2‖

,

we have the orthonormal, refinable C1 macroelement Λ = (l1, l2, r1, r2,m1,m2)T that is

equivalent to Λ and an extension of Λ. By Lemma 2, we have the scaling vector Φ oflength 4 as defined in (3) and (4) that is an extension of Φ. Since 〈φ1, φ2〉 = 0 due to theirsymmetry-antisymmetry about x = 0, Φ is an orthogonal scaling vector.

The elements of the orthogonal scaling vector constructed in the above proof are illus-trated in Figure 5.

404 KESSLER

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

φ1 φ2

φ3 φ4

Figure 5: The orthogonal scaling vector Φ = (φ1, φ2, φ3, φ4)T , where S(Φ) ⊃ S1

3 (Z).

2.3 The Associated Multiwavelets

A general construction for a multiwavelet associated with a scaling vector defined on R wasfound in [16] (see also [10]), and that technique could be used here. However, I will use anequivalent approach that will produce wavelets with symmetry properties. From Jia in [14],we know that, since Φ was of length 4, then Ψ will also have length 4(2− 1) = 4.

Define the 6-dimensional space V = f : f ∈ V−1, supp f ⊆ [0, 1] and let PV denotethe orthogonal projection onto V . Note that, in this construction, V ∩ V ⊥0 = ∅, sincethere are 6 independent orthogonality conditions affecting elements of V . Recall the originalmacroelement Λ = (l1, l2, r1, r2,m1,m2)

T . Let

li(x) = PV li(x) + aiφ1(Nx−N) + biφ2(Nx−N) for i = 1, 2,

where ai and bi are chosen so that 〈li(x), φ1(x− 1)〉 = 0 and 〈li(x), φ2(x− 1)〉 = 0. Likewise,let

ri(x) = PV ri(x) + ciφ1(Nx) + diφ2(Nx) for i = 1, 2,

where ci and di are chosen so that 〈ri(x), φj(x)〉 = 0. Note that the properties 〈li, rj〉 = 0,〈li,mj〉 = 0, and 〈rj,mj〉 = 0 for i, j ∈ 1, 2 are maintained from the original macroelement,and that a1 = c1, b1 = −d1, a2 = −c2, and b2 = d2 due to symmetry properties.

We may construct functions

f1(x) = l1(x+ 1) + r1(x) and g1(x) = −l2(x+ 1) + r2(x)

that are symmetric with respect to x = 0, and

f3(x) = −l1(x+ 1) + r1(x) and g3(x) = l2(x+ 1) + r2(x)

405AN ORTHOGONAL SCALING...

that are antisymmetric with respect to x = 0, such that all are orthogonal to V0. Set

f2 = g1 −〈g1, f1〉〈f1, f1〉

f1 and f4 = g3 −〈g3, f3〉〈f3, f3〉

f3

to handle the last remaining orthogonalities, and set ψi = fi

‖fi‖ for i = 1, . . . , 4. Then

Ψ = (ψ1, ψ2, ψ3, ψ4)T is a multiwavelet that generates W0, and is illustrated in Figure 6.

-1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

1.5

2

-1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

1.5

2

-1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

1.5

2

-1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

1.5

2

ψ1 ψ2

ψ3 ψ4

Figure 6: The multiwavelet Ψ = (ψ1, ψ2, ψ3, ψ4)T , where S(Ψ) = W0.

3 An Application of the Bases

One possible use for this smoother scaling vector is in image compression applications. Theoriginal version of JPEG used the discrete cosine transform, a non-wavelet technique, ondisjoint 8 × 8 blocks of pixels to decompose the image data. This caused noticable distor-tions, or “artifacts,” to appear in the reconstructed image at higher compression ratios. Thediscontinuous Haar wavelet basis can also be used to decompose the image data, but withsimilar results at higher compression ratios. Even continuous bases like Daubechies’s D4scaling function or the GHM scaling vector constructed in Example 2, which are not differ-entiable everywhere, leave sharp distortions in the image at higher compression ratios. Wewould expect that by using a smoother, differentiable basis like we have constructed in thispaper, we should be able to produce a smoother compressed image with no sharp artifacts.

Let ci denote the sequence of scaling function coefficients in Vi, and let di denote thesequence of wavelet coefficients in Wi. The wavelet approach to producing a compressed

406 KESSLER

image is to take a signal (or function) in V0 and find its best approximation in the smoother,nested function space V1, keeping the error in the wavelet space W1. If the function is closeto being in V1, then the wavelet coefficients will be very small (close to zero). We may thenrepeat this process with the function in V1, and then V2, etc. The process is illustrated in Fig-ure 7 for a decomposition to the fourth level. The image may be reconstructed exactly fromcn, where n is the last level of decomposition, and from the wavelet coefficients d1, . . . , dn.However, wavelet schemes typically trade accuracy of the image for more efficient storage

c0 c1 c2 c3 c4

d1 d2 d3 d4

-

@@

@R

-

@@

@R

-

@@

@R

-

@@

@R

c4 c3 c2 c1 c0

d4 d3 d2 d1

-

-

-

-

Figure 7: The decomposition of the signal c0 and the reconstruction of c0.

and/or transmission of the image. In order to achieve compression of the signal, we quantizethe wavelet coefficients, replacing coefficients with values in a certain range with a centralvalue. This creates a signal with lower entropy, a measure of the smallest average bits percharacter needed to store a signal without losing information. This measure is quantifiable,using the formula

E = −N∑

i=1

p(i) log2 p(i),

where N is the number of distinct characters in the signal, and p(i) is the relative frequency

of the ith character. The altered signal is then stored near this bit-rate using an entropyencoder. This is called lossless compression. A less accurate version of the original imagecan then be reconstructed from the quantized wavelet coefficients, as illustrated in Figure 7.This is called lossy compression.

3.1 Prefiltering

The process of turning discrete data into a function in V0 is called prefiltering. Ideally, aprefilter (usually a set of matrices) should be orthogonal (norm-preserving) and send datasampled from a polynomial of degree n to a multiple of the same polynomial in V0, up tothe approximation order of V0, using as few matrices in the prefilter as possible. Prefilteringis not an issue when using a single scaling function, as using the raw data as the basiscoefficients (the identity prefilter) accomplishes each of these goals. However, prefilteringbecomes vitally important when using multiple scaling functions: the best basis will performpoorly in applications if the data is not prefiltered efficiently. See [12] for a more thoroughdiscussion on prefiltering.

Multiple-matrix prefilters that accomplish all of the above goals can be difficult to find.We may always find an orthogonal single-matrix prefilter that preserves constant data byusing the following method. Consider a scaling vector Φ of length r that generates thefunction space V0 of approximation order n ≤ r. Let ak be the r-vector of function valuessampled uniformly over [0, 1] from fk(x) = xk, k = 0, . . . , n− 1, and let ak fill out the spaceof r-vectors for k = n, . . . , r− 1. Likewise, let αk be the r-vector of basis coefficients of Φ(x)

407AN ORTHOGONAL SCALING...

needed to construct fk(x), k = 0, . . . , n − 1, and let αk fill out the space of r-vectors fork = n, . . . , r − 1. Then use the Gram-Schmidt process on a0, . . . , ar−1 and α0, . . . , αr−1in increasing order of degrees, and normalize to get a0, . . . , ar−1 and α0, . . . , αr−1. Wemay then construct the orthogonal, single-matrix prefilter

Q =[α0 · · · αr−1

] aT0...

aTr−1

which maps ak to αk. While the prefilter Q does not preserve polynomial order aboveconstants, it gets fairly close, and has the advantages of having very short support and beingorthogonal. We have observed that this type of prefilter works well in applications, and so,we will use this type of prefilter for the GHM scaling vector and the new scaling vector in thefollowing example. We concede that better prefilters may be found for both scaling vectors.(In fact, see [12] for more elaborate prefilters for the GHM scaling vector.)

Postfiltering, turning the basis coefficients back into discrete data, is usually just aninversion of the prefiltering process. For the prefilter Q described above, we use the postfilterQT , since QTQ = I.

3.2 Comparison of Reconstructed Images

In this section, we consider a digital 512 × 512 grayscale image called “Zelda,” basically arectangular data set ranging in value from 0 to 255, requiring 256 Kb of memory (1 byte perpixel) for storage as raw data. The original image is shown in Figure 8. We decompose theimage using three different bases: the single D4 scaling function, the GHM scaling vectorconstructed in Example 2 from a C0 macroelement, and the scaling vector from Theorem 1constructed from a C1 macroelement. We quantize the wavelet coefficients uniformly, with0 at the center of the zero-bin, to achieve a moderate 25:1 compressed image (file size10.24 Kb) and an extreme 50:1 compressed image (file size 5.12 Kb), based on the entropyof the quantized signal. Particularly in the 50:1 compressed images, the reader will detectdistortions specific to the basis being used. The error in the images will be measured visuallyby the reader and with the root mean square error (RMSE), defined by

RMSE =

√∑i,j

(originali,j − newi,j

)2rows× columns

.

Reconstructions from the quantized data are shown in Figure 9 for the D4 basis, inFigure 10 for the GHM scaling vector, and in Figure 11 for the new scaling vector constructedin this paper. The reconstructions for the smooth scaling vector constructed in this paperhave the lowest RMSE’s of the three methods. Hopefully, the reader also finds the artifactsare less noticeable in the images where the new basis is used.

408 KESSLER

Figure 8: The original image 512× 512 grayscale image Zelda.

RMSE ≈ 4.88688 RMSE ≈ 6.88849

Figure 9: A 25:1 and 50:1 compression using the D4 scaling function.

4 Appendix

4.1 Scaling Vector Coefficients

The matrix coefficients of the scaling vector constructed in Section 2.2 satisfying

Φ(x) =√

21∑

i=−2

giΦ(2x− i)

409AN ORTHOGONAL SCALING...

RMSE ≈ 4.37044 RMSE ≈ 6.23848

Figure 10: A 25:1 and 50:1 compression using the GHM scaling vector.

RMSE ≈ 4.13475 RMSE ≈ 5.84525

Figure 11: A 25:1 and 50:1 compression using the new scaling vector.

are given below.

g−2 =

0 0 0.02669163201881291 0.0261887805703498840 0 −0.03940748766209954 −0.0407690355223390140 0 0 00 0 0 0

410 KESSLER

g−1 =

−0.1175124743509559 −0.10652668959666897 0.30676124077120465 0.359641678388543660.18731879383595215 0.18181954032277048 −0.4224381824999862 −0.433225475337907

0 0 0 00 0 0 0

g0 =

0.7071067811865841 0 0.3067612407712861 −0.35964167838820240 0.35355339059336993 0.42243818250022613 −0.433225475336882050 0 0.47106586494422936 0.236040933958986320 0 −0.5666158791523569 −0.2916640296146426

g1 =

−0.11751247435904282 0.10652668958949803 0.026691633234707347 −0.026188779541901065−0.1873187938600181 0.18181954030143205 0.03940749128037211 −0.0407690324618693450.6669057455800359 0 0.47106586494422936 −0.23604093395898632

0 0.4333094490577162 0.5666158791523569 −0.2916640296146426

4.2 Multiwavelet Coefficients

The matrix coefficients of the multiwavelet constructed in Section 2.3 satisfying

Ψ(x) =√

21∑

i=−2

hiΦ(2x− i)

are given below.

h−2 =

0 0 0.0266916313500821 0.026188779914217490 0 0.04374365300600389 0.064359829591981710 0 −0.013410540810749855 −0.0131578957170888480 0 −0.04006995259666773 −0.06065185761803572

h−1 =

−0.11751247140680435 −0.10652668692775463 0.3067612330856239 0.3596416693780991−0.3334538093583893 −0.42469284946317537 0.19229779050087667 −0.406706781093616240.05904119433180234 0.053521662417019424 −0.15412448873935367 −0.180692937841971550.3166000901995356 0.40882389594960616 −0.15157506646288127 0.4513983241700974

h0 =

−0.7071067870918081 0 0.30676123308570535 −0.35964166937775780 0 0.19229779050222978 0.406706781099485770 −0.9347644627208598 0.15412448873939458 −0.180692937841800070 −0.034862829700229456 0.15157506646421717 0.45139832417589415

h1 =

−0.11751247141489118 0.10652668692058369 0.02669163256597651 −0.026188778885768697−0.3334538094952175 0.424692849341863 0.043743673577174125 −0.06435981219212034−0.05904119433586537 0.05352166241341656 0.013410541421645506 −0.013157895200370542−0.31660009033465436 0.40882389582980966 0.04006997291081938 −0.060651840435570266

4.3 Prefilter Matrices

The prefilter used in the GHM scaling vector image compression examples in Section 3.2 isgiven below. This prefilter originally appeared in [12].

Q =

[1√3

+ 1√6− 1√

3+ 1√

61√3− 1√

61√3

+ 1√6

]

411AN ORTHOGONAL SCALING...

The prefilter used in the new scaling vector image compression examples in Section 3.2 isgiven below.

Q =

0.843806675746766 0.48588947587427395 −0.22260947332787062 0.04844309635543219−0.5359722073168798 0.7840831502767046 −0.30731469556949864 0.059203752609673790.015288019449165685 0.26864647184429297 0.7701708393722542 0.5783011566714802−0.022144150703698316 −0.27740602706575324 −0.5126788258279672 0.8122290035974187

Acknowledgements: Research was supported by the Kentucky Science and Engineering Foun-dation, Grant KSEF-148-502-03-57.

References

[1] M. Barnsley, “Fractal functions and interpolations,” Constr. Approx. 2, 303–329 (1986).

[2] M. Barnsley, J. Elton, D. Hardin, and P. Massopust, “Hidden variable fractal interpo-lation functions,” SIAM Journ. Math. Anal. 20:5, 1218–1242 (1989).

[3] I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. onPure and Appl. Math. 41, 909–996 (1988).

[4] G. Donovan, J. Geronimo, and D. Hardin, “Fractal functions, splines, intertwiningmultiresolution analysis, and wavelets,” Wavelet Application in Signal and Image Pro-cessing, Laine and Unser, editors, SPIE Conf. Proc., San Diego, 238–243 (1994).

[5] G. Donovan, J. Geronimo, and D. Hardin, “Intertwining multiresolution analyses andthe construction of piecewise polynomial wavelets,” SIAM Journ. Math. Anal. 27(6),1791–1815 (1996).

[6] G. Donovan, J. Geronimo, D. Hardin, and P. Massopust, “Construction of orthogonalwavelets using fractal interpolation functions,” SIAM Journ. Math. Anal. 27, 1158–1192(1996).

[7] J. Geronimo, D. Hardin, and P. Massopust, “Fractal functions and wavelet expansionsbased on several scaling functions,” J. Approx. Theory 78:3 373–401 (1994).

[8] T. N. T. Goodman, “A class of orthogonal refinable functions and wavelets,” Constr.Approx. 19:4, 525–540 (2003).

[9] B. Han and Q. Jiang, “Multiwavelets on the interval,” Appl. Comp. Har. Anal. 12:1100–127 (2002).

[10] D. Hardin and B. Kessler, “Orthogonal macroelement scaling vectors and wavelets in1-D”, (invited paper for special issue on fractals and wavelets), The Arabian Journalfor Science and Engineering, 28:1C, 73–88 (2003).

[11] D. Hardin, B. Kessler, and P. Massopust, “A multiresolution analysis based on fractalfunctions,” J. Approx. Theory 71:1, 104–120 (1992).

412 KESSLER

[12] D. Hardin and D. Roach, “Multiwavelet prefilters I: orthogonal prefilters preservingapproximation order p ≤ 2.” IEEE Trans. Circuits Syst. II: Analog Digital Signal Pro-cessing 45:8, 1106–1112 (1998).

[13] D. Hong and A. Wu, “Orthogonal multiwavelets of multiplicity four,” Comput. Math.Applic. 40, 1153–1169 (2000).

[14] R.-Q. Jia, “Refinable shift-invariant spaces: from splines to wavelets,” Approx. TheoryVIII 2, 179–208 (1995).

[15] R.-Q. Jia and Z. Shen, “Multiresolution and wavelets,” Proc. Edinburgh Math. Soc. 37,271–300 (1994).

[16] G. Strang and V. Strela, “Short wavelets and matrix dilation equations,” IEEE Trans.SP 43, 108–115 (1995).

[17] P. Vaidyanathan, “Multirate Systems and Filter Banks,” Simon and Schuster (1993).

413AN ORTHOGONAL SCALING...

The existence of tight MRA multiwavelet frames

Qun Mo

Department of Mathematical and Statistical SciencesUniversity of Alberta

Edmonton, Alberta, Canada T6G 2G1E-mail: mo@math.ualberta.ca

Abstract

We shall prove that under a mild assumption, we can construct tight

multiwavelet frames with the highest possible order of vanishing moments

from any given refinable function vector. This paper generalizes Chui, He

and Stockler’s work for dilation 2 and Chui, He, Stockler and Sun’s work

for any integer dilation larger than 2 on the existence of tight wavelet

frames derived from any given refinable function.

Keywords: tight wavelet frames, vanishing moments, transition operator,the canonical form of a matrix mask, matrix-valued Fejer-Riesz Lemma, matrixapproximation.

2000 AMS subject classification: 42C40, 42C15, 41A15, 41A25.

1 Introduction

Wavelet frames are proved to be a very useful tool in signal denoising andare being explored in other applications. Usually we derive wavelet framesfrom refinable functions. Thus the property of a wavelet frame derived froma refinable function is highly related to the property of the refinable function.However, some desirable properties cannot coexist very well at any refinablefunction. For instance, high smoothness and short support are two desirableproperties of a refinable function. But the smoothness of a refinable functionis restricted by its support. On the other hand, refinable function vectors, asgeneralizations of refinable functions, may have higher smoothness than thoserefinable functions with the same length limit of their support. Therefore, itis interesting to construct multiwavelet frames, especially tight multiwaveletframes, from refinable function vectors.

In literature, Daubechies, Han, Ron and Shen ([6]) and Chui, He and Stockler([2]) independently discovered a way to derive wavelet frames from refinablefunctions. There are many papers inspired by their results. For instance, Hanand Mo have generalized their results on the construction of pairs of dual waveletframes from the classical wavelet setting to the multiwavelet setting ([12]), have

415JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,415-433,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

given a general construction of tight wavelet frames with three (anti)symmetricgenerators derived from B-splines ([13]), and have given a characterization oftight wavelet frames with two (anti)symmetric generators derived from any givenrefinable function ([14]). Also, Chui, He, Stockler and Sun ([3]) have generalizedChui, He and Stockler’s result ([2]) on the existence of tight wavelet framesfrom dilation 2 to any integer dilation larger than 2. We can see there is aninteresting problem left unknown: under a mild assumption, can we alwaysderive tight multiwavelet frames with the highest possible vanishing momentsfrom any given refinable function vector? In this paper, we shall give a positiveanswer to the above question.

Before proceeding further, let us give some definitions first. From now on,by d we denote a dilation factor which is an integer such that |d| > 1. For afunction f ∈ L2(R), we define

fj,k := |d|j/2f(dj · −k), j, k ∈ Z.

Define

〈f, g〉 :=

∫

R

f(t)g(t) dt ∀f, g ∈ L2(R)

and ‖f‖2 := 〈f, f〉. Let ψ1, . . . , ψL be a finite set of functions in L2(R), wesay that ψ1, . . . , ψL generates a wavelet frame ψ`

j,kj,k∈Z;`=1,... ,L in L2(R)if there exist positive constants C1 and C2 such that

C1‖f‖26

L∑

`=1

∑

j∈Z

∑

k∈Z

|〈f, ψ`j,k〉|2 6 C2‖f‖2 ∀ f ∈ L2(R). (1.1)

In particular, when C1 = C2 (= 1) in (1.1), we say that ψ1, . . . , ψL generatesa (normalized) tight wavelet frame in L2(R). We shall assume that everytight wavelet frame is normalized in this paper.

A tight wavelet frame ψ`j,kj,k∈Z;`=1,... ,L can represent a function f ∈ L2(R)

as

f =

L∑

`=1

∑

j,k∈Z

〈f, ψ`j,k〉ψ`

j,k.

Furthermore, if a tight wavelet frame ψ`j,kj,k∈Z;`=1,... ,L is linearly independent,

then it is an orthonormal wavelet basis. In wavelet applications, wavelet frameshave already been proved to be very useful for signal denoising and currentlyare being explored for image compression. It is of interest in both theory andapplications to construct wavelet frames with certain properties.

An important property of a wavelet frame is its order of vanishing moments.A function f with enough decay is said to have vanishing moments of orderm if ∫

R

tkf(t) dt = 0 ∀ k = 0, . . . ,m− 1.

A wavelet frame ψ`j,kj,k∈Z;`=1,... ,L has vanishing moments of order m if all

its generators ψ1, . . . , ψL have vanishing moments of order m. The order of

416 MO

vanishing moments of a wavelet frame has a great impact on how efficient thewavelet frame is in representing a function.

We can construct the generators of a wavelet frame from refinable functionvectors. In this case, the wavelet frame is called an “MRA multiwavelet

frame”. An r × 1 function vector φ = (φ1, . . . , φr)T ∈ (L2(R))

ris a refinable

function vector, where r is called the “multiplicity”, if φ satisfies the followingrefinement equation,

φ = |d|∑

k∈Z

akφ(d · −k), (1.2)

where a is a finitely supported sequence of r × r complex-valued matrices onZ, called the “matrix mask” for φ. Under an appropriate mild condition (see[1, 5, 16, 20, 18, 22, 23]) on the matrix mask a, there exists a unique normalizeddistributional solution of the refinement equation (1.2). The refinement equationas well as the various properties of its refinable function vector has been wellstudied in the literature. To name a few, please see [1, 5, 7, 8, 10, 16, 18, 20, 21,22, 23, 26, 27, 28] and the references therein. A refinable function vector withthe multiplicity r = 1 is a scalar refinable function.

The remaining part of this paper is as follows. In Section 2, we shall givemore definitions and discuss some properties of the transition operator Ta. InSection 3, we shall build some lemmas for matrix inequalities. In Section 4, weshall prove the existence of tight MRA multiwavelet frames with the maximalpossible vanishing moments. In Section 5, we shall show one example of tightmultiwavelet frames for the case r = 2 and d = 2.

2 The transition operator Ta

Let φ = (φ1, . . . , φr)T ∈ (L2(R))r be a refinable function vector and let a be the

matrix mask of φ. We say that a mask a with multiplicity r satisfies the sum

rules of order m with respect to the lattice dZ if there exists a 1× r row vectory(ξ) of 2π-periodic trigonometric polynomials such that y(0) 6= 0 (we assume

y(0)φ(0) 6= 0 in this paper) and for k = 0, . . . , |d| − 1,

[y(d·)a(·)](j)(2πk/d) = δky(j)(0), j = 0, . . . ,m− 1, (2.1)

where δ is the Dirac sequence such that δ0 = 1 and δk = 0 for all k ∈ Z\0,and y(j)(ξ) denotes the jth derivative of y(ξ) for all j ∈ Z. As discussed in [12,Theorem 2.3], if a matrix mask a satisfies the summer rule of order m, we canassume that a takes the canonical form

a(ξ) =

[(1 + e−iξ + · · · + e−i(|d|−1)ξ

)mP1,1(ξ) (1 − e−i|d|ξ)mP1,2(ξ)

(1 − e−iξ)mP2,1(ξ) P2,2(ξ)

], (2.2)

where P1,1, P1,2, P2,1 and P2,2 are some 1 × 1, 1 × (r − 1), (r − 1) × 1 and (r −1)× (r− 1) matrices of 2π-periodic trigonometric polynomials respectively andP1,1(0) = |d|−m. We shall keep this assumption from now on.

417THE EXISTENCE OF TIGHT MRA...

Define T := R/2πR. Denote P(T)r×r the space of r × r matrices of 2π-periodic trigonometric polynomials, C∞(T)r×r the space of r × r matrices of2π-periodic C∞-functions and C(T)r×r the space of r×r matrices of 2π-periodiccontinuous functions. Given a matrix A, we denote AT the transpose of A, A∗

the transpose of the complex conjugate of A, Aadj the adjoint matrix of A, andAi,j the (i, j)-entry of A. For any A ∈ C(T)r×r , we say A > 0 (or A > 0) if forall ξ ∈ T, A(ξ) > 0 which means A(ξ) is positive definite (or A(ξ) > 0 whichmeans A(ξ) is positive semi-definite).

The transition operator Ta is defined by

(TaF )(dξ) :=

|d|−1∑

j=0

a(ξ + 2πj/d)F (ξ + 2πj/d)a(ξ + 2πj/d)∗ ∀F ∈ C(T)r×r.

(2.3)

By the definition, we know that Ta is well defined and Ta maps C(T)r×r andP(T)r×r into C(T)r×r and P(T)r×r, respectively. A special eigenfunction of Ta

is Φ, the bracket product of φ and φ. For f, g ∈ (L2(R))r , define the bracket

product (see [19]) as

[f, g](ξ) :=∑

j∈Z

f(ξ + 2πj)g(ξ + 2πj)∗ ∀ξ ∈ T.

Since f, g ∈(L2(R)

)r, [f, g] is an r × r matrix of functions in L1(T). Define

Φ :=[φ, φ

].

Then we can verify that Φ is an eigenfunction of Ta as follows.

TaΦ(dξ) =

|d|−1∑

j=0

a(ξ + 2πj/d)Φ(ξ + 2πj/d)a(ξ + 2πj/d)∗

=

|d|−1∑

j=0

∑

k∈Z

a(ξ +

2πj

d

)φ(ξ +

2πj

d+ 2πk

)φ(ξ +

2πj

d+ 2πk

)∗

a(ξ +

2πj

d

)∗

=

|d|−1∑

j=0

∑

k∈Z

φ(dξ + 2πj + 2πdk)φ(dξ + 2πj + 2πdk)∗

=Φ(dξ).

Thus we have TaΦ = Φ.Moreover, we have the following lemma.

Lemma 2.1 Let φ = (φ1, . . . , φr)T ∈

(L2(R)

)rbe a refinable function vector

with matrix mask a. Define Φ :=[φ, φ

]. It is evident that Φ∗ = Φ. If the

matrix mask a takes the canonical form (2.2), then we have

(1 − e−iξ)m | Φ(ξ)1,j = Φ(ξ)(j,1), j = 2, . . . , r

418 MO

and(1 − e−iξ)m | Φadj(ξ)1,j = Φadj(ξ)j,1, j = 2, . . . , r.

Moreover, if Φ > 0, then we have

(1 − e−iξ)m | Φ−1(ξ)1,j = Φ−1(ξ)j,1, j = 2, . . . , r

and(1 − e−iξ)2m |

[Φ−1(ξ) − a(ξ)∗Φ−1(dξ)a(ξ)

]1,1.

Proof: By assumption, we have

a(ξ) =

[(1 − e−idξ)m/(1 − e−iξ)m∗ (1 − e−idξ)m∗

(1 − e−iξ)m∗ ∗

], (2.4)

where ∗ denotes some 1×1, 1× (r−1), (r−1)×1 and (r−1)× (r−1) matricesof trigonometric polynomials. Since φ is refinable, we have

φ(dξ) = a(ξ)φ(ξ).

Taking the 0th, . . . , (m − 1)th derivatives at ξ = 0 from both sides of theequation above, since a(ξ) has the canonical form (2.4), we have

φ(`)j (0) = 0, j = 2, . . . , r; ` = 0, 1, . . . ,m− 1.

Notice that φ1 is a superfunction. We have for ` = 0, 1, . . . ,m− 1,

∑

k∈Z

k`φ1(x+ k) = p`(x),

where p` is a polynomial with degree at most `. Notice

Φ(ξ)1,j =∑

k∈Z

[∫φ1(x + k)φj(x)dx

]e−ikξ =:

∑

k∈Z

bk,je−ikξ

and for ` = 0, . . . ,m− 1, j = 2, . . . , r,

∑

k∈Z

k`bk,j =

∫ ∑

k∈Z

[k`φ1(x+ k)]φj(x)dx

=

∫p`(x)φj(x)dx

=(p`(−iD)φj

)(0) = 0.

Therefore we have

(1 − e−iξ)m | Φ(ξ)1,j = Φ(ξ)j,1, j = 2, . . . , r. (2.5)

419THE EXISTENCE OF TIGHT MRA...

By the definition of adjoint matrices, (−1)j+1Φadj1,j , j = 2, . . . , r, is the determi-

nant of a sub-matrix of Φ. Since the first row of this sub-matrix has a commonfactor (1 − e−iξ)m, we have

(1 − e−iξ)m | Φadj(ξ)1,j = Φadj(ξ)j,1, j = 2, . . . , r. (2.6)

Moreover, if Φ > 0, then detΦ(ξ) > 0 for all ξ ∈ T. Therefore, detΦ(0) 6= 0. By

the matrix identity A−1 = Aadj/detA and (2.6), we have

(1 − e−iξ)m | Φ−1(ξ)1,j = Φ−1(ξ)j,1, j = 2, . . . , r. (2.7)

As proved before, TaΦ = Φ, i.e.,

Φ(dξ) =

|d|−1∑

j=0

a(ξ + 2πj/d)Φ(ξ + 2πj/d)a(ξ + 2πj/d)∗.

By (2.4), we have

(1 − e−iξ)2m |[a(ξ + 2πj/d)Φ(ξ + 2πj/d)a(ξ + 2πj/d)∗

]1,1, j = 2, . . . , r.

Therefore,

Φ(dξ)1,1 =[a(ξ)Φ(ξ)a(ξ)∗

]1,1

+O(|ξ|2m) as ξ → 0.

Combining the equality above with (2.4) and (2.5), we have

Φ(dξ)1,1 = |a(ξ)1,1|2Φ(ξ)1,1 +O(|ξ|2m) as ξ → 0.

Hence(Φ(ξ)1,1

)−1= |a(ξ)(1,1)|2

(Φ(dξ)1,1

)−1+O(|ξ|2m) as ξ → 0. (2.8)

By (2.5), (2.7) and Φ(ξ)Φ−1(ξ) = Ir, we have

1 = Φ(ξ)1,1Φ−1(ξ)1,1 +O(|ξ|2m) as ξ → 0.

Therefore,

(Φ(ξ)1,1

)−1= Φ−1(ξ)1,1 +O(|ξ|2m) as ξ → 0. (2.9)

By (2.4), (2.7), (2.8) and (2.9), we have

Φ−1(ξ)1,1 =(Φ(ξ)1,1

)−1+O(|ξ|2m)

= |a(ξ)1,1|2(Φ(dξ)1,1

)−1+O(|ξ|2m)

= |a(ξ)1,1|2Φ−1(dξ)1,1 + O(|ξ|2m)

=[a(ξ)∗Φ−1(dξ)a(ξ)

]1,1

+O(|ξ|2m) as ξ → 0.

Therefore(1 − e−iξ)2m |

[Φ−1(ξ) − a(ξ)∗Φ−1(dξ)a(ξ)

]1,1.

420 MO

Now let us consider the eigenvectors of Ta. As been proved, we have TaΦ =Φ. Moreover, if the integer shifts of φ are stable, then the cascade algorithmassociated with the mask a converges in L2. Hence, we have that Ta has asimple eigenvalue 1 and all other eigenvalues of Ta are less than 1 in modulus(see [26]). Thus under the condition the integer shifts of φ are stable, we havethat Φ is the unique 1-eigenfunction of Ta and for all ξ ∈ T, Φ(ξ) > 0. Define aas

a(ξ) :=

[(1 − e−idξ)m 0

0 Ir−1

]−1

a(ξ)

[(1 − e−iξ)m 0

0 Ir−1

].

By our assumption, a is of the form (2.4). Therefore a is an r × r matrix oftrigonometric polynomials. Define a new operator Tea mapping C(T)r×r intoC(T)r×r by

TeaF (dξ) :=

|d|−1∑

j=0

a(ξ + 2jπ/d)F (ξ + 2jπ/d)a(ξ + 2jπ/d)∗. (2.10)

Since a is an r × r matrix of trigonometric polynomials, we denote Pea(T)r×r

a subspace containing all r × r matrices of trigonometric polynomials up toa finite degree determined by a such that it is an invariant subspace of Teaand ρ(Tea) = ρ(Tea|Pea(T)r×r) (see [11]). Similarly, since a is an r × r matrixof trigonometric polynomials, denote Pa(T)r×r a subspace containing all r × rmatrices of trigonometric polynomials up to a finite degree determined by asuch that it is an invariant subspace of Ta and ρ(Ta) = ρ(Ta|Pa(T)r×r ). It isobvious that

TeaF = λF =⇒ TaF = λF,

where F is derived from F by

F (ξ) =

[(1 − e−iξ)m 0

0 Ir−1

]F (ξ)

[(1 − e−iξ)m 0

0 Ir−1

]∗∀ξ ∈ T. (2.11)

Therefore, we have

ρ(Tea) 6 ρ(Ta). (2.12)

Now we are in the position to prove the following theorem.

Theorem 2.2 Let φ and a be defined as in Lemma 2.1. If φ has stable integershifts, then there exist a positive number ρ < 1 and some F ∈ P(T)r×r such that

F > 0 and TaF 6 ρF where F is derived from F by (2.11).

Proof: Since Tea is a linear operator acting on Pea(T)r×r which is a finitedimensional space, by the definition of spectrum, we know that there exists0 6= G(ξ) ∈ Pea(T)r×r such that TeaG = λ0G and |λ0| = ρ(Tea|Pea(T)r×r). Define

G(ξ) =

[(1 − e−iξ)m 0

0 Ir−1

]G(ξ)

[(1 − e−iξ)m 0

0 Ir−1

]∗,

421THE EXISTENCE OF TIGHT MRA...

then it is evident to see that we have TaG = λ0G.Since Φ(0)1,1 > |φ1(0)|2 6= 0, Φ(0)1,1 6= 0 and G(0)1,1 = 0 by the definition

of G. Thus for all λ ∈ C, we have Φ 6= λG. Notice that Φ is the unique 1-eigenfunction of Ta, we have |λ0| = ρ(Tea) 6= 1. Since φ has stable integer shifts,Ta has a simple eigenvalue 1 and all other eigenvalues of Ta are less than 1 inmodulus (see [26]). By (2.12), we have ρ(Tea) 6 1. Combining the fact thatρ(Tea) 6= 1, we have |λ0| = ρ(Tea) < 1.

Choose ρ1 := [1 + ρ(Tea)]/2. Then we have ρ(Tea) < ρ1 < 1. Borrowing theidea from the proof of [24, Theorem 3], since ρ(Tea/ρ1) < 1, (Id− Tea/ρ1)

−1 is awell defined operator acting on Pea(T)r×r and

(Id− Tea/ρ1)−1 = Id+ Tea/ρ1 + T 2

ea/ρ21 + · · · ,

where Id denotes the identity operator. Define F = (Id − Tea/ρ1)−1Ir, then

F = Ir + TeaIr/ρ1 + · · · . Thus F ∈ Pea(T)r×r and F > 0. By the definition of

F , we have (Id− Tea/ρ1)F > 0. Therefore, TeaF < ρ1F . Let F be derived from

F by (2.11), then we have TaF 6 ρ1F .After Theorem 2.2, our next major step will be proving that there exist a

positive number ε and some Θ ∈ P(T)r×r such that for all ξ ∈ T,

(Φ(ξ) + εF (ξ)

)−16 Θ(ξ) 6

(Φ(ξ) + ερF (ξ)

)−1.

Before going to this step, we need some lemmas to build up matrix approxima-tion first.

3 Matrix approximation

Lemma 3.1 If A,B ∈ C(T)r×r such that A∗ = A, B∗ = B and A < B. Thenthere exists some P ∈ P(T)r×r such that A < P < B.

Proof: For every ξ ∈ T, suppose λ1(ξ), ..., λr(ξ) are all the eigenvalues of(B − A)(ξ) and we have λ1(ξ) > · · · > λr(ξ). Thus B(ξ) − A(ξ) > λr(ξ)Ir .Since B(ξ) −A(ξ) > 0, we have

λ1(ξ) > · · · > λr(ξ) > 0.

Hence we have, for all ξ ∈ T,

λr(ξ) =det

(B(ξ) −A(ξ)

)

λ1(ξ) · · · λr−1(ξ)>

det(B(ξ) −A(ξ)

)[trace

(B(ξ) −A(ξ)

)]r−1 > c1,

where c1 is a suitable positive constant number we choose to satisfy the aboveinequality. Therefore, for all u ∈ Cr, u∗(B − A)u > c1u

∗u. Then we can get atrigonometric polynomial P1 ∈ P(T)r×r such that for 1 6 i, j 6 r and for allξ ∈ T,

|[P1 − (A+B)/2](ξ)i,j | < c1/(4r2).

422 MO

Define P := (P1 + P ∗1 )/2, then we have P ∗ = P and for all u ∈ Cr,

u∗(P −A)u = u∗[(A+B)/2 −A]u+ u∗[P1 − (A+B)/2]u/2

+ u∗[P ∗1 − (A+B)∗/2]u/2

> c1u∗u/2 − c1u

∗u/4/2− c1u∗u/4/2

= c1u∗u/4.

Thus P−A >c1

4 Ir > 0. Similarly, we have B−P >c1

4 Ir > 0. Thus B < P < A.

Lemma 3.2 Let A,B ∈ C∞(T)r×r be such that A∗ = A, B∗ = B and B−A =P1FP

∗1 , where P1 ∈ P(T)r×r, detP1 6≡ 0 and F ∈ C(T)r×r, F > 0. Then there

exists some P ∈ P(T)r×r such that A 6 P 6 B.

Proof: For all ξ ∈ T, define p1(ξ) := detP1(ξ). Since P1 ∈ P(T)r×r anddetP1 6≡ 0, p1 is a non-zero 2π-periodic trigonometric polynomial. Therefore,|p1|2 has finitely many roots in T. Suppose all the roots of |p1|2 in T are

−π 6 ξ1 < . . . < ξN < π

with multiplicities α1, . . . , αN , respectively. Then we have |p1|2 = p2p3, wherep2 and p3 are two 2π-periodic trigonometric polynomials such that p2(ξ) 6= 0for all ξ ∈ T and

p3(ξ) = ΠNk=1(e

−iξ − e−iξk)αk .

By the Lagrange Interpolation Theorem, for 1 6 i, j 6 r, there exist uniquepi,j ∈ P(T)1×1 such that deg pi,j < deg p3 and

p(`)i,j (ξk) = A

(`)i,j (ξk), ` = 0, . . . , αk − 1; k = 1, . . . , N.

Define Ai,j := |p1|−2(Ai,j − pi,j), P0 := [pi,j ]16i,j6r and A = [Ai,j ]16i,j6r . It

is obvious that P0 ∈ P(T)r×r and A = P0 + p1Ap∗1. Also, by the definition of

pi,j and the fact that Ai,j is a C∞-function, it is evident to see that Ai,j =

p−12

Ai,j−pi,j

p3

is a continuous function on T. Since A∗ = A and P0 is uniquely

determined by A, we have P ∗0 = P0 and A∗ = A. By F > 0, A ∈ C(T)r×r and

Lemma 3.1, there exists some P2 ∈ P(T)r×r such that

Padj1 A(P

adj1 )∗ < P2 < P

adj1 A(P

adj1 )∗ + F.

Let P := P0 + P1P2P∗1 . We have

P −A = P1[P2 − Padj1 A(P

adj1 )∗]P ∗

1 > 0.

Similarly B − P > 0. Thus A 6 P 6 B.

Recall Φ :=[φ, φ

], now we can prove the following proposition.

423THE EXISTENCE OF TIGHT MRA...

Proposition 3.3 Suppose 0 < ρ < 1, Φ > 0, F ∈ C(T)r×r, F > 0, F is

derived from F by (2.11), then there exist ε > 0 and Θ ∈ P(T)r×r such that

(Φ + εF )−16 Θ 6 (Φ + ερF )−1.

Proof: For a constant r × r matrix A > 0, we have

(I +A)−1 = I −A+A2(I +A)−1 = I −A+A(I +A)−1A.

Hence we have(I +A)−1

> I −A when A > 0.

Moreover for a given positive number λ, if 0 6 A 6 λIr, we have

(I +A)−1 = I −A+ A(I +A)−1A 6 I −A+A26 I −A+ λA.

Hence

(I +A)−16 I −A+ λA when 0 6 A 6 λIr . (3.1)

Since Φ > 0, we have Φ1 := Φ1/2 > 0. Thus we have

(Φ + ερF )−1 = Φ−11 [I + ερΦ−1

1 FΦ−11 ]−1Φ−1

1

> Φ−11 [I − ερΦ−1

1 FΦ−11 ]Φ−1

1 = Φ−1 − ερΦ−1FΦ−1,

i.e.,

(Φ + ερF )−1> Φ−1 − ερΦ−1FΦ−1. (3.2)

Choose ε > 0 small enough such that εF 6 (1 − ρ)Φ/2 since Φ > cIr for somepositive number c. By inequality (3.1), choosing λ = (1− ρ)/2, similarly to theproof of inequality (3.2), we have

(Φ + εF )−16 Φ−1 − ε(1 + ρ)Φ−1FΦ−1/2. (3.3)

Define

A0(ξ) := diag[(1 − e−iξ), 1, . . . , 1] (3.4)

andΦ := A−m

0 Φ−1Am0 .

By Lemma 2.1 we know that Φ ∈ C∞(T)r×r . By Φ > 0 we know that the

determinant of Φ is positive. By the definition of Φ, we have

[Φ−1 − ερΦ−1FΦ−1] − [Φ−1 − ε(1 + ρ)Φ−1FΦ−1/2]

= ε(1 − ρ)Φ−1FΦ−1/2

= ε(1 − ρ)Am0 [ΦA−m

0 F (A−m0 )∗Φ∗](Am

0 )∗/2

= ε(1 − ρ)Am0 [ΦF Φ∗](Am

0 )∗/2.

(3.5)

424 MO

It is obvious that ΦF Φ∗ > 0. By Lemma 3.2, there exists some P ∈ P(T)r×r

such that

Φ−1 − ε(1 + ρ)Φ−1FΦ−1/2 6 P 6 Φ−1 − ερΦ−1FΦ−1. (3.6)

Applying inequalities (3.2) and (3.3), we have

(Φ + εF )−16 P 6 (Φ + ερF )−1.

Now we can prove the following proposition.

Proposition 3.4 Let φ and a be defined as in Theorem 2.2. If φ has stableinteger shifts, then there exists Θ ∈ P(T)r×r such that Θ(0)1,1 = 1, Θ > 0 and

Θ−1 − Ta(Θ−1) > 0. (3.7)

Proof: By Theorem 2.2, there exists some positive number ρ < 1 and F > 0such that TaF 6 ρF where F is defined as in (2.11). As we discussed before,Φ satisfies TaΦ = Φ > 0. Hence by Proposition 3.3, there exist ε > 0 and someP ∈ P(T)r×r such that

0 < (Φ + εF )−16 P 6 (Φ + ερF )−1,

i.e.,

Φ + ερF 6 P−16 Φ + εF. (3.8)

Let Θ := P > 0, then by inequality (3.8) we have Θ(0)1,1 = 1 and

Θ−1 − Ta(Θ−1) > (Φ + ερF ) − Ta(Θ−1) > (Φ + ερF ) − Ta(Φ + εF ) > 0.

4 The existence of tight multiwavelet frames de-

rived from refinable function vectors

Our final major step will be proving the main theorem in this paper. We shallgo through the following well-known lemmas first.

Lemma 4.1 Suppose A is an m× n matrix and B is an n×m matrix. Thenwe have

λndet(λIm −AB) = λmdet(λIn −BA).

Proof: We have the following identities:

[Im AB λIn

] [λIm 0−B In

]=

[λIm −AB A

0 λIn

]

425THE EXISTENCE OF TIGHT MRA...

and [Im AB λIn

] [λIm −A0 In

]=

[λIm 0λB λIn −BA

].

Taking the determinants of the matrices in the above two equations, we have

λndet(λIm −AB) = λmdet(λIn −BA).

Lemma 4.2 Let A be an m×n matrix and B be an n×m matrix. If (AB)∗ =AB and (BA)∗ = BA, then we have

Im −AB > 0 ⇐⇒ In −BA > 0.

Proof: Suppose Im − AB > 0. Then all the eigenvalues of (Im − AB) arenonnegative. Hence

det[λIm − (Im −AB)] =

m∏

j=1

(λ− λj),

where for j = 1, . . . ,m, λj is nonnegative. By Lemma 4.1, we have

(λ− 1)m−ndet[λIn − (In −BA)] = (λ− 1)m−ndet[(λ− 1)In +BA)]

= det[(λ− 1)Im +AB] =

m∏

j=1

(λ− λj).

Thus all the eigenvalues of (In−BA) are nonnegative. Combining that (BA)∗ =BA, we know that In −BA > 0. Therefore, Im −AB > 0 implies In −BA > 0.Similarly, In −BA > 0 implies Im −AB > 0.

Finally, we are in the position to state the main theorem in this paper.

Theorem 4.3 Let φ ∈(L2(R)

)rbe a refinable function vector. If the integer

shifts of φ are stable and the matrix mask a of φ satisfies sum rules of order m,then there exists a tight multiwavelet frame derived from φ and it has vanishingmoments of order m.

Proof: By assumption, the matrix mask a takes the canonical form (2.4). ByProposition 3.4, there exist Θ ∈ P(T)r×r such that Θ(0)1,1 = 1, Θ > 0 andΘ−1 − Ta(Θ−1) > 0.

First we want to prove that M > 0, where M is defined by

M(ξ) :=

Θ(ξ). . .

Θ(ξ + 2π(|d|−1)

d

)

−

a(ξ)∗

a(ξ + 2π

d

)∗...

a(ξ + 2π(|d|−1)

d

)∗

Θ(dξ)

a(ξ)∗

a(ξ + 2π

d

)∗...

a(ξ + 2π(|d|−1)

d

)∗

∗

.

(4.1)

426 MO

By the fact Θ > 0 we have Θ1 := Θ1/2 > 0. Therefore, by the definition of Min (4.1), we have

Θ1(ξ). . .

Θ1

(ξ + 2π(|d|−1)

d

)

−1

M(ξ)

Θ1(ξ). . .

Θ1

(ξ + 2π(|d|−1)

d

)

−1

= I|d|r −

(aΘ−11 )∗(ξ)Θ1(dξ)

(aΘ−11 )∗

(ξ + 2π

d

)Θ1(dξ)

...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ1(dξ)

(aΘ−11 )∗(ξ)Θ∗

1(dξ)(aΘ−1

1 )∗(ξ + 2π

d

)Θ∗

1(dξ)...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ∗

1(dξ)

∗

.

Hence,

M(ξ) > 0

⇐⇒ I|d|r −

(aΘ−11 )∗(ξ)Θ1(dξ)

(aΘ−11 )∗

(ξ + 2π

d

)Θ1(dξ)

...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ1(dξ)

(aΘ−11 )∗(ξ)Θ∗

1(dξ)(aΘ−1

1 )∗(ξ + 2π

d

)Θ∗

1(dξ)...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ∗

1(dξ)

∗

> 0.

By Lemma 4.2, for all ξ ∈ T, we have that

M(ξ) > 0

⇐⇒ Ir −

(aΘ−11 )∗(ξ)Θ∗

1(dξ)(aΘ−1

1 )∗(ξ + 2π

d

)Θ∗

1(dξ)...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ∗

1(dξ)

∗

(aΘ−11 )∗(ξ)Θ1(dξ)

(aΘ−11 )∗

(ξ + 2π

d

)Θ1(dξ)

...

(aΘ−11 )∗

(ξ + 2π(|d|−1)

d

)Θ1(dξ)

> 0.

By Θ = Θ21, we have

M(ξ) > 0 ⇔ Ir − Θ1(dξ)Ta(Θ−1)(dξ)Θ1(dξ) > 0 ⇔ Θ−1(dξ) > Ta(Θ−1)(dξ),

i.e.,M > 0 ⇐⇒ Θ−1

> Ta(Θ−1).

Thus by inequality (3.7), we know M > 0.Secondly, we want to prove that we can derive a tight wavelet frame from the

given Θ. As a special case of [12, Theorem 3.1], to derive a tight wavelet framefrom the given Θ, we need to find r×r matrices a1(ξ),..., aL(ξ) of trigonometricpolynomials such that

a1(ξ)∗ · · · aL(ξ)∗

a1(ξ + 2π

d

)∗ · · · aL(ξ + 2π

d

)∗...

. . ....

a1(ξ + 2π(|d|−1)

d

)∗

· · · aL(ξ + 2π(|d|−1)

d

)∗

a1(ξ)a2(ξ)

...aL(ξ)

= M1(ξ), (4.2)

427THE EXISTENCE OF TIGHT MRA...

where

M1(ξ) :=

Θ(ξ) − a(ξ)∗Θ(dξ)a(ξ)

−a(ξ + 2π

d

)∗Θ(dξ)a(ξ)

...

−a(ξ + 2π(|d|−1)

d

)∗

Θ(dξ)a(ξ)

.

Recall M is defined as

M(ξ) =

Θ(ξ). . .

Θ(ξ + 2π(|d|−1)

d

)

−

a(ξ)∗

a(ξ + 2π

d

)∗...

a(ξ + 2π(|d|−1)

d

)∗

Θ(dξ)

a(ξ)∗

a(ξ + 2π

d

)∗...

a(ξ + 2π(|d|−1)

d

)∗

∗

.

It is evident to see that (4.2) is equivalent to

A(ξ)∗A(ξ) = M(ξ), (4.3)

where

A(ξ) :=

a1(ξ) · · · a1(ξ + 2π(|d|−1)

d

)

.... . .

...

aL(ξ) · · · aL(ξ + 2π(|d|−1)

d

)

.

Define

E(ξ) :=

Ir e−iξIr · · · e−i(|d|−1)ξIrIr e−i(ξ+2π/d)Ir · · · e−i(|d|−1)(ξ+2π/d)Ir...

.... . .

...

Ir e−i(ξ+2π(|d|−1)/d)Ir · · · e−i(|d|−1)(ξ+2π(|d|−1)/d)Ir

.

To solve (4.3), using the polyphase decomposition, define

A(dξ) := A(ξ)E(ξ)

andM(dξ) := E(ξ)∗M(ξ)E(ξ).

By direct calculation, we can verify that

M(ξ) ∈ P(T)|d|r×|d|r ⇐⇒ M(ξ) ∈ P(T)|d|r×|d|r

anda1(ξ), . . . , aL(ξ) ∈ P(T)r×r ⇐⇒ A(ξ) ∈ P(T)Lr×|d|r.

It is evident to see that (4.3) is equivalent to

A(ξ)∗A(ξ) = M(ξ). (4.4)

428 MO

We are especially interested in the case L = |d|. In this case, A(ξ) is a |d|r ×|d|r matrix. Since M(ξ) ∈ P(T)|d|r×|d|r, we know that M(ξ) ∈ P(T)|d|r×|d|r.

Moreover, by M > 0, we have M > 0. Hence by the matrix-valued Fejer-RieszLemma([9], [15], [17], [25]), there exists A ∈ P(T)|d|r×|d|r such that A(ξ)∗A(ξ) =

M(ξ). Therefore we can obtain trigonometric polynomial matrices a1(ξ), ...,

a|d|(ξ) by the relation A(ξ) = A(dξ)E(ξ)−1. Moreover, by the choice of a1,..., a|d|, we know that a1(ξ), ..., a|d|(ξ) are r × r matrices of trigonometricpolynomials and satisfy (4.2). Hence by [12, Theorem 3.1] we can derive a tightwavelet frame with generators ψ1, ..., ψ|d| such that ψ1, ..., ψ|d| are r × 1function vectors and

ψj(dξ) = aj(ξ)φ(ξ), j = 1, . . . , |d|.

Finally we want to prove that ψ1, ..., ψ|d| all have vanishing moments oforder m. By [12, Theorem 2.3] and a direct computation, we only need to provethat

(1 − e−iξ)m | aj(ξ)k,1, k = 1, . . . , r; j = 1, . . . , |d|.By (4.2), we have that

|d|∑

j=1

r∑

k=1

|aj(ξ)k,1|2 =[Θ(ξ) − a(ξ)∗Θ(dξ)a(ξ)

]1,1

=[Φ−1(ξ) +Am

0 (ξ)G(ξ)Am0 (ξ)∗ − a(ξ)∗Φ−1(dξ)a(ξ)

− a(ξ)∗Am0 (dξ)G(dξ)Am

0 (dξ)∗a(ξ)]1,1

=[Φ−1(ξ) − a(ξ)∗Φ−1(dξ)a(ξ)

]1,1

+O(|ξ|2m) as ξ → 0.

By Lemma 2.1, we have

(1 − e−iξ)2m |[Φ−1 − a(ξ)∗Φ−1(dξ)a(ξ)

]1,1.

Hence|d|∑

j=1

r∑

k=1

|aj(ξ)k,1|2 = O(|ξ|2m) as ξ → 0.

Therefore,|d|∑

j=1

r∑

k=1

|aj(ξ)k,1/ξm|2 < +∞ as ξ → 0.

Noticing the fact that the summation of nonnegative numbers is still nonnega-tive, we have

|aj(ξ)k,1/ξm| < +∞ as ξ → 0, k = 1, . . . , r; j = 1, . . . , |d|.

Thus,(1 − e−iξ)m | aj(ξ)k,1, k = 1, . . . , r; j = 1, . . . , |d|.

Hence ψ1, ..., ψ|d| all have vanishing moments of order m.

429THE EXISTENCE OF TIGHT MRA...

5 An example

−1 −0.5 0 0.5 1

−0.5

0

0.5

1

(a)−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b)

Figure 1: Generators for the tight multiwavelet frame in Example 5.1: (a) ψ1

and ψ2 (b) ψ3 and ψ4. Functions ψ1, ψ2, ψ3 and ψ4 are either symmetric orantisymmetric about the origin and have vanishing moments of order 1.

Example 5.1 Define a function vector φ := (φ1, φ2)T where

φ1 := (t+1)2(−2t+1)χ[−1,0)+(t−1)2(2t+1)χ[0,1), φ2 := t(t+1)2χ[−1,0)+t(t−1)2χ[0,1).

The function vector φ is the well known Hermite cubics and it satisfies thefollowing refinement equation

φ =

[1/2 3/4−1/8 −1/8

]φ(2 · +1) +

[1 00 1/2

]φ(2·) +

[1/2 −3/41/8 −1/8

]φ(2 · −1).

The refinable function vector φ has the following interpolation property: φ1(0) =1, φ′1(0) = 0, φ2(0) = 0 and φ′2(0) = 1. For further discussion on the Hermiteinterpolants, please see [10] and the references therein.

The mask of the Hermite cubics φ is given by

a(ξ) :=

[(eiξ + 2 + e−iξ)/4 3(eiξ − e−iξ)/8(−eiξ + e−iξ)/16 (−eiξ + 4 − e−iξ)/16

]. (5.1)

We can check out that a satisfies the sum rules of order 4 with a row vectory(ξ) = [1, eiξ/3 + 1/2 − e−iξ + e−2iξ/6]. Take m = 1. Define U(ξ) := I2. ThenU(2ξ)a(ξ)U(ξ)−1 takes the form of (2.2) with m = 1. Define

Θ(ξ) :=

[1 0

0 15 + 9√

2

]

and

a1(ξ) := d1

[(2 − e−iξ − eiξ)/4 3(e−iξ − eiξ)/8

(−29 + 16√

2)(e−iξ − eiξ)/784 (20 + 11e−iξ + 11eiξ)/112

],

430 MO

a2(ξ) := d2

[0 (3

√6 + 4

√3)(e−iξ − eiξ)/8

(6 − 5√

2)(e−iξ − eiξ)/196 3(2 − e−iξ − eiξ)/28

],

where

d1 := diag

[1,

√105 + 63

√2

], d2 := diag

[1,

√70 + 42

√2

].

Define functions ψ1, ψ2, ψ3 and ψ4 by

[ψ1(2ξ), ψ2(2ξ)]T := a1(ξ)φ(ξ), [ψ3(2ξ), ψ4(2ξ)]T := a2(ξ)φ(ξ).

By a direct computation based on [12, Theorem 3.1], one can verify that ψ1, ψ2, ψ3, ψ4generates a tight multiwavelet frame. Moreover, functions ψ1, ψ2, ψ3 and ψ4

are real-valued and symmetric, and all of them have vanishing moments of order1. For their graphs, see Figure 1.

Remark: Recently, Chui and Stockler ([4]) have constructed another tightmultiwavelet frame derived from the Hermite cubics. Their example has five(anti)symmetric generators with vanishing moments of order 4.

Acknowledgements: The autor would like to thank the anonymous refereesfor their reports to improve the presentation of this paper.

References

[1] C. Cabrelli, C. Heil, and U. Molter, Accuracy of lattice translates ofseveral multidimensional refinable functions, J. Approx. Theory 95, 5–52, (1998).

[2] C. K. Chui, W. He, and J. Stockler, Compactly supported tight andsibling frames with maximum vanishing moments, Appl. Comput. Har-mon. Anal. 13, 224–262, (2002).

[3] C. K. Chui, W. He, J. Stockler, and Q. Y. Sun, Compactly supportedtight affine frames with integer dilations and maximum vanishing mo-ments, Adv. Comput. Math. 18, 159–187, (2003).

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433THE EXISTENCE OF TIGHT MRA...

434

Face-based Hermite Subdivision Schemes

Bin HanDepartment of Mathematical and Statistical Sciences

University of AlbertaEdmonton, Alberta, Canada T6G 2G1

andThomas P.-Y. Yu

Department of Mathematical ScienceRensselaer Polytechnic Institute

Troy, New York 12180-3590

Abstract

Interpolatory and non-interpolatory multivariate Hermite type subdi-vision schemes are introduced in [8, 7]. In their applications in free-formsurfaces, symmetry properties play a fundamental role: one can essentiallyargue that a subdivision scheme without a symmetry property simply can-

not be used for the purpose of modelling free-form surfaces. The symmetryproperties defined in the article [8] are formulated based on an underly-ing conception that Hermite data produced by the subdivision processis attached exactly to the vertices of the subsequently refined tessella-tions of the Euclidean space. As such, certain interesting possibilities ofsymmetric Hermite subdivision schemes are disallowed under our vertex-based symmetry definition. In this article, we formulate new symmetryconditions based on the conception that Hermite data produced in thesubdivision process is attached to the faces instead of vertices of the sub-sequently refined tessellations. New examples of symmetric faced-basedschemes are then constructed.

Similar to our earlier work in vertex-based interpolatory and non-interpolatory Hermite subdivision schemes, a key step in our analysis isthat we make use of the strong convergence theory of refinement equationto convert a prescribed geometric condition on the subdivision scheme –namely, the subdivision scheme is of Hermite type – to an algebraic con-

dition on the subdivision mask. Our quest for face-based schemes in thisarticle leads also to a refined result in this direction.

Mathematics Subject Classification. 41A05, 41A15, 41A63, 42C40, 65T60,65F15Keywords. Refinable Function, Vector Refinability, Subdivision Scheme, Her-mite Subdivision Scheme, Shift Invariant Subspace, Symmetry, Subdivision Sur-face, Spline

435JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,435-450,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

1 Introduction

Subdivision schemes are used in both curve and surface design as well as waveletconstruction. Because of the two different applications, there comes also acertain degree of confusion in what subdivision is supposed to mean even in themost standard linear, stationary and shift-invariant case:

• [Geo] To a number of geometric modelling and computer graphics scien-tists, subdivision, in their so-called regular setting, typically consists ofthe following 4 components:

[T1] a choice of isohedral tiling of Rs;

[T2] a refinement rule that is used to transform the tiling to a similar tilingbut with a smaller tile size; this refinement rule is used repeatedly inthe subdivision process to create finer and finer tilings of Rs;

[G1] a specification of how data (measuring geometric positions) is at-tached to the individual tiles; in 2-D this comes with the choice ofattaching positional data to vertex and/or edge and/or face;

[G2] a fixed linear rule for how data attached to any specific tile is deter-mined from the data attached to the coarser scale tiles.

[T1]-[T2] are typically referred to as the “topological” part of the sub-division scheme. So far the main application of subdivision is in surfacemodelling, in which s = 2 and the isohedral tiling is usually based onequilateral triangles or squares.

(In the setting of free-form subdivision surfaces, the above is the so-calledregular part of a subdivision scheme, one needs also extraordinary andboundary subdivision rules for producing a free-form surface. We do notdiscuss the latter in this article.)

• [Wav] To a number of mathematicians working in wavelet analysis, sub-division is defined as a linear operator S := Sa,M of the form:

Sv(α) =∑

β∈Zs

v(β) a(α −Mβ), (1.1)

where a is the mask and M is a so-called isotropic dilation matrix. See[8, 7] and the references therein. This operator has a very close connectionto the refinement equation

φ(x) =∑

β

a(β) φ(Mx− β), (1.2)

which is directly related to wavelet construction under the MRA frame-work of Mallet and Meyer. Iteratively applying S to a sequence of initialdata v produces the data Snv, n = 0, 1, 2, . . . , with

(Snv)(α) ≈ f(M−nα), n large, (1.3)

436 HAN,YU

where f is the limit function – exists for a “good” mask a – of the subdi-vision process.

The geometric entities that underly (1.3) are, of course, the successivelyrefined lattices

Zs ⊂M−1Zs ⊂M−2Zs ⊂ · · · ⊂

∞⋃

n=0

M−nZs dense⊂ Rs. (1.4)

Most subdivision schemes we are aware of can be described under bothsettings. However, it seems unclear whether the two settings are exactly thesame in general. In (1.4) only discrete points are involved, these points are notconnected and hence there is no edge or face, and consequently no concept oftiling/graph/tessellation is directly involved in setting [Wav], so in this senseone may think that [Wav] is more general than [Geo]. On the other hand,the lattices in (1.4) are all isomorphic to Zs, whereas the isohedral tiling [T1]involved in [Geo] may have no structural similarity whatsoever with Zs – sofrom this point of view [Geo] is perhaps more general than [Wav].

At the analysis level, the setting [Wav] is very well-understood: there is bynow an extensive collection of analytical and computational tools available forthe analysis of (1.1)-(1.2).

In this paper, we propose face-based Hermite subdivision schemes whichwe first describe under the setting [Geo], we then translate them back to theform (1.1) and analyze them using the general theory available. To this end,we reuse the main ideas mutatis mutandis developed in [8]; see Section 2. Wefollow closely the notations and vocabularies in [8].

We recall here a couple of notations from [8] that will be used very oftenin the rest of the paper: Λr is the set of s-tuples of non-negative integers withsum no greater than r, ordered by the lexicographic ordering. S(E,Λr) is the#Λr ×#Λr matrix that measures how Hermite data of a function changes upona linear change of variable by E ∈ Rs×s: let ∂≤rf := [Dνf ]ν∈Λr

∈ R1×#Λr ,then

f ∈ Cr(Rs), g = f(E·) =⇒ ∂≤rg = ∂≤rf(E·) S(E,Λr). (1.5)

Why face-based Hermite schemes? Besides theoretical motivations, somevertex-based schemes are simply unnecessarily smooth for our purposes. Wehave been considering applications of symmetric Hermite subdivision schemesin free-form surfaces [13], in which we are primarily interested in schemes thatare C2 and have very small supports. For 2-D Hermite schemes with quincunxrefinement, if we insist on using vertex-based scheme then the smallest meaning-ful support is [−1, 1]2 and the resulted order 1 Hermite scheme is C7 [8, Section3.5]; in fact even the scalar counterpart of this scheme is much smoother than thedesired C2: the scalar quincunx scheme used in [12] (derived from the Zwart-Powell box spline) has the same [−1, 1]2 support, is a vertex-based scheme andis C4. If one wants to use the smaller support [0, 1]2 and still obtain schemes

437FACE-BASED HERMITE...

with a meaningful symmetry property then a natural approach is to changefrom vertex- to face-based scheme.

At the initial phase of the research work of this article, we speculated thatthere existed a C2 face-based Hermite subdivision scheme with the mask support[0, 1]2 (which corresponds to the stencil in Figure 2(a)) and is based on quincunxrefinement, as one could already obtain a C1 scheme in the scalar case. SeeSection 3.1. To our surprise, it does not seem to be case. Despite such a “badnews”, we shall discover in Section 3.2 a C2 Hermite face-based subdivisionscheme based on quadrisection refinement (i.e. M = 2I2) and the subdivisionstencil depicted in Figure 2(a)). The scalar counterpart of this scheme is theregular part of what Peters and Reif called “the simplest subdivision scheme

(for smoothing polyhedra)” [11].From experience it is almost always possible to gain smoothness by increas-

ing support size or by increasing multiplicity – with our experience in Section3.1 as a rare exception. For application in free-form subdivision surfaces, thelatter approach may have an advantage because of the inevitable presence ofextraordinary vertices in free-form surfaces.

Figure 1: Two topological refinement schemes based on the square tiling

2 Face-based Hermite Subdivision Schemes

In this paper, we consider subdivision schemes based on the following specifica-tions:

438 HAN,YU

[T1] The isohedral tiling of Rs is chosen to be the straightforward tiling byhypercubes:

Rs =⋃

α∈Zs

α+ [0, 1]s.

[T2] Let M be an s × s integer matrix such that M = UDUT where U isunitary and D is diagonal with diagonal entries with the same modulus σ > 1.In the rest of the paper we simply call any such matrix a dilation matrix.1 ThenM−1[0, 1]s is similar to [0, 1]s. The successively refined tilings are then given byRs =

⋃α∈Zs M−n(α + [0, 1]s). In dimension 2, two well-known examples of M

are:

2I2 =

[2 00 2

], MQuincunx =

[1 11 −1

].

See Figure 1.[G1] Subdivision data is a “jet” of Hermite data of order r at the center ofeach cube; conceptually, we have

R1×#Λr 3 vn(α) = data attached to the tile M−n([0, 1]s + α)

≈ ∂≤rf(M−n(α+ [1/2, . . . , 1/2]T )

)S(Mn,Λr). (2.1)

See [8] for the exact definition of the notations above; the vector on the right-hand side of (2.1) consists precisely of all the mixed directional derivatives of fof order up to r at the point M−n(α + [12 , . . . ,

12 ]T ) and in directions M−nej ,

j = 1, . . . , s.The informally described concept here will be made precise in definition 2.1

below.[G2] With the choice of [T1], [T2], [G1] above, there are still many choices ofsubdivision rules. We are interested in those with small supports, smooth limitsand symmetry, see the formal definitions below.

In each case above the subdivision scheme can be equivalently defined by asubdivision operator S = Sa,M of the form (1.1), i.e. there exists a subdivisionoperator S such that vn := Snv0, ∀ n ≥ 0, is precisely what [T1]-[T2] & [G1]-[G2] above would produce. Notice that S is a so-called stationary subdivisionoperator — the subdivision data vn is generated by a fixed subdivision mask aat all levels n; we mention without a rigorous justification that if we want thedata vn = Snv0 produced by a stationary subdivision process to also enjoy anatural Hermite type convergence property (c.f. Definition 2.1 below), then thescaling by S(Mn,Λr) introduced in (2.1) is essentially the only sensible choice.

In 2-D, here are some subdivision stencils that we are particularly interestedin:

1. When M = MQuincunx, the Hermite data associated with any level n +1 square is determined from a linear combination of the Hermite dataassociated with the two level n squares containing the level n+ 1 square.See Figure 2(a). In this case, supp(a) = [0, 1]× [−1, 0].

1Notice that this is more restrictive than what is usually called an isotropic dilation matrix

in the literature.

439FACE-BASED HERMITE...

2. When M = 2I2, the Hermite data associated with any level n+1 square isdetermined from a linear combination of the Hermite data associated withthe 3 or 4 level n squares closest to the level n+1 square. Figure 2(b)&(c).In this case, supp(a) = [−1, 2]2 − (−1,−1), (−1, 2), (2,−1), (2, 2) or[−1, 2]2, respectively.

(a) (b) (c)

Figure 2: Subdivision Stencils for quincunx (a) and quadrisection (b)&(c) re-finement

Following the geometric intuition we have so far, we are led to the followingdefinition: Let M be a dilation matrix with the property described in [T2]above. An order r face-based Hermite subdivision operator S := Sa,M

is a subdivision operator with multiplicity m = #Λr such that (i) for any initialsequence v ∈ [l0(Zs)]1×m there exists fv ∈ Cr(Rs) such that

limn→∞

∥∥[∂≤rfv]|M−n(Zs+[ 12,... , 1

2]T ) − vnS(Mn,Λr)

∥∥[l∞(Zs)]1×m = 0, (2.2)

where vn = Snv, (ii) fv 6= 0 for some v 6= 0. For notational convenience, wewrite

S∞v := fv.

We now recall from [8] the following definition, proposed originally for thestudy of what we now call vertex-based Hermite subdivision scheme.

Definition 2.1 ([8, Definition 1.1]) A subdivision operator S := Sa,M is ofHermite type of order r if (i) m = #Λr for some r ≥ 0, and for any initialsequence v ∈ [l0(Zs)]1×m there exists fv ∈ Cr(Rs) such that

limn→∞

||[∂≤rfv]|M−nZs − vnS(Mn,Λr)||[l∞(Zs)]1×m = 0, (2.3)

where vn = Snv, (ii) fv 6= 0 for some v 6= 0.

The two definitions are clearly equivalent. Comparing (2.2) and (2.3),of course the only difference is that the latter involves sampling of Hermite dataof at the lattices M−nZs, whereas the former involves sampling of Hermite dataat the shifted lattices M−n(Zs + [12 , . . . ,

12 ]T ). However, the two convergence

440 HAN,YU

definitions involve n tending to infinity, and the difference induced by the shiftin (2.2) becomes negligible for large n. An obvious application of triangle in-equality, combined with the assumption that fv ∈ Cr in both (2.2) and (2.3),shows that the definition of order r face-based Hermite subdivision operatorabove is identical to Definition 2.1.

So what is the real difference between vertex-based and face-based subdivi-sion schemes then? For the setup in this paper, we address only sum-rule andsymmetry conditions, as these are the two conditions we use for determining asubdivision mask a. We do not address, for instance, data-structure issues forcomputer implementation.

At first glance,

• Symmetry conditions must be different between vertex- and face-basedschemes. As expounded in the introductory section of our earlier paper[8], the meaning of symmetry relies completely on the geometric meaningof subdivision data vn(α); as the “geometric meaning” of vn(α) has beenchanged from

vn(α) ≈ ∂≤rf(M−nα

)S(Mn,Λr)

in vertex-based scheme to (2.1) in face-based scheme, it should be hardlysurprising that symmetry properties for vertex- and face-based Hermitesubdivision schemes have to be somewhat different.

As expected, this is exactly the case. See Section 2.2.

• Sum rule conditions, derived independently of symmetry conditions, arethe same for vertex- and face-based Hermite subdivision schemes, becauseof the equivalence of (2.2) and (2.3).

As it turns out, this is not the case because of a technical loophole in aresult in our earlier paper [8]. See Section 2.1.

2.1 Sum rules and the spectral quantity ν∞

(a, M)

Since a subdivision mask uniquely specifies a subdivision operator, sum rules– algebraic relations necessarily satisfied by the mask of any smooth subdivisionprocess – are very useful for construction of subdivision schemes.

In the following, let us recall the definitions of sum rules in [2, 5] and theimportant spectral quantity νp(a,M) in [5] in the setting of Hermite subdivisionschemes.

For a given sequence u ∈ (l0(Zs))m×n, its Fourier series u is defined to be

u(ξ) :=∑

β∈Zs

u(β)e−iβ·ξ, ξ ∈ Rs.

Let a be a matrix mask with multiplicity m. We say that a satisfies the sumrules of order k + 1 with respect to the dilation matrix M (see [5, Page 51]) ifthere exists a sequence y ∈ (l0(Zs))1×m such that y(0) 6= 0,

Dµ[y(MT ·)a(·)](0) = | detM |Dµy(0) and Dµ[y(MT ·)a(·)](2πβ) = 0, (2.4)

441FACE-BASED HERMITE...

for all |µ| ≤ k, β ∈ (MT )−1Zs\Zs.The quantity νp(a,M), which is defined in [5, Page 61], plays a very im-

portant role in the study of convergence of vector subdivision schemes and thecharacterization of smoothness of refinable function vectors. For the convenienceof the reader, let us recall the definition of the quantity νp(a,M) from [5, Page61] as follows. The convolution of two sequences is defined to be

[u ∗ v](α) :=∑

β∈Zs

u(β)v(α − β), u ∈ (l0(Zs))m×n, v ∈ (l0(Zs))n×j .

In terms of the Fourier series, we have u ∗ v = uv. Let y be a sequence in(l0(Zs))1×m. We define the space Vk,y associated with the sequence y by

Vk,y := v ∈ (l0(Zs))m×1 : Dµ[y(·)v(·)](0) = 0 ∀ |µ| ≤ k. (2.5)

Let 1 ≤ p ≤ ∞. For any y ∈ (l0(Zs))1×m such that y(0) 6= 0, we define

ρk(a,M, p, y) := sup

limn→∞

‖an ∗ v‖1/n(`p(Zs))m×1 : v ∈ Vk,y

, (2.6)

where an is defined to be an(ξ) = a((MT )n−1ξ) · · · a(MT ξ)a(ξ). Define

ρ(a,M, p) := infρk(a,M, p, y) : (2.4) holds for some k ∈ N0

and some y ∈ (l0(Zs))1×m with y(0) 6= 0.

We define the following quantity:

νp(a,M) := − logρ(M)

[| detM |−1/pρ(a,M, p)

], 1 ≤ p ≤ ∞, (2.7)

where ρ(M) denotes the spectral radius of the matrix M . The above quantityνp(a,M) plays a key role in characterizing the convergence of a vector cascadealgorithm in a Sobolev space and in characterizing the Lp smoothness of arefinable function vector. For example, it has showed in [5, Theorem 4.3] thata vector subdivision scheme associated with mask a and dilation matrix Mconverges in the Sobolev space W k

p (Rs) if and only if νp(a,M) > k.Since the convergence condition (2.2) for face-based Hermite subdivision

operator turns out to be exactly the same as Definition 2.1 drawn from [8],in deriving sum rule conditions for face-based Hermite subdivision operator,one can presumably simply reuse the result from [8] pertaining to sum ruleconditions, which we now recall:

Theorem 2.2 ([8, Theorem 2.2]) Let M be an isotropic dilation matrix anda be a mask with multiplicity m = #Λr. Suppose that ν∞(a,M) > r. ThenSa,M is a subdivision operator of Hermite type of order r if a satisfies the sumrules of order r + 1 with a sequence y ∈ [l0(Zs)]1×#Λr such that

(−iD)µ

µ!y(0) = eT

µ , µ ∈ Λr. (2.8)

442 HAN,YU

At the time article [8] was written, the authors questioned whether thesufficient condition (2.8) is also necessary. If the answer of this open questionwere affirmative, then, in virtue of the equivalence of (2.2) and (2.3), sum ruleconditions for vertex- and face-based Hermite subdivision schemes would beexactly the same. However, at the level of generality of Theorem 2.2, (2.8)is not necessary for an order r Hermite type subdivision operator:Theorem 2.3 below on the one hand generalizes Theorem 2.2 and on the otherhand answers the above-mentioned open question negatively.

Recall that the cascade operator Qa,M associated with mask a and dilationmatrix M is defined to be

Qa,Mf :=∑

β∈Zs

a(β)f(M · −β).

A fixed point of this operator is a solution of the refinement equation (1.2). Thisobservation is the starting point of the so-called strong convergence theory ofrefinement equation.

Theorem 2.3 Let M be an isotropic dilation matrix and a be a mask withmultiplicity m = #Λr. Suppose that ν∞(a,M) > r. Then Sa,M is a subdivisionoperator of Hermite type of order r if a satisfies the sum rules of order r + 1with a sequence y ∈ [l0(Zs)]1×#Λr such that

(c− iD)µ

µ!y(0) = eT

µ , µ ∈ Λr (2.9)

where c ∈ Rs is an arbitrary but fixed vector.

Proof: The proof follows essentially the same line of argument as in the proofof Theorem 2.2, the main new ingredient is the following observation: Let ψ bea Hermite interpolant of order r with accuracy order r + 1, then

ψc := ψ(· − c)

satisfies moment conditions with respect to a y ∈ [l0(Zs)]1×#Λr that satisfies(2.9).

By assumption, a satisfies sum rules of order r + 1 with a y that also sat-isfies (2.9), and also that ν∞(a,M) > r, then the strong convergence theory ofrefinement equation [5, Theorem 4.3] says that

limn→∞

||Qna,Mψc − φ||[Cr(Rs)]m×1 = 0

for some φ ∈ [Cr(Rs)]m×1; note that φ must be a solution of the refinementequation (1.2).

Now we show that Sa,M satisfies the Hermite property in Definition 2.1. Letv ∈ [l0(Zs)]1×#Λr , and vn = Snv. We can also write vn =

∑β v(β)an(· −Mnβ)

where an = Sn(δIm×m), m = #Λr; on the other hand, we have

Qnψc =∑

α

an(α)ψc(Mn · −α).

443FACE-BASED HERMITE...

Then

fn :=∑

α∈Zs

vn(α)ψc(Mn · −α) =

∑

α∈Zs

∑

β∈Zs

v(β)an(α−Mnβ)ψc(Mn · −α)

=∑

β∈Zs

v(β)(Qnψc)(· − β).

Therefore, if f :=∑

α v(α)φ(· − α), then limn→∞ ||fn − f ||Cr(Rs) = 0 andlimn→∞ ||Dµfn −Dµf ||L∞ = 0, ∀ µ ∈ Λr.

If we denote by ∂≤rψc(x) the #Λr × #Λr matrix with the µ-th row equalsto ∂≤r(ψc)µ(x), then since ψ is a Hermite interpolant,

∂≤rψc(α+ c) = I#Λr×#Λrδ(α), ∀ α ∈ Zs.

By (1.5), we have

∂≤rfn(M−n(α+c)) =∑

β∈Zs

vn(β)(∂≤rψc)(α+c−β)S(Mn,Λr) = vn(α)S(Mn,Λr).

But then

maxα∈Zs

||∂≤rf(M−n(α− c)) − vn(α)S(Mn,Λr)||∞

= maxα∈Zs

||∂≤rf(M−n(α− c)) − ∂≤rfn(M−n(α− c))||∞

≤ maxµ∈Λr

||Dµfn −Dµf ||L∞ → 0, as n→ ∞.

So a satisfies condition (i) of Definition 2.1 in virtue of the fact that f ∈[Cr(Rs)]m×1 and

||∂≤rf(M−nα) − vn(α)S(Mn,Λr)||∞ ≤ ||∂≤rf(M−nα) − ∂≤rf(M−n(α− c))||∞

+ ||∂≤rf(M−n(α− c)) − vn(α)S(Mn,Λr)||∞

→ 0 + 0, as n→ ∞.

The condition ν∞(a,M) > r implies, by [5, Theorem 4.3], that spanφ(·−β) :β ∈ Zs ⊇ Πr, which implies φ 6= 0. Thus condition (ii) of Definition 2.1 is alsosatisfied by a.

Theorem 2.3 does not directly tell how to choose the shift vector c. Fromthe proof, however, it seems the most reasonable to choose

c = [0, . . . , 0] and c = [1/2, . . . , 1/2]

for vertex and face-based schemes, respectively. We speculate that after one fixesthe symmetry condition on mask a – the topic of the next section – then therecorresponds a unique shift vector c such that condition (2.9) is both necessaryand sufficient for Sa,M being a Hermite type subdivision operator.

444 HAN,YU

2.2 Symmetry

Let G be a finite group of linear maps leaving the cube tiling of Rs invariant.For a technical reason, we assume also that G is compatible to dilation matrixM in the following sense ([3]):

MEM−1 ∈ G, ∀ E ∈ G. (2.10)

In our bivariate examples, we use exclusively the following symmetry group:

D4 =

±

[1 00 1

],±

[0 −11 0

],±

[1 00 −1

],±

[0 11 0

]. (2.11)

Note that D4 is compatible to either 2I2 or MQuincunx.

Definition 2.4 Let G be a symmetry group compatible with dilation matrix M .An order r Hermite subdivision operator has a face-based symmetry propertywith respect to G if the following condition is satisfied: ∀ f ∈ Cr(Rs) andE ∈ G, if v := [∂≤rf ]|M−n(Zs+[ 1

2,... , 1

2]T ), w := [∂≤rg]|M−n(Zs+[ 1

2,... , 1

2]T ) where

g := f(E·), then S∞w = (S∞v)(E·).

Theorem 2.5 An order r Hermite subdivision operator Sa,M has a face-basedsymmetry property with respect to G if

1. 1 is a simple and dominant eigenvalue of the matrix

J0 :=∑

β∈Zs

a(β)/| detM |

and the first entry of its nonzero eigenvector for the eigenvalue 1 is nonzero;

2. The following symmetry condition on the mask a holds

a(E(α − Ca) + Ca) = S(M−1EM,Λr) a(α) S(E−1,Λr), ∀ E ∈ G(2.12)

where e := [12 , . . . ,12 ]T and

Ca := (M − Is) e. (2.13)

Proof: If φ = [φ1, . . . , φ#Λr]T is the “impulse response” of Sa,M , then φ

satisfies the refinement equation (1.2) (that is, Qa,Mφ = φ) and

S∞v =∑

α∈Zs

v(α)φ(· − α).

We first show that the symmetry condition on Sa,M is implied by the followingcondition on φ:

φ(x) = S(E−1,Λr) φ(E(x − e) + e). (2.14)

445FACE-BASED HERMITE...

(The converse implication is also true and can be easily obtained by adaptingpart of the proof of [8, Proposition 2.3].)

Assume (2.14). Let f , E, g, v and w be as in Definition 2.4. Then

S∞w =∑

α

∂≤rg(α+ e) φ(· − α)(1.5)=

∑

α

∂≤rf(E(α+ e)) S(E,Λr) φ(· − α)

=∑

β

∂≤rf(β + e) S(E,Λr) φ(· − (E−1(β + e) − e))

(2.14)=

∑

β

∂≤rf(β + e) φ(E · −β) = S∞v(E·).

Therefore, the order r Hermite subdivision operator Sa,M has a face-basedsymmetry property with respect to G.

Below we show that (2.14) follows from the assumptions of the theorem; theargument is essentially a time-domain version of the frequency-domain proof of[4, Proposition 2.1].

Let φE := S(E−1,Λr)φ(E(· − e) + e), E ∈ G. It follows from (2.12) and thedefinition of φMEM−1 that

Qa,MφMEM−1 =∑

α∈Zs

a(α)φMEM−1 (M · −α)

= S(E−1,Λr)∑

α∈Zs

S(E,Λr)a(α)S(ME−1M−1,Λr)φ(MEM−1(M · −α− e) + e)

= S(E−1,Λr)∑

α∈Zs

a(MEM−1(α− Ca) + Ca)φ(ME · −MEM−1α−MEM−1e+ e)

= S(E−1,Λr)∑

α∈Zs

a(α)φ(ME · −α+ Ca −MEM−1Ca −MEM−1e+ e)

= S(E−1,Λr)∑

α∈Zs

a(α)φ(M(E · +M−1Ca − EM−1Ca − EM−1e+M−1e) − α)

= S(E−1,Λr)φ(E · +M−1Ca − EM−1Ca − EM−1e+M−1e).

By Ca = (M − Is)e, we deduce that

M−1Ca − EM−1Ca − EM−1e+M−1e = (Is − E)M−1Ca − EM−1e+M−1e

= (Is − E)M−1(M − Is)e− EM−1e+M−1e

= (Is − E)(Is −M−1)e− EM−1e+M−1e

= e− Ee−M−1e+ EM−1e− EM−1e+M−1e

= e− Ee.

We conclude that

Qa,MφMEM−1 = S(E−1,Λr)φ(E(· − e) + e) = φE ∀ E ∈ G.

In other words, we have Qa,MφE = φM−1EM for all E ∈ G. Therefore,Qn

a,MφE = φM−nEMn for all n ∈ N and E ∈ G.

446 HAN,YU

Since G is a finite group and M−nEMn ∈ G for all n ∈ N, there mustexist a positive integer ` such that M−`EM ` = E. Consequently, we haveQ`

a,MφE = φM−`EM` = φE for some positive integer `. Since 1 is a simpledominant eigenvalue of the matrix J0, the same can be said to Jn

0 for all n ∈ N.It is known ([10] and references therein) that if Jn

0 has 1 as a simple dominanteigenvalue, then φ is the unique solution, up to a scalar multiplicative constant,to the refinement equation Qn

a,Mφ = φ.

On the other hand, it follows from the refinement equation that φ(0) and

φE(0) are eigenvectors of the matrices J0 and J`0 , respectively. Since 1 is a

simple eigenvalue of J`0 , we must have φE(0) = cφ(0) for some complex number

c ∈ C since J`0φE(0) = φE(0) and Jn

0 φ(0) = φ(0). By the definition of φE ,

we have φE(0) = S(E−1,Λr)φ(0) by | detE| = 1. By our assumption, the first

entry in the vector φ(0) is nonzero; that is, eT1 φ(0) 6= 0. Note that the the

first row of the matrix S(E−1,Λr) is eT1 . We see that eT

1 φE(0) = eT1 φ(0) 6= 0.

Therefore, it follows from φE(0) = cφ(0) that c must be 1. Hence, we conclude

that we must have φE = φ by Q`a,MφE = φE and φE(0) = φ(0). In other words,

(2.14) holds.

3 Examples

In this section, we explore examples in the three cases depicted in Figure 2. Ineach case, we are interested in bivariate Hermite schemes of order r = 1, and wereuse exactly the computational framework developed in [8, Section 3] and theassociated Maple based solver together with the smoothness optimization codedeveloped in [6]. Underlying this smoothness optimization code is a method byJia and Jiang [9] which gives the critical L2 regularity of the refinable functionvector φ associated with a subdivision mask in the case when φ has stable shifts,and a lower bound for the critical L2 regularity in the absence of stability.

The definition of sum rules, which is given in (2.4) from the frequency do-main, can be equivalently rewritten in the time domain as follows: There existsa set of m× 1 vectors Yµ : µ ∈ Ns

0, |µ| ≤ k with Y0 6= 0 (see [2, Page 22] and[8, Equation (3.9), Section 3]) such that

∑

0≤ν≤µ

(−1)|ν|Yµ−νJa

α(ν) =∑

ν∈Λk

S(M−1,Λk)µ,νYν ∀ µ ∈ Λk, α ∈ Zs,

where (ν1, . . . , νs) ≤ (µ1, . . . , µs) means νj ≤ µj for all j = 1, . . . , s, andJa

α(ν) :=∑

β∈Zs a(α+Mβ)(β+M−1α)ν/ν! with (ξ1, . . . , ξs)(ν1,... ,νs) := ξν1

1 · · · ξνss .

The precise relation between the above definition and the definition of sum rulesin (2.4) is

Yµ =(−iD)µy(0)

µ!.

For face-based Hermite subdivision schemes of order r = 1 in dimensions = 2:

447FACE-BASED HERMITE...

• The sum rules are now with respect to a y that satisfies (2.8) with c = [ 12 ,12 ]T .

For s = 2, r = 1 and c = [12 ,12 ]T , (2.8) is equivalent to

Y(0,0) = [1, 0, 0], Y(1,0) = [1/2, 1, 0], Y(0,1) = [1/2, 0, 1].

• Symmetry conditions are those in (2.12)-(2.13).

3.1 M = MQuincunx, supp(a) = [0, 1] × [−1, 0], G = D4

The highest order of sum rule achievable in this case is 3, and the mask wefound by our solver has one degree of freedom:

a(0, 0) =

12

−4t+12

4t−12

0 14

14

− 18 t −t

and a(0,−1), a(1,−1), a(1, 0) are given by symmetry conditions (2.12)-(2.13).We found, first by empirical observation and then by an explicit calculation,that when t = −1/4, the refinable function vector has L2 smoothness equals to2.5 and consists of piecewise quadratic C1 spline functions. By optimizing theL2 smoothness over the parameter t, we found that when t = −0.1875, the L2

smoothness of the subdivision scheme is 2.57793, only slightly higher than thatof the spline scheme.

From a smoothness point of view, one gains almost nothing by going fromscalar scheme r = 0 to Hermite scheme r = 1 – this is atypical when comparedto all the examples obtained in [7, 6, 8]. When we choose r = 0 with the supportsize and symmetry group unchanged, the only mask that satisfies the first sumrule is

a(0, 0) = a(1, 0) = a(0,−1) = a(1,−1) = 1/2, (3.1)

its refinable function is a box-spline by Zwart and Powell which is a piecewisequadratic C1 spline function – same as what the above vector scheme giveswhen t = −1/4. A major difference is that the ZP element has unstable shiftswhereas, from various observations, we believe that the refinable function vectorof the above spline scheme has stable shifts.

Despite the bad news, we seem to be saved by the good news below.

3.2 M = 2I2, supp(a) ⊆ [−1, 2]2, G = D4

While (the regular part of) Peters and Reif’s mid-edge scheme [11] is essen-tially the quincunx subdivision scheme (3.1), their scheme actually operatesbased on quadrisection refinement instead of quincunx refinement; this is madepossible by the following observation: notice that M2

Quincunx = 2I2, so if b =

S2a,MQuincunx

δ, then, in our notations, we have

S2a,MQuincunx

= Sb,2I2 .

448 HAN,YU

Notice also that supp(b) = [−1, 2]2 − (−1,−1), (−1, 2), (2,−1), (2, 2). Thisscheme is C1 but not C2. Here, we construct a Hermite version of Sb,2I2 whichturns out to be C2.

Using our solver, we found the following 3-parameter mask with the samesupport as b above which satisfies sum rules of order 4:

a(0, 0) =

3/4 − 4c3 −c1 + 2c2 −c1 + 2c2−3/32 + c3 c1/2 + 5/16 −c2 − 1/16−3/32 + c3 −c2 − 1/16 c1/2 + 5/16

,

a(2, 0) =

1/8 + 2c3 c1 −2c2

c3 1/8 + c1/2 −c2−1/32 1/16 1/16

(3.2)

with the other entries given by symmetry conditions. By our smoothness op-timization code, the L2 smoothness of this scheme occurs to be the highestwhen (c1, c2, c3) ≈ (−7/16,−3/32, 3/64), at which the L2 smoothness is 3.5,thus the Holder smoothness is at least 2.5, meaning that the scheme is C2. Theassociated refinable function vector φ is depicted in Figure 3.

support order of sum rules # free parameters highest L2 Smoothness12 points 4 (highest possible) 3 3.516 points 4 7 3.516 points 5 (highest possible) 2 3.0

Table 1: Sum Rules and smoothness attained by some small support face-basedHermite subdivision schemes with M = 2I2

The now-classical Doo-Sabin scheme [1] is based on the tensor productquadratic B-spline, which the latter can be viewed as a face-based scalar subdi-vision scheme with the 16 point support [−1, 2]2. We have explored symmetricface-based Hermite schemes with the same support, which corresponds to thestencil in Figure 2(c). A higher order of sum rules – but not higher smoothness– can be achieved when compared to the 12 point support. See Table 1.

Figure 3: Refinable φ = [φ(0,0), φ(1,0), φ(0,1)]T associated with the subdivision

mask in (3.2)

449FACE-BASED HERMITE...

Acknowledgements. Thomas Yu would like to thank Yonggang Xue for tech-nical assistance and Peter Oswald for helpful discussions. Bin Han research wassupported in part by NSERC Canada under Grant G121210654. Thomas Yu’sresearch was supported in part by an NSF CAREER Award (CCR 9984501.)

References

[1] D. Doo and M. Sabin. Analysis of the behavoir of recursive division surfacesnear extraordinary points. Comp. Aid. Geom. Des., 10:6 (1978), 356–360.

[2] B. Han. Approximation properties and construction of Hermite interpolantsand biorthogonal multiwavelets. J. Approx. Theory, 110:1 (2001), 18–53.

[3] B. Han. Computing the smoothness exponent of a symmetric multivariaterefinable function. SIAM J. Matrix Anal. and Appl., 24:3 (2003), 693–714.

[4] B. Han. Symmetric multivariate orthogonal refinable functions. Appliedand Computational Harmonic Analysis, 17 (2003), 277–292.

[5] B. Han. Vector cascade algorithms and refinable function vectors in Sobolevspaces. Journal of Approximation Theory, 124:1 ( 2003), 44–88.

[6] B. Han, M. Overton, and T. P.-Y. Yu. Design of Hermite subdivisionschemes aided by spectral radius optimization. SIAM Journal on ScientificComputing, 25:2 (2003), 643–656.

[7] B. Han, T. P.-Y. Yu, and B. Piper. Multivariate refinable Hermite inter-polants. Mathematics of Computation, 73 (2004), 1913–1935.

[8] B. Han, T. P.-Y. Yu, and Y. Xue. Non-interpolatory Hermite subdivisionschemes. Mathematics of Computation 74 (2005), 1345–1367.

[9] R. Q. Jia and Q. T. Jiang. Spectral analysis of the transition operator andits applications to smoothness analysis of wavelets. SIAM J. Matrix Anal.and Appl., 24:4 (2003), 1071–1109.

[10] Q. Jiang and Z.W. Shen. On existence and weak stability of matrix refinablefunctions. Constr. Approx., 15 (1999), 337–353.

[11] J. Peters and U. Reif. The simplest subdivision scheme for smoothingpolyhedra. ACM Transactions on Graphics, 16:4 (October 1997), 420–431.

[12] L. Velho and D. Zorin. 4-8 subdivision. Comp. Aid. Geom. Des., 18:5(2001), 397–427.

[13] Y. Xue, T. P.-Y. Yu, and T. Duchamp. Hermite subdivision surfaces:Applications of vector refinability to free-form surfaces. In preparation,2003.

450 HAN,YU

On the Construction of Linear Prewavelets Over

a Regular Triangulation

Don Hong and Qingbo Xue

Department of Mathematical Sciences

Middle Tennessee State University

Murfreesboro, TN 37614, USA

Abstract

In this article, all the possible semi-prewavelets over uniform refine-

ments of a regular triangulation are constructed. A corresponding theorem

is given to ensure the linear independence of a set of different pre-wavelets

obtained by summing pairs of these semi-prewavelets. This provides effi-

cient multiresolution decompositions of the spaces of functions over vari-

ous regular triangulation domains since the bases of the orthogonal com-

plements of the coarse spaces can be constructed very easily.

AMS 2000 Classifications: 41A15, 41A63, 65D07, 68U05.Key Words: Multiresolution analysis, pre-wavelets, regular trangulations,

splines.

1 Introduction

Piecewise linear prewavelets with small support are useful tools in approxima-tion theory and in the numerical solution of partial differential equations aswell as in the computer graphics and practical largescale data representations.Basically speaking, a multiresolution is a decomposition of a function space intomutually orthogonal subspaces, each of which is endowed with a basis. Thebasis functions of each subspace are called wavelets if they are mutually orthog-onal and prewavelets otherwise. The subspaces are called wavelet spaces andprewavelet spaces accordingly.

Kotyczka and Oswald [10] constructed piecewise linear prewavelets withsmall support in 1995. Floater and Quak [7] published their results on piecewiselinear prewavelets with small support on arbitrary triangulations in 1999. Lateron, they simplified the above results by introducing the idea of semi-wavelets,which can be used to construct wavelets (these semi-wavelets and wavelets areactually semi-prewavelets and prewavelets). Using this idea, Floater and Quakinvestigated the Type-1 triangulation in [8] and Type-2 in [7] respectively. Hong

451JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,451-471,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

and Mu [6] have discussed the piecewise linear prewavelets with minimal supportover Type-1 trangulation. Some recent results on piecewise linear pre-waveletsand orthogonal wavelets can be found in [1] and [4]. Very recently, a hierarchicalbasis for C1 cubic bivariate splines is used for surface compression in [5].

In this article, we study all the possible semi-prewavelets over uniform re-finements of a regular triangulation. A corresponding theorem is given to ensurethe linear independence of any set of different pre-wavelets obtained by sum-ming pairs of these semi-prewavelets. This means that the multiresolutions ofthe linear function spaces over various regular triangulation domains can bedone conveniently, since the bases of the orthogonal complements of the coarsespaces can be constructed very easily.

If not clearly claimed, we shall only consider the dyadic refinement. Thereader will find that most symbols in this paper have commonly been used inrelated monographs. The paper is organized as follows. Preliminaries are intro-duced in Section 2. In Section 3, we construct the semi-prewavelets and obtainthe main theorem on the construction of prewavelets over a regular triangulation

2 Multiresolution of Linear Spline Spaces over

r-Triangulations

From a mathematical point of view, a computer graphic is nothing else but afunction defined on a given region. On the other hand, a graphic on a domainΩ can be represented by functions in different level of function spaces Sj (j =0, 1, 2, · · · ). The difference between them is that the function from a fine spacegives more detail of the original graphic than the one from a coarse space does.In an ideal situation, we can easily “switch” a function from one space intoanother when it is necessary. The key to choosing another function space isto use a different basis of functions. Surprisingly, in a multivariate setting,the relation between the bases of these nested spaces makes it difficult to do so.Even the case of continues piecewise linear wavelets construction is unexpectedlycomplicated, see [3]–[4] and [6]–[10] and the references therein.

In the following we shall discuss the multiresolution of the linear spline func-tion space defined on any r-triangulation.

Definition 2.1 A set of triangles T = T1, ..., TM is called a triangulationof some subset Ω of R

2 if Ω = ∪Mi=1Ti and

(i) Ti ∩ Tj is either empty or a common vertex or a common edge, i 6= j,(ii) the number of boundary edges incident on a boundary vertex is two,(iii) Ω is simply connected.A triangulation T 0 = T1, ..., TM over some domain Ω is a regular triangu-lation or simply an r-triangulation if all the elements of T 0 are equilateraltriangles.

Figure 1 gives an example of an r-triangulation over a triangle shaped region.We denote by V the set of all vertices v ∈ R

2 of triangles in T and by E the set

452 HONG,XUE

Figure 1: An r-triangulation

of all edges e = [v, w] of triangles in T . By a boundary vertex or boundary edgewe mean a vertex or edge contained in the boundary of Ω. All other verticesand edges will be called interior vertices and interior edges. A boundary edgebelongs to only one triangle, and an interior edge to two.

For a vertex v ∈ V , the set of neighbours of v in V is

Vv = w ∈ V : [v, w] ∈ E.

Suppose T is a triangulation. Given data values fv ∈ R for v ∈ V , thereis a unique function f : Ω −→ R which is linear on each triangle in T andinterpolates the data: f(v) = fv, v ∈ V . The function f is piecewise linear andthe set of all such f constitute a linear space S with dimension |V |. For eachv ∈ V , let φv : Ω −→ R be the unique ‘hat’ or nodal function in S such that forall w ∈ V ,

φv(w) =

1, w = v;0, otherwise.

The set of functions Φ = φvv∈V is a basis for the space S and for any functionf ∈ S,

f(x) =∑

v∈V

f(v)φv(x), x ∈ Ω. (2.1)

The support of φv is the union of all triangles which contain v:

supp(φv) := ∪v∈T∈T T.

For a given triangulation T 0 = T1, · · · , TM, a refined triangulation is atriangulation T 1 such that every triangle in T 0 is the union of some trianglesin T 1. The result of this process is called a refinement of T 0.

Obviously, there are various kinds of refinements. If not clearly claimed, weshall only consider the following uniform or dyadic refinement. We shall use

453ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 2: The first refinement of the r-triangulation in Figure 1

[u, v] to denote the edge incident to two vertices u, v. A triangle with verticesu, v, w will be denoted as [u, v, w].

For a given triangle T = [x1, x2, x3], let y1 = (x2 + x3)/2, y2 = (x1 + x3)/2,and y3 = (x1 + x2)/2 denote the midpoints of its edges. Then the set of fourtriangles

TT = [x1, y2, y3], [y1, x2, y3], [y1, y2, x3], [y1, y2, y3]

is a triangulation and a refinement of the coarse triangle T . The set of trianglesT 1 = ∪T∈T 0TT is evidently a triangulation and a refinement of T 0. Similarly, awhole sequence of triangulations T j , j = 0, 1, · · · , can be generated by furtherrefinement steps. Let V j be the set of vertices in T j , and define Ej , Sj, φj

v, Vjv ,

and supp(φjv) accordingly. A straightforward calculation shows that

φj−1v = φj

v +1

2

∑

w∈Vj

v

φjw, v ∈ V j−1, (2.2)

and therefore we obtain a nested sequence of spaces

S0 ⊂ S1 ⊂ S2 ⊂ · · · . (2.3)

Clearly, a refinement T 1 of T 0 is still an r-triangulation (see Figure 2).Continuing the refinement process on Ω leads us to a nested sequence of spacesof linear splines defined on the domain Ω with r-triangulations.

As usual, we use the following standard definition of the inner product oftwo continuous functions on T 0,

〈f, g〉 =∑

T∈T 0

1

a(T )

∫

T

f(x)g(x) dx, f, g ∈ C(Ω),

where a(T ) is the area of the triangle T .

454 HONG,XUE

Let c be the area of any triangle in the r-triangulation T 0. Since all thetriangles are congruent, the inner product reduces to the scaled L2 inner product

〈f, g〉 =1

c

∫

Ω

f(x)g(x) dx. (2.4)

With this inner-product, the spaces Sj become inner-product spaces. Let W j−1

denote the relative orthogonal complement of the coarse space Sj−1 in the finespace Sj, so that

Sj = Sj−1 ⊕W j−1. (2.5)

We have the following decomposition:

Sn = S0 ⊕W 0 ⊕W 1 ⊕ · · · ⊕Wn−1 (2.6)

and the dimension of W j−1 is |V j | − |V j−1| = |Ej−1|.In the following, we shall try to construct a basis for the unique orthog-

onal complement W j−1 of Sj−1 in Sj. Each of these basis functions will becalled a prewavelet and the space W j−1 a prewavelets space. By combiningprewavelet bases of the spaces W k with the nodal bases for the spaces Sk, weobtain the framework for a multiresolution analysis (MRA). Thus any functionfn in Sn can be decomposed into its n+ 1 mutually orthogonal components:

fn = f0 ⊕ g0 ⊕ g1 ⊕ · · · ⊕ gn (2.7)

where f0 ∈ S0 and gj ∈ W j (j = 0, 1, ..., n − 1). We shall restrict our workfor the construction of bases of W k to the first refinement level since uniformrefinement has been used.

Let b be any given non-zero real number, a1 and a2 be two neighboringvertices in V 0, and denote by u ∈ V 1 \ V 0 their midpoint. We define the semi-prewavelet σa1,u ∈ S1 as the element with support contained in the support ofφ0

a1and having the property that, for all v ∈ V 0,

〈φ0v , σa1,u〉 =

−b, if v = a1;b, if v = a2;0, otherwise

(2.8)

whereσa1,u(x) =

∑

v∈N1a1

rvφ1v(x)

andN1

a1= a1 ∪ V

1a1

denotes the fine neighborhood of a1. The only nontrivial inner products be-tween σa1,u and coarse nodal functions φ0

v occur when v belongs to the coarseneighborhood of a1:

N0a1

= a1 ∪ V0a1.

455ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Thus the number of coefficients and conditions are the same and, as we willsubsequently establish, the element σa1,u is uniquely determined for any givennon-zero value of b in (2.8).

Since the dimension of W 0 is equal to the number of fine vertices in V 1,i.e. |V 1| − |V 0|, it is natural to associate one prewavelet ψu per fine vertexu ∈ V 1 \ V 0. Since each u is the midpoint of some edge in E0 connecting twocoarse vertices a1 and a2 in V 0, the element of S1,

ψu = σa1,u + σa2,u (2.9)

is a prewavelet since it is orthogonal to all nodal functions φ0v, v ∈ V 0.

The following theorem gives a sufficient condition that all the different pre-wavelets obtained in this way are linearly independent and hence form a basisof W 0.

Theorem 2.2 The set of prewavelets ψuu∈V 1\V 0 defined by (2.9) is a linearlyindependent set if

ψu(u) >∑

w∈V 1\V 0,w 6=u

|ψu(w)|, ∀u ∈ V 1 \ V 0

Proof: Let|V 0| = m, |V 1 \ V 0| = n,

andu1, u2, · · · , um, um+1, · · · , um+n

be any permutation of all the vertices in V 1 such that uj ∈ V 0 (1 ≤ j ≤m) and uj ∈ V 1 \ V 0 (m + 1 ≤ j ≤ m + n). The corresponding nodalfunctions are

φj(x) = φuj(x), j = 1, 2, · · · ,m+ n.

Then the prewavelets in ψuu∈V 1\V 0 can be written as

ψi(x) = ψum+i(x) =

m+n∑

j=1

rijφj(x), (1 ≤ i ≤ n)

where rij = ψi(uj) (1 ≤ i ≤ n, 1 ≤ j ≤ m+ n). Consider the matrix

R =

r11 r12 · · · r1,m+n

r21 r22 · · · r2,m+n

. . . . . . . . . . . . . . . . . . . . . . .rn1 rn2 · · · rn,m+n

and its sub matrix

R1 =

r1,m+1 r1,m+2 · · · r1,m+n

r2,m+1 r2,m+2 · · · r2,m+n

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .rn,m+1 rn,m+2 · · · rn,m+n

.

456 HONG,XUE

We claim that R1 is diagonally dominant. Actually, keeping in mind thatri,m+j = ψi(um+j) = ψum+i

(um+j) (1 ≤ i, j ≤ n), we know that

ri,m+i >∑

1≤j≤n,j 6=i

|ri,m+j |, (1 ≤ i ≤ n)

is equivalent to

ψum+i(um+i) >

∑

1≤j≤n,j 6=i

|ψum+i(um+j)|, (1 ≤ i ≤ n)

orψu(u) >

∑

w∈V 1\V 0,w 6=u

|ψu(w)|, ∀u ∈ V 1 \ V 0.

This completes the proof of the theorem.

3 Semi-prewavelets

We are going to establish the uniqueness of the semi-prewavelets for W 0 withregard to (2.8) and to find their coefficients. To simplify our calculation, Floaterand Quak’s result on inner products of nodal functions [7], which we state hereas a Lemma, will be used.

Lemma 3.1 Let t(e) denote the number of triangles (one or two) in T 0 con-taining the edge e ∈ E0 and t(v) the number of triangles (at least one) containingthe vertex v ∈ V 0. If v ∈ V 0 and w ∈ V 1 are contained in the same triangle inT 0 then

96〈φ0v, φ

1w〉 =

6t(v), if v = w;10t(e), if w is the midpoint of e;t(e), if e = [v, w];4, if otherwise.

(3.1)

Let σa1,u be a semi-prewavelet where the fine vertex u is the midpoint ofa1 and another coarse vertex a2. We call a1 the center (vertex) of the semi-prewavelet. The degree of a vertex in a triangulation is the number of neighborvertices of the vertex in that triangulation. Trivially, every coarse vertex is alsoa vertex in the fine triangulation and it has the same degree in both coarse andfine r-triangulations. Let k = |V 0

a1| = |V 1

a1| be the degree of a1. If a1 is an

interior vertex then k = 6. If a1 is a boundary vertex then the value of k couldrange from 2 to 6. Hence there are six possible topological structures of thesupport of σa1,u, which are identical to the support of φ0

a1.

For convenience, in the later steps to construct semi-prewavelets, we shalluse the following permutations of the vertices in the coarse neighborhood N0

a1

and the fine neighborhood N1a1

of a1:

N0a1

: v1 = a1, v2, · · · , vk,

457ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 3: Interior vertex a1 and its support

Figure 4: Boundary vertices and their supports

458 HONG,XUE

N1a1

: u1 = a1, u2, · · · , uk,

where v2 through vk are all the neighbor vertices of a1 in the coarse space labeledconsecutively in the counterclockwise order, and uj is the midpoint of the edge[a1, vj ], j = 2, 3, ..., k.

Let us simply rewrite φluj

(x) as φlj(x), where j = 1, 2, ..., k and l = 0, 1.

Then,

σa1,u(x) =

k∑

j=1

rjφ1j (x). (3.2)

Let A = (aij) be the k × k matrix such that aij = 96〈φ0i , φ

1j 〉. We have

〈φ0i , σv1,u〉 =

⟨

φ0i ,

k∑

j=1

rjφ1j

⟩

=k

∑

j=1

rj〈φ0i , φ

1j 〉

=

k∑

j=1

1

96aijrj =

1

96[ai1, ai2, · · · , aik]~r,

where~r = [r1, r2, · · · , rk]T .

Therefore, the “semi-orthogonal condition” (2.8) is equivalent to

A~r = ~b, (3.3)

where ~r = [r1, r2, · · · , rk]T ,

A =

a11 a12 · · · a1k

a21 a22 · · · a2k

. . . . . . . . . . . . . . . . . . .ak1 ak2 · · · akk

,

and~b = [b1 = −b, 0, ..., 0, bj = b, 0, ...0]T (here j satisfying u = uj).

Thus, if A is invertible then (3.3) has a unique solution of the coefficients. Fig-ure 3 and Figure 4 give all the cases of the supports of possible semi-prewaveletsup to a symmetric permutation.

Using Lemma 1, we can verify that A is invertible in every case. We choosethe value of b in (2.8) as 66240 so that we can get integer coefficients rj for allthe semi-prewavelets, except the boundary one with the center vertex of degree6.

Case 1, a1 is an interior vertex, SPW(I6) We obtain that

A =

36 20 20 20 20 20 202 20 4 0 0 0 42 4 20 4 0 0 02 0 4 20 4 0 02 0 0 4 20 4 02 0 0 0 4 20 42 4 0 0 0 4 20

.

459ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 5: Semi-prewavelet SPW(I6)

A is non-singular with the inverse

A−1 =1

1152

42 −30 −30 −30 −30 −30 −30−3 65 −11 5 1 5 −11−3 −11 65 −11 5 1 5−3 5 −11 65 −11 5 1−3 1 5 −11 65 −11 5−3 5 1 5 −11 65 −11−3 −11 5 1 5 −11 65

.

Let ~b =[

−b b 0 0 0 0 0]T

. Then

~r = A−1~b =[

−4140 3910 −460 460 230 460 −460]T.

Thus, a semi-prewavelet with its center a1 as an interior vertex has been uniquelydetermined as shown in Figure 5.

Simply, by turning Figure 5 around its center in the counterclockwise di-rection step-by-step, we can obtain all the other symmetric semi-prewaveletswhich share the same center a1. We shall see this effect in the following casefrom another point of view.

Case 2, a1 is a boundary vertex with degree 2, SPW(B2)In this case,

A =

6 10 101 10 41 4 10

.

and thus,

A−1 =1

192

42 −30 −30−3 25 −7−3 −7 25

.

460 HONG,XUE

Figure 6: Semi-prewavelet SPW(B2)

We obtain~r = A−1~b =

[

−24840 9660 −1380]T

for ~b =[

−b b 0]T, and

~r = A−1~b =[

−24840 −1380 9660]T

for ~b =[

−b 0 b]T

, respectively.As we can see in Figure 6, the two subcases of SPW(B2) are symmetric.

Note that they are the same up to symmetry. In the following cases of otherboundary semi-prewavelets we shall not mention this repeatly.

Case 3, a1 is a boundary vertex with degree 3, SPW(B3)We have

A =

12 10 20 101 10 4 02 4 20 41 0 4 10

and

A−1 =1

1920

210 −150 −150 −150−15 221 −35 29−15 −35 125 −35−15 29 −35 221

.

Therefore,

~r = A−1~b =[

−12420 8142 −690 1518]T

if ~b =[

−b b 0 0]T, and

~r = A−1~b =[

−12420 −690 4830 −690]T

if ~b =[

−b 0 b 0]T

.Case 4, a1 is a boundary vertex with degree 4, SPW(B4)

461ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 7: Semi-prewavelet SPW(B3[1]), SPW(B3[2])

In this case,

A =

18 10 20 20 101 10 4 0 02 4 20 4 02 0 4 20 41 0 0 4 10

and thus,

A−1 =1

576

42 −30 −30 −30 −30−3 65 −11 5 1−3 −11 35 −5 5−3 5 −5 35 −11−3 1 5 −11 65

.

Therefore, ~r =[

−8280 7820 −920 920 460]T

for~b =[

−b b 0 0 0]T,

and ~r =[

−8280 −920 4370 −230 920]

for ~b =[

−b 0 b 0 0]T

,correspondingly.

Case 5, a1 is a boundary vertex with degree 5, SPW(B5)We have

A =

24 10 20 20 20 101 10 4 0 0 02 4 20 4 0 02 0 4 20 4 02 0 0 4 20 41 0 0 0 4 10

462 HONG,XUE

Figure 8: Semi-prewavelet SPW(B4[1]), SPW(B4[2])

and

A−1 =1

88320

4830 −3450 −3450 −3450 −3450 −3450−345 9883 −1765 667 155 283−345 −1765 5275 −805 475 155−345 667 −805 5083 −805 667−345 155 475 −805 5275 −1765−345 283 155 667 −1765 9883

.

Accordingly, we obtain ~r =[

−6210 7671 −1065 759 375 471]T

if ~b =[

−b b 0 0 0 0]T

, ~r =[

−6210 759 −345 4071 −345 759]T

if

~b =[

−b 0 0 b 0 0]T

, and ~r =[

−6210 −1065 4215 −345 615 375]T

if ~b =[

−b 0 b 0 0 0]T

.Case 6, a1 is a boundary vertex with degree 6, SPW(B6)In the final case, we have

A =

30 10 20 20 20 20 101 10 4 0 0 0 02 4 20 4 0 0 02 0 4 20 4 0 02 0 0 4 20 4 02 0 0 0 4 20 41 0 0 0 0 4 10

.

463ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 9: Semi-prewavelet SPW(B5[1]), SPW(B5[2]), and SPW(B5[3])

464 HONG,XUE

A is non-singular with the inverse

A−1 =1

192

7160

−132

−132

−132

−132

−132

−132

−1320

11779105792

−2173105792

739105792

131105792

259105792

227105792

−1320

−2173105792

6259105792

−1021105792

499105792

179105792

259105792

−1320

739105792

−1021105792

6019105792

−973105792

499105792

131105792

−1320

131105792

499105792

−973105792

6019105792

−1021105792

739105792

−1320

259105792

179105792

499105792

−1021105792

6259105792

−2173105792

−1320

227105792

259105792

131105792

739105792

−2173105792

11779105792

.

We can solve for the vectors of coefficients as the following.

~r = A−1~b =276

551

[

−9918 15137 −2303 1337 577 737 697]T

for~b =

[

−b b 0 0 0 0 0]T,

~r = A−1~b =276

551

[

−9918 −2303 8237 −863 1037 637 737]T

for~b =

[

−b 0 b 0 0 0 0]T,

and

~r = A−1~b =276

551

[

−9918 1337 −863 7937 −803 1037 577]T

for~b =

[

−b 0 0 b 0 0 0]T.

Now we can get any possible prewavelets of an r-triangulation, since theabove semi-prewavelets included all the possible semi-prewavelets with the ex-ception of symmetries. By (2.9), to get a prewavelet, we only need to “sum”two semi-prewavelets together in such a way that the fine vertex u (which hasbeen circled in each figure of the semi-prewavelet) is the midpoint of the centersof these two semi-prewavelets. Some prewavelets which could be often used aregiven by Figure 11 through Figure 17. In these figures we denote the prewaveletobtained by summing an interior semi-prewavelet, SPW(I6), and a boundaryone, say SPW(Bj), by PW(I6, Bj). We denote the prewavelet obtained bysumming two boundary semi-prewavelets, SPW(Bi) and SPW(Bj), by PW(Bi,Bj).

With the above results on semi-prewavelets, we are now ready to state ourmain result on r-triangulations.

465ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 10: Semi-prewavelet SPW(B6[1]), SPW(B6[2]) and SPW(B6[3]) (q =276551 )

466 HONG,XUE

Figure 11: Pre-wavelet PW(I6,B3)

Figure 12: Pre-wavelet PW(B4,B2)

Figure 13: Pre-wavelet PW(B4,B4)

467ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Figure 14: Pre-wavelet PW(I6,B4)

Figure 15: Pre-wavelet PW(B5,B2)

468 HONG,XUE

Figure 16: Pre-wavelet PW(B5,B3)

Figure 17: Pre-wavelet PW(I6,B5)

469ON THE CONSTRUCTION OF LINEARPREWAVELTS...

Theorem 3.2 For any level of the refinements of any an r-triangulation, allthe possible prewavelets can be constructed by simply summing up the two semi-prewavelets illustrated in Figure 5 through Figure 10.

Proof: Each semi-prewavelet σa1,u(x) illustrated in Figure 5 through Fig-ure 10, which are all the possible cases of semi-prewavelets in r-triangulations,satisfies

σa1,u(u) >∑

w∈V 1\V 0,w 6=u

|σa1,u(w)|, ∀u ∈ V 1 \ V 0.

A prewavelet is the sum of two semi-prewavelets as expressed in (2.9). Thesum can be done in such a way that the fine vertex u (which has been circledin each figure of semi-prewavelets) is the midpoint of the centers, a1 and a2, ofthese two semi-prewavelets in (2.9). Since the intersection of N1

a1and N1

a2is

the single element set u, the only overlapped values are the values of the twosemi-prewavelets functions at vertex u. Therefore, the condition of Theorem 2.1is satisfied and this completes the proof.

Theorem 3.2 can be used on various shaped domains and all the possibleprewavelets can be found easily by simply checking if any figure in Figure 3 andFigure 4 is a subset of the given refined r-triangulation.

Acknowledgments: This work was supported in part by NSF-IGMS 0408086and 0552377 for Hong.

References

[1] J.S. Cao, D. Hong, and W.J. Huang, Parameterized Piecewise Linear Pre-Wavelets over Type-2 Triangulations, in Interactions between Wavelets andSplines (M.J. Lai ed.), Nashboro Press, Nashville, 2006, pp. 75–92.

[2] C.K. Chui, Multivariate Splines, SIAM, Philadelphia, 1988.

[3] G.C. Donovan, J.S. Geronimo, and D.P. Hardin, Compactly supported,piecewise affine scaling functions on triangulations, Constructive Approxi-mation 16, 201–219 (2000).

[4] D.P Hardin and D. Hong, On Piecewise Linear Wavelets and Prewaveletsover Triangulations, J. Comput. and Applied Math., 155, 91-109 (2003).

[5] D. Hong and L.L. Schumaker, Surface compression using hierarchical basesfor bivariate C1 cubic splines, Computing, 72 (2004), 79-92.

[6] D. Hong and Y. Mu, On construction of minimum supported piecewise lin-ear prewavelets over triangulations, in Wavelets and Multiresolution Meth-ods (T.X. He ed.), Marcel Decker Pub., New York, 2000, pp. 181–201.

[7] M.S. Floater and E.G. Quak, Piecewise linear prewavelets on arbitrarytriangulations, Numer. Math. 82, 221–252 (1999).

470 HONG,XUE

[8] M.S. Floater and E.G. Quak, A semiprewavelet approach to piecewise linearprewavelets on triangulations, in Approximation Theory IX, Vol 2: Com-putational Aspects (C.K. Chui and L.L. Schumaker eds.), Vanderbilt Uni-versity Press, Nashville, 1998, pp. 63–70.

[9] M.S. Floater and E.G. Quak, Linear independence and stability of piecewiselinear prewavelets on arbitrary triangulations, SIAM J. Numer. Anal., 38,58–79 (2000).

[10] U. Kotyczka and P. Oswald, Piecewise linear prewavelets of small support,in Approximation Theory VIII, Vol. 2 ( C.K. Chui and L.L. Schumaker,eds.), World Scientific, Singapore, 1995, pp. 235–242.

471ON THE CONSTRUCTION OF LINEARPREWAVELTS...

472

Spline Wavelets on Refined Grids and

Applications in Image Compression

Lutai Guan and Tan YangDept. of Sci. Computation & Computer Application

Sun Yat-sen (Zhongshan) UniversityGuangzhou, Guandong 510275, P.R. China

Abstract

In this paper, one kind of new spline-wavelets on refined grids is con-

structed. The scale functions are natural splines on refined grids that

were studied by authors for interpolation surfaces recently. Because of

the advantages of these splines to describe details well in some areas, the

spline-wavelets on refined grids will be applied in image compression and

information processing.

2000 Mathematics Subject Classification: Primary 90C325, 41A29; Sec-ondary 65D15, 49J52.

Keywords: image compression, refined grids, spline-wavelets.

1 Introduction

Wavelets play an important role in quite different fields of science and technol-ogy. There are many wavelets books available now, some of them are destinedto be classics in the field, for example, [2], [4], and [9]. Image or surface com-pression is one of the most active fields in the applications of wavelets (see [8],[10], [11]).

In general, wavelets need a uniform grid subdivision, but for image process-ing, sometimes we only need detail information in some given regions and forother parts of the image we could represent them in a rough scale.

In [7], a method for compressing surfaces associated with C1 cubic splinesdefined on triangulated quadrangulations is described. The refinement methodfor triangular regions is efficient for some applications in surface compression.

Classical natural spline interpolation can be found in [1]. Some recent re-search work on natural spline interpolation for scattered data has been done in[3], [5], and [6]. In [6], one kind of surface design called natural splines overrefined grid points is given. The method is simple and easy to be implemented

473JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,473-483,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

in computers. The designed surfaces have no gaps in the connecting patchesand are smoother. A change of only one de Boor point in refined grid will affectthe surface in a very small region.

In this paper, based on the spline interpolation over refined grid points,we consider a new kind of wavelets called addition spline–wavelets correspond-ing to the splines over refined grid points, and discuss their vanishing momentproperties and give algorithms for their application in image compression.

2 Compactly Supported Spline over Refined Grids

Following the notion of [6], we first define the refined grids and interpolatingnatural spline.

Definition 2.1 For given partitions π1 : a = x0 < x1 < · · · < xk+1 = b andπ2 : c = y0 < y1 < · · · < yl+1 = d, and subdivisions π3 : xi = xi,0 < xi,1 < · · · <xi,r+1 = xi+1 and π4 : yj = yj,0 < yj,1 < · · · < yj,s+1 = yj+1, the grid points:

xp, yq, p = 1, · · · , k; q = 1, · · · , `,

and the sub-grid points:

xi,µ, yj,ν, µ = 1, · · · , r; ν = 1, · · · , s

are called the refined grid points (or for short, refined grids).

Definition 2.2 The function σ(x, y) ∈ Hm,n(R) is called the interpolating nat-ural spline if it satisfies

‖Tσ‖2 = minu∈X,Au=z

‖Tu‖2,

where

‖Tu‖2 =

∫∫

R

(u(m,n)(x, y))2dxdy+

n−1∑

ν=0

∫ b

a

(u(m,ν)(x, c))2dx+

m−1∑

µ=0

∫ d

c

(u(µ,n)(a, y))2dy,

and (xi, yi, zi), i = 1, · · · , N are given scattered data for z = (z1, · · · , zN) and alinear operator A : Hm,n(R) 7→ R

N defined as

Au = (u(x1, y1), · · · , u(xN , yN )).

When the scattered data are over the refined grid points xp, yq, xi,µ, yj,ν,µ = 1, · · · , r; ν = 1, · · · , s; p = 1, · · · , k; q = 1, · · · , `, we call σ(x, y) an interpo-lating spline over refined grid points. We can write the data over refined grid asxp, yq, zp,q, xi,µ, yj,ν, zi,j,µ,ν, µ = 1, · · · , r; ν = 1, · · · , s; p = 1, · · · , k; q =1, · · · , `.

The following result on bivariate natural local basis interpolation over refinedgrid points is given in [6].

474 GUAN,YANG

Lemma 2.3 The natural spline σ(x, y) over refined grid points has the expres-sion:

σ(x, y) =k∑

µ=1

∑

ν=1

λµ,νBµ(x)Bν(y) +r∑

µ=1

s∑

ν=1

γµ,νBi,µ(x)Bj,ν(y),

where Bµ(x), Bν(y), Bi,µ(x), and Bj,ν(y) are B-splines of order 2m − 1 for xand of order 2n− 1 for y.

Similar to B-spline interpolation, we can add the boundary knots for the con-struction of natural B-splines, and the invariable natural B-spline interpolationsatisfying

σ(`)(a) = σ(`)(b) = 0, ` = m,m+ 1, · · · , 2m− 1.

3 Spline-wavelet on Refined Grids

It is not trivial at all to select the compactly supported splines over refined gridsmentioned in last section as the scaling functions for constructing wavelets. Forthis purpose, we apply the tensor product method.

First, we notice that the basis of a spline space over refined grids can bedivided into two sets: Bµ(x)Bν(y)k `

µ=1,ν=1 and Bi,µ(x)Bj,ν(y)r sµ=1,ν=1. Sec-

ond, for equal division (uniform grid), the basis of a spline space can be grouped

as B(x−µ)B(y− ν)k `µ=1,ν=1 and Bi(x−µ)Bj(y− ν)r s

µ=1,µ=1. They are twosets of tensor product spline scaling functions.

Definition 3.1 (Addition Wavelet Space) If V0 = V 10

⋃V 2

0 and there existswavelet spaces W 1

1 and W 21 satisfying:

V 10 = V 1

1 ⊕W 11 , V

20 = V 2

1 ⊕W 21 ,

then we call W 11

⋃W 2

1 an addition wavelet space, and V 11

⋃V 2

1 an additionscaling space.

Theorem 3.2 For a function f0 ∈ V0, we can find a decomposition f0 = f11 +

f21 + g1

1 + g21 where f i

1 ∈ V i1 , g

i1 ∈W i

1 , i = 1, 2, and f11 is orthogonal to g1

1 and f21

orthogonal to g21.

Proof. Since V0 = V 10

⋃V 2

0 , f0 ∈ V0, there exist f10 and f2

0 satisfyingf0 = f1

0 + f20 .

By the wavelet decomposition:

V 10 = V 1

1 ⊕W 11 , V

20 = V 2

1 ⊕W 21

We can find f11 ∈V 1

1 and g11 ∈W 1

1 , such that f10 = f1

1 + g11 and f1

1 is orthogonalto g1

1 .There also exist f2

1 ∈ V 21 and g2

1 ∈ W 21 , such that f2

0 = f21 + g2

1 and f21 is

orthogonal to g21. Hence,

f0 = f10 + f2

0 = f11 + f2

1 + g11 + g2

1.

475SPLINE WAVELETS...

Definition 3.3 (Addition Wavelet Decomposition) To an addition wavelet spaceW 1

1

⋃W 2

1 and an addition scaling space V 11

⋃V 2

1 , if for f0 ∈ V0, there existf i1 ∈ V i

1 and gi1 ∈ W i

1, i = 1, 2 such that

f0 = f11 + f2

1 + g11 + g2

1,

Then this equation is called an addition wavelet decomposition.

For an addition space V0 = V 10

⋃V 2

0 , if there exits wavelet decompositionsfor V 1

0 and V 20 , respectively, then there exists addition wavelet decomposition

for V0 and the decomposition is the sum of two decompositions.

Theorem 3.4 (Vanishing Moments Property) For an addition wavelet spaceW1 = W 1

1

⋃W 2

1 , let ψ1(x) be the wavelet function of W 11 having N1 vanishing

moments and ψ2(x) the wavelet function of W 21 having N2 vanishing moments.

Then ψ(x) ∈ W1 has N vanishing moments:∫

R

xmψ(x)dx = 0, 0 ≤ m ≤ N := minN1, N2.

Proof. Assume that∫

R

xmψj,i(x)dx = 0, 0 ≤ m ≤ Nj , j = 1, 2, i ∈ Z,

where the functions ψj,i, i ∈ Z are basis functions of the wavelet spaces W ji ,

j = 1, 2.By the definition of addition wavelet spaces, for any function ψ(x) ∈ W1,

there exist coefficients λi and µi satisfying:

ψ(x) =∑

i

λiψ1,i(x) +∑

i

µiψ2,i(x)

Hence, for 0 ≤ m ≤ minN1, N2

∫

R

xmψ(x)dx =

∫

R

xm

(∑

i

λiψ1,i(x) +∑

i

µiψ2,i(x)

)dx

=∑

i

λi

∫

R

xmψ1,i(x)dx +∑

i

µi

∫

R

xmψ2,i(x)dx

= 0

Let scaling spaces V 10x, V

10y , V

20x, and V 2

0y be generated by the scaling functions

φ1(x) = B(x), φ1(y) = B(y), φ2(x) = Bi(x), and φ2(y) = Bj(y), respectively.For one dimension spaces, we have wavelet space decompositions:

V i0x = V i

1x ⊕W i1x, V i

0y = V i1y ⊕W i

1y i = 1, 2.

The wavelet functions are: ψi1(x), ψ

i1(y) for W i

1x and W i1y, scaling functions are:

φi1(x), φ

i1(y) for V i

1x and V i1y, i = 1, 2 correspondingly.

476 GUAN,YANG

For two dimensional spaces, there exist tensor production wavelet decompo-sitions:

V i0 = V i

0x ⊗ V i0y = (V i

1x ⊕W i1x) ⊗ (V i

1y ⊕W i1y)

= (V i1x ⊗ V i

1y) ⊕ (V i1x ⊗W i

1y) ⊕ (W i1x ⊗ V i

1y) ⊕ (W i1x ⊗W i

1y)

:= V i1 ⊕W i

1 i = 1, 2

Definition 3.5 (Spline–wavelets on Refined Grids) For uniform B-spline, the

basis of a spline space over refined grids can be divided into two sets Bµ(x)Bν(y) k lµ=1,ν=1

and Bi,µ(x)Bj,ν(y) r sµ=1,ν=1, let

V 10 = SpanBµ(x)Bν(y) k l

µ=1,ν=1

V 20 = SpanBi,µ(x)Bj,ν(y) r s

µ=1,ν=1.

The addition wavelet space for V0 = V 10

⋃V 2

0 is called a spline-wavelet space onrefined grids.

Theorem 3.6 (Spline–wavelet Space Decomposition on Refined Grids) The spline-wavelet space on refined grids is an addition wavelet space:

W1 = W 11

⋃W 2

1

= [(V 11x ⊗W 1

1y) ⊕ (W 11x ⊗ V 1

1y) ⊕ (W 11x ⊗W 1

1y)]⋃

[(V 21x ⊗W 2

1y)⊕

(W 21x ⊗ V 2

1y) ⊕ (W 21x ⊗W 2

1y)]

The corresponding addition scaling space is:

V1 = V 11

⋃V 2

1 =(V 1

1x ⊗ V 11y

)⋃(V 2

1x ⊗ V 21y

).

Proof. By the definition of a spline-wavelet space on refined grids:

V0 = V 10

⋃V 2

0

and using the tensor product wavelet decompositions, we have

V0 = V 11 ⊕W 1

1

⋃V 2

1 ⊕W 21

=(V 1

1x ⊗ V 11y

)⊕(V 1

1x ⊗W 11y

)⊕(W 1

1x ⊗ V 11y

)⊕(W 1

1x ⊗W 11y

)⋃(

V 21x ⊗ V 2

1y

)⊕(V 2

1x ⊗W 21y

)⊕(W 2

1x ⊗ V 21y

)⊕(W 2

1x ⊗W 21y

)

=[(V 1

1x ⊗ V 11y

)⋃(V 2

1x ⊗ V 21y

)]⋃[(V 1

1x ⊗W 11y

)⊕(W 1

1x ⊗ V 11y

)⊕(W 1

1x ⊗W 11y

)]

⋃[(V 2

1x ⊗W 21y

)⊕(W 2

1x ⊗ V 21y

)⊕(W 2

1x ⊗W 21y

)]

=(V 1

1

⋃V 2

1

)⋃(W 1

1

⋃W 2

1

)

= V1

⋃W1.

477SPLINE WAVELETS...

Theorem 3.7 (Spline–wavelet Decomposition on Refined Grids) If ψi1(x) ∈

W i1x and ψi

1(y) ∈ W i1y are wavelet functions and φi

1(x) ∈ V i1x and φi

1(y) ∈ V i1y

are scaling functions for i = 1, 2, then for f(x, y) ∈ V0, we have spline-waveletdecomposition on refined grids:

f(x, y) =∑

i

2∑

j=1

αijφj1,i(x)φ

j1,i(y) +

2∑

j=1

(∑

i

λijφj1,i(x)ψ

j1,i(y) +

∑

i

µijψj1,i(x)φ

j1,i(y) +

∑

i

νijψj1,i(x)ψ

j1,i(y)

),

where the coefficients αij, λij, µij, νij are determined in the tensor prod-uct wavelet decompositions.

Proof. By the definition of splines over refined grids,

f(x, y) = f1(x, y) + f2(x, y), f1(x, y) ∈ V 10 , f

2(x, y) ∈ V 20 .

From the tensor product wavelet decomposition, there exist coefficients αij,λij, µij, νij such that

f j(x, y) =∑

i

αjijφ

j1,i(x)φ

j1,i(y) +

∑

i

λijφj1,i(x)ψ

j1,i(y) +

∑

i

µijψj1,iφ

j1,i(y) +

∑

i

νijψj1,i(x)ψ

j1,i(y),

for j = 1, 2. The conclusion follows from Theorem 3.6.

Corollary 3.8 For a function f(x, y) ∈ V0, there exists a spline-wavelet de-composition on refined grids:

f(x, y) = Φ(x, y) + Ψ(x, y)

where Φ(x, y) is the scaling approximation part of f(x, y) and Ψ(x, y) is thewavelet detail part of f(x, y). That is, there exist coefficients αij, λij,µij, νij such that

Φ(x, y) =∑

i

2∑

j=1

αijφj1,i(x)φ

j1,i(y),

Ψ(x, y) =

2∑

j=1

(∑

i

λijφj1,i(x)ψ

j1,i(y) +

∑

i

µijψj1,i(x)φ

j1,i(y) +

∑

i

νijψj1,i(x)ψ

j1,i(y)).

Theorem 3.9 (Vanishing Moments Property of Spline-wavelet on the RefinedGrids) For any function ψ(x, y) ∈ W1, the spline-wavelet space on refined grids,we have the following vanishing moments property:∫ ∫

R×R

xmynψ(x, y)dxdy = 0, 0 ≤ m ≤ minM1,M1, 0 ≤ n ≤ minN1, N2,

478 GUAN,YANG

where the wavelet functions ψi1(x) ∈ W i

1x and ψi1(y) ∈ W i

1y have vanishingmoments Mi and Ni, i = 1, 2 respectively.

Proof. For ψi1(x) ∈W i

1x and ψi1(y) ∈ W i

iy , we have

∫

R

xmψi1,k(x)dx = 0, 0 ≤ m ≤Mi, i = 1, 2, k = 0,±1,±2, · · ·

∫

R

ynψi1,k(y)dy = 0, 0 ≤ n ≤ Ni, i = 1, 2, k = 0,±1,±2, · · · .

By Corollary 3.8 there exists coefficients λij, µi,j,νij, such that

ψ(x, y) =

2∑

j=1

(∑

i

λijφj1,i(x)ψ

j1,i(y) +

∑

i

µijψj1,i(x)φ

j1,i(y) +

∑

i

νijψj1,i(x)ψ

j1,i(y)

).

Hence

∫∫

R×R

xmynψ(x, y)dxdy =

∫∫

R×R

xmyn

2∑

j=1

(∑

i

λijφj1,i(x)ψ

j1,i(y)

+∑

i

µijψj1,i(x)φ

j1,i(y) +

∑

i

νijψj1,iψ

j1,i(y)

)]dxdy

=

2∑

j=1

∑

i

λij

∫∫

R×R

xmynφj1,i(x)ψ

j1,i(y)dxdy

+

2∑

j=1

∑

i

µij

∫∫

R×R

xmynψj1,i(x)φ

j1,i(y)dxdy

+

2∑

j=1

∑

i

νij

∫∫

R×R

xmynψj1,i(x)ψ

j1,i(y)dxdy

=2∑

j=1

∑

i

λij

(∫

R

xmφj1,i(x)dx

)(∫

R

ynψj1,i(y)dy

)

+

2∑

j=1

∑

i

µij

(∫

R

xmψj1,i(x)dx

)(∫

R

ynφj1,i(y)dy

)

+

2∑

j=1

∑

i

νij

(∫

R

xmψj1,i(x)dx

)(∫

R

ynψj1,i(y)dy

)

= 0.

479SPLINE WAVELETS...

4 Image Compression of Spline-wavelets on Re-

fined Grids

There are many applications of wavelets in image compression. The goal ofimage compression is to take advantage of proper structures in the image toreduce image storages. There usually are three steps in the process: (1) thetransform step, (2) the quantization step, and (3) the coding step. Symmetry,vanishing moments and size of the filters are three things to be considered withchoosing a filter for the transform step.

Because of the high vanishing moments of the spline-wavelets on refinedgrids, we try to choose them as filters. Using spline interpolation, we can extractthe part of smooth image easily. And by coding technique, we can get the non-smooth details using wavelets. Adding these two parts, we can reach the goalof compression of the images. We provide the following algorithm to show thewhole process.

algorithm 4.1 1. Input data of an image.

2. Design a basic grid (rough grid) for the image and select some regions inthe image that need to have more details, then apply some refined grids onthose regions.

3. To the rough grid, use the tensor product spline-wavelet decomposition toget coefficients αi1, λi1, µi1 and νi1, such that:

f1(x, y) =∑

i

αi1φ11,i(x)φ

11,i(y) +

∑

i

λi1φ11,i(x)ψ

11,i(y)

∑

i

µi1ψ11,i(x)φ

11,i(y) +

∑

i

νi1ψ11,i(x)ψ

11,i(y).

4. Let f2(x, y) = f(x, y) − f1(x, y). On refined grids, use the tensor prod-uct spline–wavelet decomposition to get coefficients αi2, λi2, µi2 andνi2, such that

f2(x, y) =∑

i

αi2φ21,i(x)φ

21,i(y) +

∑

i

λi2φ21,i(x)ψ

21,i(y)

+∑

i

µi2ψ21,i(x)φ

21,i(y) +

∑

i

νi2ψ21,i(x)ψ

21,i(y)

5. Choose a coding technique to find the wavelet detail part:

g1(x, y) =

2∑

j=1

(∑

i

λijφj1,i(x)ψ

j1,i(y) +

∑

i

µijψj1,i(x)φ

j1,i(y) +

∑

i

νijψj1,i(x)ψ

j1,i(y)

)

and the scaling approximation part of the image/sub-image:

f1(x, y) =

2∑

j=1

∑

i

αijφj1,i(x)φ

j1,i(y).

480 GUAN,YANG

6 If the approximation and compression meet the criteria, stop; else, returnto step 2.

5 General Cases and Image Compression Exam-

ple

In the following, we show an example of image compression using spline-waveleton refined grids, for a woman face image in high resolution (1600×1200 pixels).First, we build a rough grid and on this grid we get a low resolution imageshown in Figure 1 (a) (200 × 150 pixels). In the “Face Recognition”, the eyesand mouth are important parts which should be described in details, speciallyat canthus and the corners of the mouth. We put our six refined grids on thoseparts (Figure 1 (b)) which uniformly refine those regions to 32 × 32 grids.

(a) (b)

Figure 1: The woman face image with a rough grid (a) and refined grids (b).

On refined grids, we sample the original image and obtain details usingspline-wavelet on refined grids. The two sets of data combined together form anew image file with compression.

In this example, there are six parts that need to be refined. We apply thealgorithm and finish them one by one.

Only the necessary details have been reserved. So the image’s storage spaceis observably reduced.

We can compare the details in rough grids (Figure 2 (a)) and in refined grids(Figure 2 (b)) for the woman’s right eye’s outboard canthus (marked in Figure 1(b)). The information brought in by addition spline-wavelet improves localimage quality and the two sets of data can be quantized in different resolutions,respectively.

Acknowledgments

This work is supported by Guangdong Provincial Natural Science Founda-tion of China (036608) and the Foundation of Zhongshan University Advanced

481SPLINE WAVELETS...

(a) (b)

Figure 2: Details over rough grids (a) and details over refined grids (b).

Research Center, Hong Kong. The authors would like to thank the editors,professors Don Hong and Yu Shyr of this special issue, and anonymous refereesfor a number of comments which led to important improvements to this paper.

References

[1] C. de Boor, A Practical Guide in Splines(second edition). Springer, NewYork, 2002.

[2] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.

[3] C.K. Chui and Lutai Guan, Multivariate polynomial natural splines forinterpolation of scattered data and other application, in Workshop onComputational Geometry (A.Conte et al. eds.), World Scientific, Singa-pore, (1993).

[4] I. Daubechies, Ten Lectures on Wavelets, Philadelphia, CBMS–NSF Seriesin Appl.Math. (SIAM), 1991.

[5] Lutai Guan, Bivariate polynomial natural spline interpolation algorithmswith local basis for scattered data, J. of Computational Analysis andAppl., 5 (2003), 77–100.

[6] Lutai Guan and Bin Liu, On Bivariate N-spline Interpolation Local Ba-sis for Scattered Data over Refined Grid Points, Mathematica NumericaSinica 25 (2003), 375–384.

[7] D. Hong and L.L. Schumaker, Surface Compression Using a Space of C1

Cubic Splines with a Hierarchical Basis, Computing, 72 (2004), 79-92.

482 GUAN,YANG

[8] S. Mallat, A Wavelet Tour of Signal Processing (Second Edition), Acad-emic Press, Boston, 1999.

[9] Y. Meyer, Wavelets and Operators, Cambridge University Press, 1992.

[10] N.T. PanKaj, Wavelets Image and Compression, Boston, Klumer Acad-emic Publisher, 1998.

[11] D.F. Walnut, An Introduction to Wavelet Analysis, Boston, Birkhauser,2002.

483SPLINE WAVELETS...

484

Variational method and Wavelet Regression

Jianzhong Wang

Department of Mathematics and Statistics

Sam Houston State University

Huntsville, TX 77341

email: mth jxw@shsu.edu

Abstract

Wavelet regression is a new nonparametric regression approach. Com-

pared with traditional methods, wavelet regression has the advantages of

ideal minimax property, spatial inhomogeneous adaptivity, high dimen-

sion expansibility and fast algorithm. In this paper, we provide a brief

review of wavelet shrinkage. We also apply the variational method for

wavelet regression and show the relation between the variational method

and wavelet shrinkage.

Keywords. Wavelets, Orthonormal wavelets, Wavelet regression, Varia-

tional Methods.

1 Introduction

Regression analysis is an important statistical technique for investigating andmodeling the relationship between variables. The goal of regression is to geta model of the relationship between one variable y and one or more variablest. Let y = f(t) be the observation (also called the observed function), whichresponds to the regressor t :

yi(:= f(ti)) = f(ti) + εi i = 1, 2, ..., N (1)

where ti, i = 1, 2, · · · , N, are sampling points, f(ti) are values of an unknownfunction f(t) (called the underlying function), and εi are the statistical errors(called noise), which are often formulated as independent random variables. Inthis paper, we assume the sampling points are equal-spaced between 0 and 1.The goal of a regression is to estimate the underlying function f(t) from thesampling data (ti, yi)

ni=1 (called the observed data), i.e., to establish a method

to find the estimator f , which achieves the minimal risk

R(f , f) = N−1E||f − f ||22,N .

The classical nonparametric regression methods mainly include orthogonal se-ries estimators, kernel estimators and smoothing splines. Theoretical discus-sions about these methods can be found in [17] and [35]. Recently, a variety

485JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,485-503,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

of spatially adaptive methods has been developed in the statistical literature.Among them, wavelet adaptive methods have received great attention. Refer-ences are given in Bock and Pliego [5], Vidakovic [36], Ogden [33] , Donoho andJohnstone [13], [14], [15], [16], and their references. In wavelet regression, animportant issue is to select a subset of wavelet coefficients, which well representthe underlying function f while removing most of noise.

Since 1980s, developing numerical methods for removing noise from an imagewhile preserving edge also became an active research area. Mumford and Shah[28] proposed an energy functional for images so that the image processing suchas noise removal and image segmentation can be formulated as a variationalproblem associated with the functional. Perona and Malik [34] proposed ananisotropic diffusion model for removing noise while enhancing edge. It is proventhat the anisotropic diffusion equation in [34] was the steepest decent methodfor solving the variational problem associated with a certain energy functional[38]. References in this aspect can be found in the book [3].

The goal of this paper is to apply the variational method in the waveletregression. The main idea is the following. Since the underlying function f isunknown, we cannot obtain the formula for R(f , f). Hence to directly minimize

R(f , f) is impossible. We shall create a substitution of the risk functional, say

GR(f , f), in which the underlying function is not involved. Thus, we can use

variational method to find f . The wavelet representation of data will enable us toestablish the variational problems with a very simple structure. We outline thepaper as follows. In Section 2, we briefly introduce orthonormal wavelet basesand adaptive wavelet regression (also called wavelet shrinkage). In Section 3, wedescribe the variational model for wavelet regression and establish the relationbetween it and wavelet shrinkages. In Section 4, we give some examples.

2 Preliminarily

2.1 Orthonormal Wavelet Bases

The regular orthonormal wavelet basis of L2(R) is constructed from a multires-olution analysis (MRA) (see [24], [25], [26], [22], and [23]).

Definition 1 A multiresolution analysis of L2(R) is a sequence (Vj)j∈Z ofclosed subspaces of L2(R) such that the followings hold:

(1) Vj ⊂ Vj+1 j ∈ Z(2) ∪j∈ZVj = L2(R) and ∩j∈ZVj = 0(3) f(x) ∈ Vj <=> f(2x) ∈ Vj+1

(4) f(x) ∈ Vj => f(x− 2−jk) ∈ Vj j, k ∈ Z(5) There exists a function φ ∈ V0 such that φ(x − n)n∈Z forms an un-

conditional basis of V0, i.e., φ(x− n)n∈Z is a basis of V0 and there exist twoconstants, A,B > 0 such that, ∀(cn) ∈ l2, the following inequality holds

A∑

|cn|2 ≤ ||

∑

cnφ(· − n)||2 ≤ B∑

|cn|2.

486 WANG

The function φ in Definition 1 is called an MRA generator. Furthermore,if φ(x − n)n∈Z is an orthonormal basis of V0, then φ is called an orthonor-mal MRA generator (also called an orthonormal scaling function). Assume φsatisfies the two-scale equation

φ(x) = 2∑

k∈Z

h(k)φ(2x− k), (h(k))k∈Z ∈ l2. (2)

Define the wavelet function ψ by

ψ(x) = 2∑

k∈Z

(−1)kh(1 − k)φ(2x − k). (3)

For a function f, we write

fj,k(x) = 2j

2 f(2jx− k), j, k ∈ Z. (4)

Then φj,kk∈Z forms an orthonormal basis of Vj . Let Wj ⊂ L2(R) be thesubspace spanned by ψj,kk∈Z . Then Wj is called the wavelet subspace at levelj. It is clear that Wj⊥Vj and Wj ⊕ Vj = Vj+1. Hence, ⊕∞

j=−∞Wj = L2(R) and

ψj,kj,k∈Z forms an orthonormal basis of L2(R), called a standard orthonormalwavelet basis. Recall that we also have

L2(R) = VJ0⊕ (⊕j≥J0

Wj) .

Hence, φJ0,kk∈Z ∪ ψj,kj≥J0,k∈Z also forms an orthonormal basis of L2(R),which is called a hybrid orthonormal wavelet basis. Using this basis we candecompose a function f ∈ L2(R) into the following form

f = fJ0+

∑

j≥J0

gj , fJ0∈ VJ0

, gj ∈ Wj ,

fJ0=

∑

aJ0,kφJ0,k, gj =∑

wj,kφj,k.

Particularly, a function f ∈ VJ has the decomposition

f = fJ0+

J−1∑

j=J0

gj, fJ0∈ VJ0

, gj ∈Wj .

Hybrid orthonormal wavelet bases are very useful in many applications be-cause in the decomposition above fJ0

provides an estimator of f while gj , j =J0, · · · , J − 1, preserve the details of f at different levels (see Figures 1-3). Formore discussions of wavelet bases, we refer to [12] and [24].

Daubechies ([11], [12]) created compactly supported orthonormal waveletsbases, which are very useful in wavelet regression. In regression the sample dataare finite. In order to deal with finite data, we need the multiresolution analysis,scaling functions, wavelets, and wavelet bases on intervals. The details of thesemodifications can be found in [10]. In wavelet regression, the wavelet bases on

487VARIATIONAL METHOD...

final intervals are used in the following way. Let Vj∞j=0 be the MRA defined

on the interval [0, 1] and Wj∞j=0 be the corresponding wavelet subspaces. For

a fixed J, we assume (φJ,k) forms an orthonormal basis of VJ , where φJ,k hasbeen modified from its original version 2J/2φ(2Jx − k) as in [10],[13]. Afterthe modification, the subscript (J, k) indicates the spatial location of φJ,k. Afunction f ∈ VJ is now decomposed to

fJ(t) =∑

aJ,kφJ,k(t), aJ,k =< f, φJ,k > . (5)

where the sum in (5) is a finite one. Similarly, we have VJ = VJ0

⊕J−1j=J0

Wj and

fJ(t) = fJ0(t) +

J−1∑

j=J0

gj(t) =∑

aJ0,kφJ0,k(t) +

J-1∑

j=J0

∑

wj,kψj,k(t), (6)

where aJ0,k =< fJ , φJ0,k > and wj,k =< f, ψj,k > .For convenience, we write aJ=(aJ,k),aJ0

= (aJ0,k), wj = (wj,k) , va =[aJ0

, wJ0, · · · ,wJ-1], and denote the transform from aJ to va by W :

va = WaJ .

We call W the discrete wavelet transform (DWT) and call its inverse W−1 theinverse discrete wavelet transform (IDWT). It is easy to see that when φ is anorthonormal MRA generator and ψ is the corresponding orthonormal wavelet,W is an orthonormal matrix. It follows that W−1 = WT . Without loss ofgenerality, later we shall always assume J0 = 0.

Based on pyramidal algorithms, Mallat in [18] developed fast algorithms forcomputing DWT and IDWT, which are also called Mallat’s algorithms. Theyhave the computational complexity of O(n logn), where n =length(aJ ).

2.2 Adaptive Wavelet Regression

We now return to the model (1). Since the sample data are finite, without lossof generality, we assume N in (1) is equal to the dimension of the space VJ

for a certain level J. (If it is not, we can extend the sample data to let it be.)Then we can identify the data in (1) to a function of the space VJ . Waveletregression adopts a wavelet estimator to estimate the underlying function . Theestimator is select from a subspace of VJ and the selection is dependent on thesample data. Hence, wavelet regression usually is a nonlinear regression. For thegiven sample data in (1), the wavelet regression can be outlined in the followingdiagram.

Sample Data → DWT → Wavelet Coefficients →Adaptive

Operation→

Modified Coefficients → IDWT → Wavelet Estimator

488 WANG

Let W be the DWT. Let Ta : Rn → Rn be the adaptive operator, whichchanges wavelet coefficients for a certain purpose. Then the algorithm thatperforms wavelet regression can be written as follows.

vy = Wy,

vy = Tavy,

y = W−1vy .

We write

W = WTTaW.

Then W denotes the wavelet regression operator. If the adaptive operator Ta isa compression one, then the adaptive wavelet regression is a wavelet shrinkagesince the number of non-vanished entries of vy is smaller than that one of vy.

The wavelet regression is based on the following facts.

• The orthonormal transform of white noise ∼ N(0, σ2) is still a white noise.∼ N(0, σ2). (See [13].)

• If f is a noise-free function, then most of its wavelet coefficients vanish,only a very few of them have non-neglected values, which represent thedetails of the function.

• If a function f carries noise, then its smooth component fJ0(t) in (6) is not

influenced by noise very much, while the wavelet components keep noise.

Figures 1 and 2 show the wavelet decompositions of noise-free functions, andFigures 3 and 4 show them of noisy functions.

By these properties, in wavelet regression, the adaptive operator Ta is se-lected so that it does not change a0, i.e.,

Ta =

[

I 0

0 T

]

,

where T is the operator on W := ⊕J−1j=0Wj . Let w = [w0, · · · ,wJ−1] and assume

the length of w is n. We simply denote w = (wi)ni=1. Thus,

w = θ + z, wi = θi + zi, (7)

where θi is the wavelet coefficient of the underlying function f and zi is thewavelet transform of εi. If εi is assumed to be the white noise ∼ N(0, σ2), so iszi. Then the wavelet regression is essentially to find a w that minimizes the risk

R(w, θ) =1

nE(||w − θ||2). (8)

Since small wavelet coefficients mostly contribute to noise while large ones tosignal, wavelet regression removes the small wavelet coefficients from w.

489VARIATIONAL METHOD...

Donaho and Johnstone in [13] used two different shrinkages to design T. Fora given threshold λ > 0, the hard threshold function is

ηh(x;λ) =

0, |x| ≤ λx, |x| > λ

, (9)

and the soft threshold function is

ηs(x;λ) =

0, |x| ≤ λsign(x)(|x| − λ), |x| > λ

. (10)

For a vector w, we write ηh(w;λ) = (ηh(wi;λ)) and ηs(w;λ) = (ηs(wi;λ)) re-spectively. Then the adaptive operators corresponding to ηh(w;λ) and ηs(w;λ)are Thw =ηh(w;λ) and Tsw =ηs(w;λ) respectively.

The choice of the threshold λ is a fundamental issue in wavelet shrinkage. Alarge threshold cuts too many coefficients and will result in an oversmoothingestimator. Conversely, a small threshold does not remove noise well and willproduce a wiggly, undersmoothing estimator. The proper threshold ought totake a careful balance. A lot of work have been done in this aspect. (See [1],[2], [7], [12],[13], [14], [15], [16], [29], [30], [31], and [32].)

0 500 1000 1500−5

0

5

10

15Blocks

0 500 1000 1500−4

−2

0

2

4Wavelet level 1

0 500 1000 1500−4

−2

0

2

4Wavelet level 2

0 500 1000 1500−4

−2

0

2

4Wavelet level 3

0 500 1000 1500−4

−2

0

2

4Wavelet level 4

0 500 1000 1500−5

0

5

10

15Smooth level

Figure 1. Wavelet decomposition of Block. Up to Level 4.

490 WANG

0 500 1000 15000

10

20

30Bumps

0 500 1000 1500−0.2

−0.1

0

0.1

0.2Wavelet level 1

0 500 1000 1500−1

0

1

2Wavelet level 2

0 500 1000 1500−1

0

1

2Wavelet level 3

0 500 1000 1500−1

0

1

2Wavelet level 4

0 500 1000 1500−10

0

10

20Smooth level

Figure 2. The wavelet decomposition of Bumps.

0 500 1000 1500−10

0

10

20

30 Blocks SNR=5

0 500 1000 1500−10

−5

0

5Wavelet level 1

0 500 1000 1500−5

0

5Wavelet level 2

0 500 1000 1500−5

0

5Wavelet level 3

0 500 1000 1500−5

0

5Wavelet level 4

0 500 1000 1500−10

0

10

20

30Smooth level

Figure 3. Wavelet decomposition of Blocks with noise.

491VARIATIONAL METHOD...

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0

10

20 Bumps SNR=5

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−2

0

2

4Wavelet level 1

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−1

0

1

2Wavelet level 2

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−1

0

1

2Wavelet level 3

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−1

0

1

2Wavelet level 4

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0

10

20Smooth level

Figure 4. Wavelet decomposition of Bumps with noise.

3 Variational Method for Wavelet Regression

Wavelet shrinkage is based on the principle that each wavelet coefficient whoseabsolute value is smaller than the threshold contributes to noise; otherwise itcontributes to the signal. The threshold in wavelet shrinkage is used to bal-ance the oversmoothing and undersmoothing. The value of the threshold isdependent of the noise level, the size of the sampling data, and the smoothnessrequired by the regression. The authors of [13] and [14] established several rulesfor determining thresholds, including Universal thresholding, MiniMax thresh-olding, SureShrink, Heursure and so on. (See also [15], [16], [29], [30].) Thesemethods are very effective when noise is white and its level is known. How-ever, in many applications, the noise type is unknown and the noise level is notuniform over all sampling areas. It is necessary to study wavelet regression ina general framework. By definition, regression is the minimization of the riskfunction, where the key issue is to remove noise while keeping the features ofthe data. Hence, we discuss the variational method for wavelet regression andreveal the relation between the variational method and the wavelet shrinkage.

Following the idea of [28], to balance the smoothness and the sharpness ofan estimator, we introduce two energy functions. The first energy measuresthe distance between the estimator and the sampling data. It is clear that theestimator should be close to the sample data. Recall that in wavelet regression,we do not change the function fJ0

∈ VJ0. Hence, the distance can be measured

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by the following wavelet deviation energy

A(u) =||u − w0||2, (11)

where w0 represents the vector of wavelet coefficients of the observation. Bythe properties of wavelets, A(u) controls the undersmoothing. Since w0 car-ries noise, the estimator will be undersmoothing if A(u) is close to 0. On theother hand, the wavelet components of a function represent its wiggle and noise.Therefore, we can create a weighted wavelet energy to measure the wiggle to-gether with the noise for data u:

F (u)= uTGu, (12)

where G(u) is a positive definite (or semi-definite) matrix. Under the assump-tion that noises are independent random variables, we may assume G is a diago-nal matrix, in which each entry on the diagonal is a weight function of ui. Recallthat small wavelet coefficients contribute to noise, and large ones contribute tosignal. For the purpose of regression, large weights ought to assign to smallcoefficients and vice versa. Hence, we have the following.

Definition 2 Let G be a nonnegative diagonal matrix defined by

G= diag(ci(ui))ni=1. (13)

where each ci is an even function satisfying the conditions: (1) ci(s) ≥ 0 andlims→∞ ci(s) = 0; (2) ci(|s|) is decreasing on [0,∞). Then

F (u)= uTGuu

is called the weighted wavelet energy and G is called the weight matrix.

Since G is a diagonal matrix, F (u) can be simplified to the quadratic form∑n

i=1 s2ci(s). Write gi(s) = s2ci(s). Then (gi(s)) is the energy density. The

weighted wavelet energy controls oversmoothing. If F (u) = 0, then all waveletcomponents vanish, which indicates that the estimator is in VJ0

, i.e., the regres-sion will cause oversmoothing.

We now combine G(u) and A(u) to make an energy function

Γ(u) = G(u) + λA(u), (14)

where λ is the parameter used to balance oversmoothing and undersmoothing.Thus, wavelet regression can be formulated to the following variational problem:Find w such that

w = arg(min Γ(u)). (15)

Assume each ci(s) is differentiable. Then the solution of (15) satisfies the Euler-Lagrangian equation

g′(w) + 2λ(w − w0) = 0,

493VARIATIONAL METHOD...

where g′(w) = (g′i(wi)). It follows that

2λw + g′(w) = 2λw0, (16)

i.e.,

2λwi + g′(wi, w0i ) = 2λw0

i , i = 1, · · · , n. (17)

If the matrix G is not a diagonal one, then a nonlinear matrix equation willreplace (17).

We now establish the relation between the variational method and the waveletshrinkage. A general linear wavelet shrinkage is formulated as

ΓDS(wi, δi) = (δiwi)ni=1, δi ∈ [0, 1].

We show that if in F (u) =∑

gi(u), each density gi is linear on [0,∞), then thesolution of the variational problem (15) leads to a linear shrinkage.

Theorem 3 If a variational problem (15) satisfies

gi(s) = µi|s|, µi ≥ 0 (18)

then it yields a linear wavelet shrinkage.

Proof. Without loss of generality, we can assume the balance parameter λin Γ(u) is 1/2. Otherwise, we use λµi/2 to replace µi. We have

ρ′i(s) = µi sgn(s),

which leads to the Euler-Lagrangian equation

s+ µi sgn(s) = w0i , i = 1, · · · , n. (19)

Since ρ′i(0) does not exist, 0 is also a critical point. When |w0i | < µi, the equation

(19) has no solution; and when |w0i | ≥ µi, the equation has the unique solution

s = sgn(w0i )

(∣

∣w0i

∣

∣ − µi

)

. (20)

Then it is easy to verify that the vector w = (wi) with

wi = sgn(w0i )

(∣

∣w0i

∣

∣ − µi

)

+=

(∣

∣w0i

∣

∣ − µi

)

+

|w0i |

w0i , i = 1, · · · , n, (21)

minimizes Γ(u), where x+ = max(x, 0). Let δi =(|w0

i |−µi)+

|w0i |

. Then δi ∈ [0, 1].

The theorem is proven. .As applications of Theorem 3, we discuss the wavelet shrinkages by using

hard threshold and soft threshold respectively.Example 1. [Hard thresholding] Let δ be a given threshold. We assume

in w0 each sample value whose absolute value is less than δ represents noise.

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Otherwise it does not carry on noise. The assumption suggests the followingweight:

ci(s) =

δ/ |s| ,∣

∣w0i

∣

∣ ≤ δ,0,

∣

∣w0i

∣

∣ > δ,(22)

which leads

ρi(s) =

δ |s| ,∣

∣w0i

∣

∣ ≤ δ,0,

∣

∣w0i

∣

∣ > δ.

and

ρ′i(s) =

δ sgn |s| ,∣

∣w0i

∣

∣ ≤ δ,0,

∣

∣w0i

∣

∣ > δ.

By (20), the solution w of the variational problem (15) is

wi =

(1 − 12λ )+w

0i ,

∣

∣w0i

∣

∣ ≤ δ,w0

i ,∣

∣w0i

∣

∣ > δ.(23)

Hence, when λ ≤ 12 , the solution provides the wavelet shrinkage using the hard

threshold δ.Example 2. [Soft thresholding] We now assume in w0 each sample value

carries on noise∼ N(0, σ2).. Hence, we adopt the following weight

ci(s) = δ/ |s| ,

which leads ρi(s) = δ|s| and therefore

ρ′i(s) = δ sgn(s).

By (21), the solution w of the variational problem (15) is

wi = sgn(w0i )

(

∣

∣w0i

∣

∣ −1

2λδ

)

+

,

which provides the wavelet shrinkage using the soft threshold δ/(2λ).

4 Examples

In this section, we give several examples to illustrate the function of λ in thewavelet regression. We select four functions to test the regression: “Blocks”isa step function, which has several step jumps. According to [20] and [21], Lip-schitz order of a step jump is 0. “Heavy Sine”is a broken sine wave, which hastwo jumps. “Bumps”is a continuous function but has a lot of non-differentiablepoints, which are classified to be with Lipschitz orders in (0, 1). “Doppler”issmooth everywhere except at the origin, where the function has infinite oscilla-tions. The white noise is added to the sample data of them. Let σ(f) and σ(n)

495VARIATIONAL METHOD...

denote the standard deviation of the function and the noise respectively. In

Figure 5 – Figure 8, we set σ(f)σ(n) = 5. In the regression, we set δ =

√

2σ(n) lnN,

where N is the size of the data. We also set λ = 2, 1, 1/2, 1/4, 1/10, which yieldthe thresholds δ/2, δ, 2δ, 4δ, 10δ respectively. In Figure 9, we choose λ = 2,which yield the threshold 4δ. The results show that the regression is not verysensitive to small λ, say λ ≤ 1/2, but sensitive to λ > 1. To regress the func-tions with singularity as in “Doppler”, a larger λ should be chosen. Other testsshow that the regression does not very much rely on the choices of particularwavelets. ( The figures are not included in this paper.) Using different waveletsin a regression causes little differences, provided that the size of sampling datais large (say ≥ 210 when wavelet level is ≤ 4).

When the sampling data carry on the noise with uniform distribution, thewavelet regression still works well. Figure 10 – Figure 13 show the waveletregression of the four functions with uniform noise. Again, the estimator of“Doppler”produces a larger error.

Acknowledgment. The author thanks the referees for their valuable com-ments.

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20Blocks

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10

20λ=1

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0

10

20λ=1/2

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0

10

20λ=1/4

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0

10

20λ=1/10

Figure 5. Wavelet regression for Blocks.

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−10

0

10Heavy Sine

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−10

0

10SNR=5

0 500 1000 1500−20

−10

0

10λ=1

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−10

0

10λ=1/2

0 500 1000 1500−20

−10

0

10λ=1/4

0 500 1000 1500−20

−10

0

10λ=1/10

Figure 6. Wavelet regression for Heavy Sine.

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5

10

15

20Bumps

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0

10

20SNR=5

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0

10

20λ=1

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0

10

20λ=1/2

0 500 1000 1500−10

0

10

20λ=1/4

0 500 1000 1500−10

0

10

20λ=1/10

Figure 7. Wavelet regression for Bumps.

497VARIATIONAL METHOD...

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−5

0

5

10Doppler

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−10

0

10

20SNR=5

0 500 1000 1500−10

−5

0

5

10λ=1

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−5

0

5

10λ=1/2

0 500 1000 1500−10

−5

0

5

10λ=1/4

0 500 1000 1500−10

−5

0

5

10λ=1/10

Figure 8. Wavelet regression for Doppler.

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−5

0

5

10

15

20Blocks

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0

5

10

15

20Bumps

0 500 1000 1500−15

−10

−5

0

5

10Heavy Sine

0 500 1000 1500−10

−5

0

5

10Doppler

Figure 9. Wavelet regression for λ = 2, with σ(f)σ(n) = 5.

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0 500 1000 1500−5

0

5

10Blocks

0 500 1000 1500−5

0

5

10SNR=5

0 500 1000 1500−5

0

5

10λ=1

0 500 1000 1500−5

0

5

10λ=1/2

0 500 1000 1500−5

0

5

10λ=1/4

0 500 1000 1500−5

0

5

10λ=0.1

Figure 10. Wavelet regression of Block with uniform noise.

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−5

0

5Heavy Sine

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−5

0

5SNR=5

0 500 1000 1500−10

−5

0

5λ=1

0 500 1000 1500−10

−5

0

5λ=1/2

0 500 1000 1500−10

−5

0

5λ=1/4

0 500 1000 1500−10

−5

0

5λ=0.1

Figure 11. Wavelet regression of Heavy Sine with uniform noise.

499VARIATIONAL METHOD...

0 500 1000 15000

2

4

6Bumps

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0

2

4

6SNR=5

0 500 1000 1500−2

0

2

4

6λ=1

0 500 1000 1500−2

0

2

4

6λ=1/2

0 500 1000 1500−2

0

2

4

6λ=1/4

0 500 1000 1500−2

0

2

4

6λ=0.1

Figure 12. Wavelet regression of Bumps with uniform noise.

0 500 1000 1500−0.5

0

0.5Doppler

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−0.5

0

0.5

1SNR=5

0 500 1000 1500−1

−0.5

0

0.5

1λ=1

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−0.5

0

0.5

1λ=1/2

0 500 1000 1500−1

−0.5

0

0.5

1λ=1/4

0 500 1000 1500−1

−0.5

0

0.5

1λ=0.1

Figure 13. Wavelet regression of Doppler with uniform noise.

Acknowledgments: This work was supported in part by ER-Grant 2003 ofSam Houston State University.

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References

[1] Abramovich, F., Sapatinas, F., and Silverman, B.W, “Wavelet thresholdingvia a Bayesian approach”, J. Roy. Statist. Soc. B, 60, pp 725-749, 1998.

[2] Abramovich, F. and Benjamini, Y., “Thresholding of wavelet coefficients

as multiple hypotheses testing procedure”, Wavelets and Statistics, (Eds.Antoniadis A. & Oppenheim G.), New York: SpringerVerlag, pp 5-14, 1995.

[3] Aubert, G. and Kornprobst, P., “Mathematical Problems in ImageProcessing–Partial Differential Equations and the Calculus of Variations”,Springer, 2001.

[4] Aubert, G. and Vese, L., “A variational method in image recovery,”SIAMJournal of Numerical Analysis, 34(5): pp 1948-1979, 1997.

[5] Bock, M.E. and Pliego, G., “Estimating functions with wavelets part II:Using a Daubechies wavelet in nonparametric regression”, Statistical Com-puting and Statistical Graphics Newsletter, 3 (2): pp 27-34, November1992.

[6] Chan, A.K. and Goswami, J.C., “Fundamentals of wavelets: theory, algo-rithms, and application”, John Wiley & Sons, Inc, 1999.

[7] Chipman, H., Kolacyzk, E.D., and McCulloch, R.E., “Adaptive Bayesianwavelet shrinkage”, J. Amer. Statist. Ass., 92, pp 1413-1421. 1997.

[8] Chui, C.K., “An Introduction to Wavelets”, Academic Press, Boston, 1992.

[9] Chui, C.K. and Wang, J.Z., “A study of asymptotically optimal time-frequency localization by scaling functions and wavelets”, Annals of Nu-merical Mathematics 4, pp 193-216, 1997.

[10] Cohe, A., Daubechies I., Jawerth, B., and Vial, P., “Multiresolution analy-sis, wavelets, and fast algorithms on an interval”, Comptes Rendus AcadSci. Paris A 316, pp 417-421, 1993.

[11] Daubechies, I., “Orthonormal bases of compactly supported wavelets”,Comms. PureAppl. Math., 41, pp 909-996, 1988.

[12] Daubechies, I., “Ten Lectures on Wavelets”, Society for Industrial and Ap-plied Mathematics, Philadelphia, PA, 1992.

[13] Donoho, D.L. and Johnstone, I.M., “Ideal spatial adaptation by waveletshrinkage”, Biometrika, 81, pp 425-455, 1994.

[14] Donoho, D.L. and Johnstone, I.M., “Adapting to unknown smoothness viawavelet shrinkage”, J. Amer. Statist. Assoc., 90, pp 1200-1224, 1995.

[15] Donoho, D.L. and Johnstone, I.M., Kerkyacharian, G. and Picard, D.,“Wavelet shrinkage: Asymptopia?”, Journal of the Royal Statistical Soci-ety, Series B 57, pp 301-369, 1995.

501VARIATIONAL METHOD...

[16] Donoho, D.L. and Johnstone, I.M., “Minimax estimation via waveletshrinkage”, 1998.

[17] Eubank, R.L., “Spline Smoothing and Nonparametric Regression”, MarcelDekker, 1999.

[18] Hall, P., McKay, I., and Turlach, B.A., “Performance of wavelet methodsfor functions with many discontinuities”, Annals of Statistics, 24(6), pp2462-2476, 1996.

[19] Krishnaiah, P.R. and Miao, B.Q., “Review about estimation of change-points”, Handbook of Statistics, 7. P.R.Krishnaiah and C.R. Rao, eds., pp375-402, Amsterdam:Elsevier, 1988.

[20] Mallat, S. and Hwang, W.L., “Singularity detection and processing withwavelets”, IEEE Trans. Information Theory, 38, pp 617-643, 1992.

[21] Mallat, S. and Zhong, Z., “Characterization of signals from multiscaleedges”, IEEE Trans. Pattern Anal. and Machine Intel., 14 (7), pp 710-732, 1992.

[22] Mallat, S., “Multiresolution approximations and wavelet orthonormal basesof L2(R)”, Transactions of the American Mathematical Society, 315(1),1989.

[23] Mallat, S., “A theory of multiresolution signal decomposition: the waveletrepresentation”, IEEE Trans. Pattern Anal. Machine Intell. 11, pp 674-693, 1989.

[24] Meyer, Y., “Principe d’incertitude, bases hilbertiennes et algebresd’operateurs”, Bourbaki seminar, 662, 1985-1986.

[25] Meyer, Y., “Ondelettes, functions splines at analyses graduees”, RapportCeremade, 8703, 1987.

[26] Meyer, Y., “Wavelets: Algorithms and Applications”, Philadelphia: SIAM,1993.

[27] Misiti, M., Misiti, Y., Oppenheim, G. and Poggi, J.M., “Wavelet toolbox:

for use with MATLAB”, The MathWorks Inc., 1996.

[28] Mumford, D. and Shah, J., “Optimal approximations by piesewise smoothfunctions and associated variational problems”, Comm. on Pure and Appl.Math., XLII no.5 (1989).

[29] Nason, G.P., “Wavelet shrinkage using cross-validation”, J. Roy. Statist.Soc. B, 58, pp 463-479, 1996.

[30] Nason, G.P. and Silverman, B.W., “The discrete wavelet transform in S”,J. Comput. Graph. Statist., 3, pp 163-191, 1994.

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[31] Ogden, R.T. and Parzen, E., “Change-point approach to data analyticwavelet thresholding”, Statistics and Computing 6, pp 93-99, 1996.

[32] Ogden, R.T. and Parzen, E., “Data dependent wavelet thresholding innonparametric regression with change-point applications”, ComputationalStatistics and Data Analysis 22, pp 53-70, 1996.

[33] Ogden, R.T., “Essential wavelets for statistical applications and data analy-sis”, Birkhauser, Boston, 1997.

[34] Perona, P. and Malik, J., “Scale-space and edge detection using anisotropicdiffusion”, IEEE PAML, 12 (1990) pp. 629-639.

[35] Schimek, M.G., “Smoothing and Regression: Approaches, Computation,and Application”, Wiley, 2000.

[36] Vidakovic, B., “Statistical modeling by wavelets”, Wiley Series in Proba-bility and Statistics, 1999.

[37] Wang, Y.Z., “Jump and sharp cusp detection by wavelets”, Biometrika, 82

(2), pp 385-97, 1995.

[38] You, Y., Xu, W., Tannenbaum, A., and Kaveh, M., “Behavioral analysis ofanisotropic diffusion in image processing”, IEEE Trans. Image Processing,5 (1996) pp 1539-1553.

503VARIATIONAL METHOD...

504

Wavelet Applications in Cancer Study

Don Hong

Department of Mathematical Sciences

Middle Tennessee State University

Murfreesboro, Tennessee

and

Yu Shyr

Department of Biostatistics

Vanderbilt University

Nashville, Tennessee

Abstract

In this survey article, we review wavelet methodology in many biosta-

tistical applications of cancer study: microcalcifications in digital mam-

mography, functional MRI data analysis, genome sequence, protein struc-

ture and microarray data analysis, and MALDI-TOF MS data analysis.

AMS 2000 Classifications: 42C40, 65T60, 92C55.Key Words: Splines, statistics, mass spetrometry data processing, medical

image compression, wavelets.

1 Introduction

Wavelet theory has developed now into a methodology used in many disciplines:mathematics, physics, engineering, signal processing and image compression,numerical analysis, and statistics, to list a few. Wavelets are providing a richsource of useful tools for applications in time-scale types of problems. Waveletsbased methods are developed in statistics in areas such as regression, densityand function estimation, modeling and forecasting in time series analysis, andspatial analysis. The attention of wavelets was attracted by statisticians whenMallat (see [35] and [36]) established a connection between wavelets and sig-nal processing and Donoho and Johnstone (see [11] – [13]) found that waveletthresholding has desirable statistical optimality properties. Since then, waveletshave proved very useful in nonparametric statistics, and time series analysis.Wavelets based Bayesian concepts and modeling approaches have wide applica-tions in data denoising. Wang [51] in this special issue presents a brief reviewof wavelet shrinkage and the application of the variational method for waveletregression and shows the relation between the variational method and wavelet

505JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,505-521,2006,COPYRIGHT 2006 EUDOXUS PRESS,LLC

shrinkage. In recently years, wavelets have been applied to a large variety ofbiomedical signals (see [1], [33] for example) and there is a growing interest inusing wavelets in the analysis of sequence and functional genomics data. It isimportant to point out that wavelets are very useful, but of course are not, andwill never be a panacea. However, in many statistical data analyses, waveletmethods can be tried and tested in practice. In the following, we first give anintroduction to wavelets in a relatively accessible format, then briefly mentiontechniques of wavelets in medical imaging and signal processing such as in de-tection of microcalcifications in digital mammography and functional MagneticResonance Imaging (fMRI), and biostatistical applications in molecular biol-ogy areas such as in genome sequence analysis, protein structure investigation,and gene expression data analysis. Finally, we present matrix-assisted laserdesorption-ionization time-of-flight mass spectrometry (MALDI-TOF MS) dataanalysis using wavelet-based methods.

2 Wavelets

In the following, we review some essential concepts of wavelets and describe themethod of wavelet analysis for statistical data analysis. Roughly speaking, awavelet is a little wavy function with certain useful properties. A set of wavelets,usually is constructed through a single “mother” wavelet, to form a basis of L2

space and provide “building blocks” that can be used to describe any functionin L2. A simple example of wavelets is the Haar function and Haar wavelets[17]:

Haar began with the function

h(x) =

1, x ∈ [0, 1/2);

−1, x ∈ [1/2, 1);

0, otherwise.

For n ≥ 1, let n = 2j + k, j ≥ 0, 0 ≤ k < 2j and define

hn(x) = 2j/2h(2jx− k) = ψj,k(x).

It is easy to see that

supp (hn) = [k

2j,k + 1

2j].

Let

h0(x) =

1, x ∈ [0, 1)

0, otherwise.

Then the Haar waveletsh0, h1, · · · , hn, · · ·

form an orthonormal basis for L2[0, 1].

506 HONG,SHYR

The uniform approximation of f by

Sn(f) = 〈f, h0〉h0 + · · · + 〈f, hn〉hn

is nothing but the classical approximation of a continuous function by stepfunctions. Clearly, Haar wavelets are not continuous. However, Haar waveletsallow information to be encoded according to “levels of detail.”

In general, wavelets construction starts with a function equation:

φ(·) =2N−1∑

k=0

pkφ(2 · −k) (2.1)

The equation (2.1) is called the two-scale equation and the function φ is called ascaling function (also referred to as a “father” wavelet). Under some restrictions(multiresolution analysis conditions), a “mother” wavelet ψ can be obtained byusing the scaling function:

ψ(x) =

2N−1∑

k=0

(−1)kp1−kφ(2x− k). (2.2)

In the late 1980s, Daubechies constructed a family of wavelets, now calledDaubechies wavelets. The easiest example is

φD2(x) =3

∑

k=0

pkψ(2x− k),

where p0 = 1+√

3

4, p1 = 3−

√3

4, p2 = 3+

√3

4, and p3 = 1−

√3

4. Correspondingly,

the Daubechies wavelet ψD2 is the solution of the equation (2.1) determinedby these coefficients (see [9] for details and more orthonormal wavelets). FromHaar wavelets to continuous wavelets took about 80 years!

A wavelet must be localized in time, in the sense that ψ(t) → 0 quickly as|t| → ∞. The wavelet should be also oscillating about zero so that

∫ ∞

−∞ ψ(t)dt =

0 and even the first m moments are also zero:∫ ∞

−∞tkψ(t)dt = 0 for k =

0, 1, · · · ,m− 1. The oscillation property makes the function a wave. Togetherthe localized feature, the function ψ becomes a wavelet. Semi-orthonormalspline wavelets, often called Chui-Wang wavelets were introduced by CharlesK. Chui and Jianzhong Wang (See [6] for details). These wavelets trade thetranslation orthogonality with an explicit expression of the wavelets in termsof B-spline functions. These wavelets are compactly supported. They are veryeasy to compute and to work with.

In some applications, we need to use wavelets with symmetry, orthogonality,compact support, and high approximation order, simultaneously. This is impos-sible to achieve according to the above construction scheme. When the coeffi-cients in the scaling equation (2.1) and wavelet equation (2.2) are matrices andφ and ψ are vectors, then there are two or more scaling functions and an equalnumber of wavelets. We call them multi-scaling functions and multiwavelets,

507WAVELET APPLICATIONS...

respectively. These multiwavelets open new possibilities for wavelets to possessthose properties simultaneously. In [24], a set of orthogonal multiwavelets ofmultiplicity four was constructed. More recent discussion on multiwavelets canbe found in [28] and [18] and the references therein.

The continuous wavelet transform (CWT) is a linear signal transformationthat uses templates ψa,b = a1/2ψ( t−ba ) which are shifted (index b) and dilated(index a) of a given wavelet function using the mother wavelet function ψ(x).The wavelet transform of a signal f can be written as

Tψf(a, b) = 〈f, ψa,b〉.

The CWT maps a one-dimensional signal to a two-dimensional time-scale jointrepresentation. The resulting wavelet coefficients are highly redundant.

A basic requirement is that the transform is reversible, i.e., the signal fcan be reconstructed from its wavelet coefficients. The distinction between thevarious types of wavelet transforms depends on the way in which the dilationand shift parameters are discretized. This is more desirable in molecular biologyand genetics.

A discrete wavelet transform (DWT) decomposes a signal into several vectorsof wavelet coefficients. Different coefficient vectors contain information aboutthe signal function at different scales. Coefficients at coarse scale capture grossand global features of the signal while coefficients at fine scale contain detailedinformation. In practice, it is often more convenient to consider the wavelettransform for some discretized values of a = 2j (the dyadic scales) and b = k (theinteger shifts). The transform is reversible if and only if the wavelet coefficientsdefine a wavelet frame A‖f‖2 ≤

∑

a,b |〈f, ψa,b〉|2 ≤ B‖f‖2, where A and B are

called bounds of wavelet frame. In particular, if A = B = 1, the set of templatesdefines an orthogonal wavelet basis. The important point is that the waveletdecomposition provides a one-to-one representation of the signal in terms of itswavelet coefficients. This makes a wavelet basis well suited for any tasks forwhich block transforms (Fourier, Discrete Cosine Transform, etc.) have beenused traditionally.

In statistical practice, the discrete wavelet transform plays a role analogousto the fast Fourier transform in conventional frequency-domain analysis. Unliketheir Fourier cousins, wavelet methods make no assumptions concerning peri-odicity of the data at hand. As a result, wavelets are particularly suitable forstudying data exhibiting sharp changes or even discontinuities.

Recognize that the wavelet transform on a finite sequence of data pointsprovides a linear mapping to the wavelet coefficients: wn = Wfn, where thematrix W = Wn×n is orthogonal and wn and fn are n-dimensional vectors. Thewavelet approximation to a signal function f is built up over multiple scales andmany localized positions. For the given family

φJ,k(t) = 2−J/2φ(t

2j− k), ψj,k = 2−j/2ψ(

t

2j− k), j = 1, 2, · · · , J.

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The coefficients are given by the projections:

sJ,k =

∫

f(t)φJ,k(t)dt, dj,k =

∫

f(t)ψj,k(t)dt

so that

f(t) =∑

k

sJ,kφJ,k(t) +∑

k

J∑

j=1

dj,kψj,k(t).

The large J refers to the relatively small number of coefficients for the lowfrequency, smooth variation of f , the small j refers to the high frequency detailcoefficients.

When the sample size n, the number of observations, is divisible by 2, sayn = 2J , then the number of coefficients, n can be grouped as, n/2 coefficientsd1,k at the finest level, n/4 coefficients d2,k at the next finest level, · · · , n/2J

coefficients dJ,k and n/2J coefficients sJ,k at the coarsest level. In the following,we demonstrate this by using Haar wavelet, the simplest and crudest memberof a large class of wavelet functions.

We describe a scheme for transforming large arrays of numbers into arraysthat can be stored and transmitted more efficiently. For this purpose, we de-scribe a method for transforming stings of data using Haar wavelets, calledaveraging and differencing. For instance, let us start with a row vector

(2854, 6334, 3226, 2277, 2030, 1323, 367, 2890).

Table 1 shows the results of three steps in the transform process respectively.The first row in the table is the original data string, which consists of four pairsof numbers. The first four numbers in the second row are the averages of thesepairs. Similarly, the first two numbers in the third row are the averages of thosefour averages, taken two at a time, and the first entry in the fourth and last rowis the average of the preceding two computed averages. The remaining numbersmeasure deviations from the various averages. For example, the first four entriesin the second half of the second row are the result of subtracting the first fouraverages from the first elements of the pairs that generate them: 2854− 4594 =−1740, 3226− 2751.5 = 474.5, 2030− 1676.5 = 353.5, 367 − 1628.5 = −1261.5.These are called detail coefficients, they are repeated in each subsequent row ofthe table. The following matrix representation illustrates this three-step processclearly and precisely.

The first step involved averaging and differencing four pairs of numbers andit can be expressed as the matrix equation

(4594, 2751.5, 1676.5, 1628.5,−1740, 474.5, 353.5,−1261.5)

= (2854, 6334, 3226, 2277, 2030, 1323, 367, 2890)H1,

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2854 6334 3226 2277 2030 1323 367 2890

4594 2751.5 1676.5 1628.5 -1740 474.5 353.5 -1261.5

3672.8 1652.5 921.3 24 -1740 474.5 353.5 -1261.5

2662.6 1010.1 921.3 24 -1740 474.5 353.5 -1261.5

Table 1: Results of a vector after three steps of Haar wavelet transformation.

where H1 is the matrix

1/2 0 0 0 1/2 0 0 01/2 0 0 0 −/2 0 0 00 1/2 0 0 0 1/2 0 00 1/2 0 0 0 −1/2 0 00 0 1/2 0 0 0 1/2 00 0 1/2 0 0 0 −1/2 00 0 0 1/2 0 0 0 1/20 0 0 1/2 0 0 0 −1/2

.

The transitions of the second step and the third step are equivalent to the matrixequations

(3672.8, 1652.5, 921.3, 24,−1740, 474.5, 353.5,−1261.5)

= (4594, 2751.5, 1676.5, 1628.5,−1740, 474.5, 353.5,−1261.5)H2

and

(2662.6, 1010.1, 921.3, 24,−1740, 474.5, 353.5,−1261.5)

= (3672.8, 1652.5, 921.3, 24,−1740, 474.5, 353.5,−1261.5)H3,

where H2 and H3, respectively are the matrices

1/2 0 1/2 0 0 0 0 01/2 0 −/2 0 0 0 0 00 1/2 0 1/2 0 0 0 00 1/2 0 −1/2 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

,

1/2 1/2 0 0 0 0 0 01/2 −/2 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

.

Thus, the combined effect of the Haar wavelet transform can be achievedwith the single equation:

(2662.6, 1010.1, 921.3, 24,−1740, 474.5, 353.5,−1261.5)

= (2854, 6334, 3226, 2277, 2030, 1323, 367, 2890)W,

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where W = H1H2H3. Noticing that the columns of Hi’s are orthogonal, eachof these matrices are invertible. The inverses simply reverse the three averagingand differencing steps. In any case, we can recover the original string from thetransformed version by means of the inverse Haar wavelet transform:

(2854, 6334, 3226, 2277, 2030, 1323, 367, 2890)

= (2662.6, 1010.1, 921.3, 24,−1740, 474.5, 353.5,−1261.5)W−1,

where W−1 = H−1

3 H−1

2 H−1

1 . It becomes a routine matter to construct thecorresponding 2n × 2n matrices Hi, i = 1, · · · , n needed to work with stringsof length 2n. In general, there is no loss of generality in assuming that eachstring’s length is a power of 2.

In image compression, not only the row transformations but also columntransformations are applied to image matrices (see [16] and [22] for example).

Applications of wavelets in statistics can be found in [3], [39], [45], and thereferences therein. In the following, we only emphasize to review biostatisticalwavelet applications of medical science, particularly in cancer study.

3 Wavelet-Based Detection of Microcalcifications

in Digital Mammography

There are many applications of wavelets in medical image analysis. For example,quantitative neuroimaging by position emission tomography (PET) is a majorresearch tool to investigate functional brain activity in vivo dynamic processesoccurring in discrete brain region. The main difficulty with the analysis ofPET data is the absence of reliable anatomical markers and the fact that theshape and size of the brain may very substantially from one subject to another.In [49], the authors proposed an image-based statistical method for detectingdifferences between subject groups. The two important processing steps in theirmethodology are (i) the registration of the individual brain images, and (ii) thesubsequent statistical analysis in order to detect differences in functional activitybetween subject groups. Wavelet transform plays an important role in each step,but for different reasons. In [30], wavelet compression technique is used for thelow-dose chest CT data analysis. Wavelet image decomposition for the detectionof myocardial viability in the early postinfarction period with the wavelet-basedmethod for texture characterization is discussed in [40]. Ultrasonic liver tissuesclassification by fractal feature vector based on M-band wavelet transform canbe found in [31].

Breast cancer is the main cause of death for women between the ages of 35and 55. It has been shown that early detection and treatment of breast cancerare the most effective methods of reducing mortality. The detection of micro-calcifications plays a very important role in breast cancer detection becausethese microcalcifications can be the only mammographic sign of nonpalpablebreast disease. Screen-film mammography is widely recognized as being theonly effective imaging modality for early detection of breast cancer. Though

511WAVELET APPLICATIONS...

advances in screen-film mammographical technology have resulted in significantimprovements in image resolution and film contrast, images provided by screen-film mammography remain very difficult to interpret. Mammograms are of lowcontrast, and features in mammograms indicative of breast disease, such as themicrocalcifications, are often very small. The theory of wavelets provides a com-mon framework for numerous techniques developed independently for varioussignal and image processing applications. For example, multiresolution imageprocessing, subband coding, and wavelet series expansions are the different viewsof this theory. In [52], an approach is presented for detecting microcalcificationsin digital mammograms employing wavelet-based subband image decomposition.The authors proposed a system for microcalcification detection under the hy-pothesis that the microcalcifications present in mammograms can be preservedunder a transform which can localize the signal characteristics in the originaland the transform domain. Therefore, microcalcifications can be extracted frommammograms by suppressing the subband of the wavelets-decomposed imagethat carries the lowest frequencies and contains smooth background information,before the reconstruction of the image. Multiresolution-based segmentation forthe detection and enhancement of the microcalcifications is also presented in[42]. Very recently, the authors in [26] present the implementation and evalu-ation of a novel enhancement technique for improved interpretation of digitalmammography with wavelet processing. A new algorithm for enhancement ofmicrocalcifications in mammograms is given in [20] and highly regular waveletsare used for the detection of clustered microcalcifications in mammograms in[32]. More wavelet applications of medical imaging can be found in [48].

4 Wavelet-Based Functional MRI Data Analysis

Functional Magnetic Resonance Imaging (fMRI) is a technique for determiningwhich part of the brain are activated by different types of physical sensation oractivity, such as light, sound or the movement of a subject’s fingers. This “brainmapping” is achieved by setting up an advanced MRI scanner in a special wayso that the increased blood flow to the activated areas of the brain shows up onfunctional MRI scans. Functional MRI is based on the increase in blood flow tothe local vasculature that accompanies neural activity in the brain. This resultsin a corresponding local reduction in deoxyhemoglobin because the increase inblood flow occurs without an increase of similar magnitude in oxygen extraction.Since deoxyhemoglobin is paramagnetic, it alters the T2* weighted magneticresonance image signal. Thus, deoxyhemoglobin is sometimes referred to as anendogenous contrast enhancing agent, and serves as the source of the signalfor fMRI. Using an appropriate imaging sequence, human cortical functionscan be observed without the use of exogenous contrast enhancing agents on aclinical strength (1.5 T) scanner. Functional activity of the brain determinedfrom the magnetic resonance signal has confirmed known anatomically distinctprocessing areas in the visual cortex, the motor cortex, and Broca’s area ofspeech and language-related activities. Further, a rapidly emerging body of

512 HONG,SHYR

literature documents corresponding findings between fMRI and conventionalelectrophysiological techniques to localize specific functions of the human brain.Consequently, the number of medical and research centers with fMRI capabilitiesand investigational programs continues to escalate.

The main advantages to fMRI as a technique to image brain activity re-lated to a specific task or sensory process include 1) the signal does not requireinjections of radioactive isotopes, 2) the total scan time required can be veryshort, i.e., on the order of 1.5 to 2.0 min per run (depending on the para-digm), and 3) the in-plane resolution of the functional image is generally about1.5 × 1.5 mm although resolutions less than 1 mm are possible. To put theseadvantages in perspective, functional images obtained by the earlier method ofpositron emission tomography, PET, require injections of radioactive isotopes,multiple acquisitions, and, therefore, extended imaging times. Further, the ex-pected resolution of PET images is much larger than the usual fMRI pixel size.Additionally, PET usually requires that multiple individual brain images arecombined in order to obtain a reliable signal. Consequently, information on asingle patient is compromised and limited to a finite number of imaging ses-sions. Although these limitations may serve many neuroscience applications,they are not optimally suitable to assist in a neurosurgical or treatment planfor a specific individual.

Wavelet transformation is a time-frequency representation which decomposesthe signal according to a set of functions through translation and dilation op-erations. Such a time-frequency representation could provide a more efficientsolution than the usual Fourier-transform or other methods presently availableto process MRI data. One of the first application of wavelet transform in med-ical imaging was for noise reduction in MRI [53]. The approach proposed therewas to compute an orthogonal wavelet decomposition of the image and applythe following soft thresholding rule on the wavelet coefficients ci,k = 〈f, ψi,k〉:

˜ci,k =

ci,k − ti, ci,k ≥ ti,

0, |ci,k| ≤ ti,

ci,k + ti, ci,k ≤ −ti,

where ti is a threshold that depends on the noise level at the ith scale. Theimage is then reconstructed by the inverse wavelet transform of the ˜ci,k’s.

Ruttimann et al. [41] have proposed to use the wavelet transform for thedetection and localization of activation patterns in fMRI. The idea is to ap-ply a statistical test in the wavelet domain to detect the coefficients that aresignificantly different from zero. In [14], the authors improved the method byreplacing the original z-test by a t-test that takes into account the variability ofeach wavelet coefficient separately. A key issue is to find out which wavelet andwhich type of decomposition is best suited for the detection of a given activationpattern. Various brands of fractional spline wavelets are applied and an exten-sive series of tests using simulated data are performed. An interesting practicalfinding is that performance is strongly correlated with the number of coefficientsdetected in the wavelet domain, at least in the orthonormal and B-spline cases.

513WAVELET APPLICATIONS...

Therefore, it is possible to optimize the structural wavelet parameters simplyby maximizing the number of wavelet counts, without any prior knowledge ofthe activation pattern.

A new method is proposed in [25] for activation detection in event-relatedfMRI. The method is based on the analysis of selected resolution levels in trans-lation invariant wavelet transform domain. The power in different resolutionlevels in wavelet domain is analyzed and an optimal set of resolution levels isselected. A randomization-based statistical test is then applied in wavelet do-main for activation detection. The problem of detecting significant changes infMRI time series that are correlated to a stimulus time course is addressed in[37]. The fMRI signal is described as the sum of two effects: a smooth trendand the response to the stimulus. The trend belongs to a subspace spanned bylarge scale wavelets. The wavelet transform provides an approximation to theKarhunen-Loeve transform for the long memory noise. A scale space regressionis developed that permits carrying out the regression in the wavelet domainwhile omitting the scales that are contaminated by the trend.

In [43], two iterative procedures obtained from the application of wavelettransform to noise-free magnetic resonance spectroscopy (MRS) signals com-posed of one and two resonances, respectively, are tested on simulated and realbiomedical MRS data.

5 Wavelet-Based Molecular Biology Data Analy-

sis

In this section, we briefly review the most interesting applications of wavelets ingenome sequence analysis, protein structure investigation, and gene expressiondata analysis.

The Human Genome Project (HGP) was one of the great feats of explorationin history, an inward voyage of discovery rather than an outward exploration ofthe planet or the cosmos. An international research effort to sequence and mapall of the genes - together known as the genome - of members of our species,Homo sapiens, the HGP was completed in April 2003. Now we can, for the firsttime, read nature’s complete genetic blueprint for building a human being.

Integral to the HGP are similar efforts to understand the genome of variousorganisms commonly used in biomedical research, such as mice, fruit flies androundworms. Such organisms are called ”model organisms,” because they canoften serve as research models for how the human organism behaves.

In order to fully understand these genomes, the HGP also develops technolo-gies for genomic analysis, and trains scientists who will be able to use the toolsand resources created by the HGP to perform research that will improve humanhealth. Recognizing the profound importance and seriousness of this venture,another mission of the HGP - and NHGRI - is to examine the ethical, legal andsocial implications of human genetic research.

Wavelets can be useful in detecting patterns in DNA sequences. In [34], it

514 HONG,SHYR

was shown that wavelet variance decomposition of bacterial genome sequencescan reveal the location of pathogenicity islands. The findings show that waveletsmoothing and scalogram are powerful tools to detect differences within andbetween genomes and to separate small (gene level) and large (putative patho-genicity islands) genomic regions that have different composition characteristics.An optimization procedure improving upon traditional Fourier analysis perfor-mance in distinguishing coding from noncoding regions in DNA sequences wasintroduced in [2]. The approach can be taken one step further by applyingwavelet transforms.

Proteins are macromolecules (heteropolymers) made up from 20 differentL-α-amino acids, also referred to as residues. A certain number of residuesis necessary to perform a particular biochemical function, and around 40-50residues appears to be the lower limit for a functional domain size. Proteinsizes range from this lower limit to several hundred residues in multi-functionalproteins. Very large aggregates can be formed from protein subunits, for ex-ample many thousand actin molecules assemble into a an actin filament. Largeprotein complexes with RNA are found in the ribosome particles, which are infact ’ribozymes’.

To find the similarities between two or more protein sequences is of greatimportance for protein sequence analysis. In [10], a comparison method basedon wavelet decomposition of protein sequences and a cross-correlation study wasdevised that is capable of analyzing a protein sequence “hierarchically”, i.e., itcan examine a protein sequence at different spatial resolutions. A sequence-scale similarity vector is generated for the comparison of two sequences feasibleat different spatial resolutions (scales).

In the process of protein construction, buried hydrophobic residues tend toassemble in a core of a protein. Methods used to predict these cores involve useor no use of sequential alignment. In the case of a close homology, predictionwas more accurate if sequential alignment was used. If the homology was weak,prediction would be unreliable. A hydrophobicity plot involving the hydropathyindex is useful for purposes of prediction. In [21], wavelet analysis is used forhydrophobicity plots to quantitatively decompose to high and low frequencies.The prediction of hydrophobic cores of proteins was made with low frequencyextracted from the hydrophobicity plot.

The cosine Fourier series and discrete wavelet transforms are applied in [38]for describing replacement rate variation in genes and proteins, in which theprofile of relative replacement rates along the length of a given sequence isdefined as a function of the site number. The new models are applicable totesting biological hypotheses such as the statistical identity of rate variationprofiles among homologous protein families.

Microarray technology allows us to analyze the expression pattern of hun-dreds of genes. The use of statistics is the key to extracting useful informationfrom this technology. Recently, “False Discovery Rate” (FDR) becomes a veryuseful inferential approach in the analysis of microarray data [47]. The FDR isthe expected proportion of rejected hypotheses that are falsely rejected. Whenthe model is sparse, FDR-like selection yields estimators with strong large sam-

515WAVELET APPLICATIONS...

ple adaptivity properties. One natural application of FDR is threshold selectionin wavelet denoising. In wavelets thresholding, FDR is the proportion of waveletcoefficients erroneously included in the reconstruction among those included.This approach might lead to other applications of wavelets in microarray dataanalysis.

6 Wavelet Applications in MALDI-TOF MS Data

Analysis

In recent years there has been a great amount of interest in investigating thebiochemical events involved in the metabolism of peptides, primarily in thebrain and gut of mammals, encompassing the enzymatic breakdown of thesepeptides, their production from peptide and protein precursors, and the dis-ruption of these processes by certain xenobiotics. Modern mass spectrometrictechniques are used in these studies, including electrospray and matrix-assistedlaser desorption-ionization mass spectrometry (MALDI MS) (for example, [5],[29], [50], and the references therein).

Matrix-assisted laser desorption-ionization, time-of-flight (MALDI-TOF) massspectrometry (MS) is emerging as a leading technology in the proteomics revolu-tion. Indeed, the Nobel prize in chemistry in 2002 recognized MALDI’s ability toanalyze intact biological macromolecules. Though MALDI-TOF MS allows di-rect measurement of the protein “signature” of tissue, blood, or other biologicalsamples, and holds tremendous potential for disease diagnosis and treatment,key challenges remain in signal normalization and quantitation (for example,see [4], [7], [19], [46], and the references therein). Wavelet methods promise aprincipled approach for evaluating these complicated biological signals.

Mass spectrometry, especially surface enhanced laser desorption and ion-ization (SELDI), is increasingly being used to find disease-related proteomicpatterns in complex mixtures of proteins derived from tissue samples or fromeasily obtained biological fluids. In [8], a wavelet-based method was proposed tothe low-level processing of SELDI spectra data. This method was motivated bythe idea that the total ion current is a surrogate for the total amount of proteinin the sample being measured. If the baseline correction algorithm starts withthe raw spectrum and ensures that the corrected signal never becomes negative,then electronic noise contributes a substantial portion of the total ion current,and one normalizes, in part, to the noise. Statistically, the low-level processingof mass spectra reduces to decomposing the observed signal into three compo-nents: true signal, baseline, and noise:

f(t) = B(t) +N · S(t) + ε(t),

where, f(t) is the observed signal, B(t) is the baseline, S(t) is the true signal,N is the normalization factor, and ε(t) is the noise. It has been shown that thenonorthogonal discrete wavelet transform can be used to denoise MS SELDI dataand the peaks in the intensity can be rapidly identified and precisely quantified.

516 HONG,SHYR

Lung cancer kills more Americans each year than the next four leading cancerkillers– cancers of the colon, breast, prostate and pancreas – combined.

What eludes medical science is a reliable method to screen those high-riskpeople for lung cancer so that the disease is caught early enough to make adifference in survival. Proteomics-based approaches complement the genomeinitiatives and may be the next step in attempts to understand the biology ofcancer. It is now clear that the behavior of individual non-small cell lung can-cer tumors cannot be understood through the analysis of individual or smallnumber of genes, so cDNA microarray analysis has been employed with somesuccess to simultaneously investigate thousands of RNA expression levels andbegin to identify patterns associated with biology. However, mRNA expressionsare poorly correlated with protein expression levels, and it cannot detect im-portant post-translation modifications of proteins, all very important processesin determining protein function. Accordingly, comprehensive analysis of pro-tein expression patterns in tissues might improve our ability to understand themolecular complexities of tumor cells.

MALDI-TOF MS can profile proteins up to 50 kDa in size in tissue. Thistechnology can not only directly assess peptides and proteins in sections of tumortissue, but also can be used for high resolution image of individual biomoleculespresent in tissue sections. The protein profiles obtained can contain thousandsof data points, necessitating sophisticated data analysis algorithms.

Wavelets are providing a rich source of useful tools for applications in time-scale types of problems. In analysis of signals, the wavelet representations allowus to view a time-domain evolution in terms of scale components. The adaptivityproperty of wavelets certainly can help us to determine the location of peakdifference(s) of MALDI-TOF MS protein expressions between cancerous andnormal tissues in terms of molecular weights.

Wavelets, as building blocks of models, are well localized in both time andscale (frequency). Signals with rapid local changes (signals with discontinu-ities, cusps, sharp spikes, etc) can be precisely represented with just a fewwavelet coefficients. In [23], we apply wavelet transform together with cluster-ing techniques [15] and weighted flexible compound covariate method [44] tothe MALDI-TOF MS data from lung cancer patients. Class-prediction modelsbased on wavelet coefficients were found to classify lung cancer histologies anddistinguish primary tumors.

Very recently, wavelets are used for neural classification of lung sounds analy-sis in [27].Acknowledgments: This work was supported in part by Lung Cancer SPORE(Special Program of Research Excellence) (P50 CA90949), Breast Cancer SPORE(1P50 CA98131-01), GI (5P50 CA95103-02), and Cancer Center Support Grant(CCSG) (P30 CA68485) for Shyr and by NSF-IGMS 0408086 and 0552377 forHong.

517WAVELET APPLICATIONS...

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Instructions to Contributors

Journal of Concrete and Applicable Mathematics A quartely international publication of Eudoxus Press, LLC, of TN.

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TABLE OF CONTENTS,JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.4,NO.4,2006 SPECIAL ISSUE ON “WAVELETS AND APPLICATIONS”,EDITED BY GUEST EDITORS D.HONG,Y.SHYR PREFACE,D.HONG,Y.SHYR,……………………………………………..................391 AN ORTHOGONAL SCALING VECTOR GENERATING A SPACE C^1 CUBIC SPLINES USING MACROELEMENTS,B.KESSLER,……………………………....393 THE EXISTENCE OF TIGHT MRA MULTIWAVELET FRAMES,Q.MO,………. 415 FACE-BASED HERMITE SUBDIVISION SCHEMES,B.HAN,T.YU,……………..435 ON THE CONSTRUCTION OF LINEAR PREWAVELETS OVER A REGULAR TRIANGULATION,D.HONG,Q.XUE,……………………………………………….451 SPLINE WAVELETS ON REFINED GRIDS AND APPLICATIONS IN IMAGE COMPRESSION,L.GUAN,T.YANG,…………………………………………………473 VARIATIONAL METHOD AND WAVELET REGRESSION,J.WANG,………….. 485 WAVELET APPLICATIONS IN CANCER STUDY,D.HONG,Y.SHYR,…………...505

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