Spline Functions and Multivariate Interpolations

Spline Functions and Multivariate Interpolations

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 248

Spline Functions and Multivariate Interpolations

by

B. D. Bojanov Department of Mathematics, University of Sofia, Sofia, Bulgaria

H. A. Hakopian Department of Mathematics, Yerevan University, -Yerevan, Armenia

and

A. A. Sahakian Department of Mathematics, Yerevan University, Yerevan, Armenia

Springer-Science+Business Media, B.Y.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-4259-0 ISBN 978-94-015-8169-1 (eBook) DOI 10.1007/978-94-015-8169-1

Printed on acid-free paper

All Rights Reserved © 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993. Softcover reprint of the hardcover 1 st edition 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

CONTENTS

Preface

Chapter 1. Interpolation by Algebraic polynomials 1.1. Lagrange Interpolation Formula 1.2. The Hermite Interpolation Problem 1.3. Divided Differences 1.4. Birkhoff Interpolation 1.5. Budan-Fourier Theorem

Notes and References

Chapter 2. The Space of Splines 2.1. Polynomial Spline Functions 2.2. The Closure ofthe Spline Space 2.3. Splines with Multiple Knots


Chapter 3. B-Splines 3.1. Peano's Kernel 3.2. Definition of B-Splines 3.3. B-Spline Basis 3.4. Recurrence Relations

3.4.1. The basic recurrence relation 3.4.2. Differentiation of splines 3.4.3. Tschakaloff's formula

3.5. Variation Diminishing Property Notes and References

Chapter 4. Interpolation by Spline Functions 4.1. Total Positivity 4.2. Hermite Interpolation 4.3. Birkhoff Interpolation

4.3.1. B-splines with Birkhoff's knots 4.3.2. Sign changes of a spline function 4.3.3. Main interpolation theorem

4.4. Total Positivity of the Truncated Power Kernel Notes and References

ix

1 1 2 4

12 14 18

19 19 21 25 27

28 28 29 33 36 36 38 39 41 43

45 45 49 52 53 60 60 63 66

vi Contents

Chapter 5. Natural Spline Functions 67 5.1. Interpolation by Natural Spline Functions 67

5.1.1. Definition 67 5.1.2. Interpolation 68 5.1.3. Holladay's theorem 72

5.2. Best Approximation of Linear Functionals 75 5.3. Extremal Property of the Natural Spline Interpolation 78

Notes and References 81

Chapter 6. Perfect Splines 82 6.1. Favard's Interpolation Problem 82 6.2. Oscillating Perfect Splines 90

6.2.1. Splines with preassigned integrals over subintervals 90 6.2.2. Interpolation at the extremal points 99 6.2.3. Perfect splines of least uniform norm 100

6.3. Optimal Recovery of Functions 102 6.3.1. The best method of recovery 102 6.3.2. Characterization of the optimal nodes 104

6.4. Smoothest Interpolant 105 Notes and References 107

Chapter 7. Monosplines 109 7.1. Monosplines and Quadrature Formulae 109 7.2. Zeros of Monosplines 111 7.3. The Fundamental Theorem of Algebra for Monosplines 114


Chapter 8. Periodic Splines 117 8.1. Basis 117

8.1.1. Periodic B-splines 118 8.1.2. Representation by the Bernoulli polynomials 119

8.2. Hermite Interpolation 124 8.3. Favard's Problem 128


Chapter 9. Multivariate B-Splines and Truncated Powers 132 9.1. A Geometric Interpretation of Univariate B-Splines and Truncated

Powers 132 9.2. Multivariate B-Splines and Truncated Powers 137 9.3. Recurrence Relations for B-Splines 142 9.4. Ridge Functions 147


Chapter 10. Multivariate Spline Functions and Divided Differences 149 10.1. Multivariate Spline Functions 149

vii

10.2. Multivariate Divided Differences 156 10.3. Polyhedral Splines 159


Chapter 11. Box Splines 163 11.1. Definition and Basic Properties 163 11.2. Integer Translates of a Box Spline 168 11.3. A System of Partial Differential Equations Connected with V(X) 172 11.4. Further Properties of the Spaces V(X) and P(X) 175 11.5. Linear Independence of Translates of a Box Spline 188 11.6. Interpolation by Translates of a Box Spline 192


Chapter 12. Multivariate Mean Value Interpolation 198 12.1. Mean Value Interpolation of Lagrange Type 198 12.2. Kergin Interpolation and the Scale of Mean Value Interpolations 203


Chapter 13. Multivariate Polynomial Interpolations Arising by Hyperplanes 206 13.1. Pointwise Interpolation 206 13.2. Polynomial Interpolation by Traces on Manifolds 209 13.3. Special Cases and Consequences 224

13.3.1. Interpolation on the sphere by homogeneous polynomials 224 13.3.2. Hermite interpolation 226 13.3.3. Tensor-product interpolation 226 13.3.4. Finite element interpolations 227 Notes and References 230

Chapter 14. Multivariate Pointwise Interpolation 231 14.1. Birkhoff Interpolation 231 14.2. Shifts of Sets and Differentiation of the Vandermonde

Determinant d'H(z) 236

14.3. Quadratic Transformations 245 14.4. Hermite Interpolation 247 14.5. The Birkhoff Diagonal Interpolation 258 14.6. Uniform Hermite Interpolation 260


References 265

Index 273

Notation 275

PREFACE

Spline functions entered Approximation Theory as solutions of natural extremal problems. A typical example is the problem of drawing a function curve through given n + k points that has a minimal norm of its k-th derivative. Isolated facts about the functions, now called splines, can be found in the papers of L. Euler, A. Lebesgue, G. Birkhoff, J. Favard, L. Tschakaloff. However, the Theory of Spline Functions has developed in the last 30 years by the effort of dozens of mathematicians. Recent fundamental results on multivariate polynomial interpolation and multivariate splines have initiated a new wave of theoretical investigations and variety of applications.

The purpose of this book is to introduce the reader to the theory of spline functions. The emphasis is given to some new developments, such as the general Birkoff's type interpolation, the extremal properties of the splines and their prominant role in the optimal recovery of functions, multivariate interpolation by polynomials and splines.

The material presented is based on the lectures of the authors, given to the students at the University of Sofia and Yerevan University during the last 10 years. Some more elementary results are left as excercises and detailed hints are given.

Borislav Bojanov, Hakop Hakopian, Artur Sahakian

December 1992

ix

Chapter 1

INTERPOLATION BY ALGEBRAIC POLYNOMIALS

In a letter to Leibniz, dated October 24, 1676, Newton alluded to his "expeditious method of passing a parabolic curve through given points". The meaning attached to these words was that he could construct explicitly the algebraic polynomial of arbitrary degree n which assumes preassigned values 10, ... , In at given points Xo < ... < X n . Newton described this method in 1687 in the third book of his famous "Principia". This is the way classical interpolation theory was born.

§ 1.1. Lagrange Interpolation Formula

We denote in this book by 1I'n the class of all algebraic polynomials of degree less than, or equal to, n.

Let Xo < ... < Xn be fixed points on the real line lR and let the function I be defined by them. Then there exists a unique algebraic polynomial Ln(f; x) from 1I'n, which satisfies the interpolation conditions

k = O, ... ,n. (1.1.1)

Newton expressed this polynomial in terms of the so-called divided differences. Much later, Lagrange presented Ln(f;x) in the following way:

where

n

Ln(f;x) = LI(xk)/nk(x), k=O

n

II x - Xi Ink(x) := .

i=O Xk - Xi i#

(1.1.2)

Another useful representation of the fundamental polynomials {Ink} is given by

k = 0, ... ,n, (1.1.3)

where wn(x) := (x - xo) ... (x - xn). Formula (1.1.2) is called the Lagrange interpolation formula.

1

2 Interpolation by Algebraic Polynomials [Ch. 1, § 1.2

§ 1.2. The Hermite Interpolation Problem

The next generalization of (1.1.1) is known as the Hermite interpolation problem: Let tl < ... < tn be given points and Vl, ... ,Vn be positive integer numbers. Set

N := Vl + ... + Vn - 1. For an arbitrary sufficiently smooth function f, construct a polynomial p from 7f' N which satisfies the interpolation conditions

k = 1, ... , n, A = 0, ... , Vie - 1. (1.2.1)

THEOREM 1.1. For a given f there exists a unique polynomial p from 7f' N which satisfies (1.2.1). This polynomial may be written in the form

n IIk-l

p(x) ="L "L f(>\)(tle) . HIe)..(x), (1.2.2) Ie=l A=O

where

and f!(x) := (x - tl)"1 ... (x - tn ),,".

Proof. Every polynomial of degree N

is defined by its coefficients {aj}. Thus (1.2.1) is a linear system of N + 1 equations in N + 1 unknowns: ao, ... , aN. We have to show that it has a unique solution, i.e., that the corresponding homogeneous system admits only the trivial solution ao = ... = aN = O. Assume the contrary. Then there exists a nonzero algebraic polynomial po(x) E 7rN , which vanishes at h, ... , tn with multiplicities Vb .. ·, Vn ,

respectively. But Vl + ... + Vn = N + 1. Therefore Po(x) == O. The contradiction shows that the Hermite interpolation problem has a unique solution.

Let p be the solution of (1.2.1). Clearly p may be written in the form (1.2.2), where the polynomials H leA from 7r N are defined by the interpolation conditions

i=I, ... ,n, j=O, ... ,v;-I, (1.2.3)

(8mn being the symbol of Kroneker: 8mn = 0 for m =/: nand 8mn = 1 for m = n). It remains to verify that the polynomials Hie).. defined in the theorem satisfy (1.2.3). In order to this, let us denote by Tm(gj x) the m-th partial sum of Taylor's expansion of g at x = tie, i.e.,

m

Tm(gj x) = "Lg(j)(tle)(x - tle)j Ii! j=O

Ch. 1. § 1.2) The Hermite Interpolation Problem 3

It is seen that T!.!)(g;tk) = g(j)(tk) for j = 0, ... ,m. Then, by Leibniz' rule for the differentiation of a product,

for j = 0, ... , m. In particular, for g(x) := (x - tlc)"k /fl(x),

ddii {_(I )T"k-).-I(g;X)}1 = bio,

x g x :r:=tk j = 0, ... ,Vk - A-I,

and therefore

for j = 0, ... ,Vic - 1. The other equalities asserted in (1.2.3) follow immediately from the expression of

HIc)' because of the factor fl(x). The theorem is proved.

Exercise 1.2.1. Let Xl < ... < xn. Denote by D(VI, ... , vn) the matrix of the interpolation system

{ao + alx + ... + aNxN } ().)I:r:=:r:k = flc)', k = 1, ... , n, A = 0, ... , Vic - 1,

with N + 1 := VI + ... + Vn. Show that

n "k-l det D(VI, ... , vn) = II II A! II(xi - Xi)"j";.

1c=0 ).=0 i<i

Hint. Let

q(x) = (x - Xt}"l ... (x - xn)",,-l = bi + b2x + ... + bNxN- I + xN.

Add to the last column of D(VI' ... , vN ) the sum of the columns of numbers 1, ... , N, multiplied by bb ... , bN , respectively. Then the last column will take the form [0, ... 0, q(",,-I)(Xn)]T. Thus

det D(vl, ... ,vn) = q<",,-l)(xn)· det D(VI' ... ,Vn - 1)

and the result follows by induction on N.

Exercise 1.2.2. Show that the polynomial

_ ~ . (x - b)m+l(x - a)i ~(m+ i) (~)i P(x) - ~A3 ·!(a _ b)m+l ~ i b - a

3=0 J .=0

+ ~B.(x-a)n+l(x-b)i ~(n+1)(b-X)i ~ 3 j!(b - a)n+l ~ i b - a 3=0 .=0

4 Interpolation by Algebraic Polynomials (Ch. 1, § 1.2

satisfies the interpolation conditions

p(j)(a) = Ai, j = 0, ... , n, P(j)(b) = Bi' j = 0, ... ,m.

Exercise 1.2.3. Let Xo < ... < Xn and w(x) := (x - xo) ... (x - xn). Show that the polynomial

~ {wll(Xl:) } { w(x) }2 P(x) = L.J/l: 1 - -,-( -) (x - Xl:) ( ) '( )

l:=0 w Xl: X - Xl: w Xl:

~ '{ w(x) }2 + L.J/l: ( )'() (x - Xl:) l:=0 x - Xl: W Xl:

satisfies the interpolation conditions P(Xl:) = IA:. P'(Xl:) = I~ for k = 0, ... , n.

§ 1.3. Divided Differences

The notion of divided difference /[xo, ... ,xn] of a function 1 at the points Xo, ... ,Xn is defined inductively:

I[ ] .- I[xl. ... , xn] - /[xo, ... ,Xn-l] Xo, .. ·, Xn .- .

Xn - Xo (1.3.1)

It is supposed here that the points {Xj} are distinct, i.e., Xi:l Xj for i:l j. For the sake of convenience, we set /[Xi] := I(Xi).

THEOREM 1.2. The divided difference /[Xo, ... , xn] of the function 1 at the points Xo, ... , Xn coincides with the coefficient of xn in the polynomial Ln{/; x) from 1I"n, which interpolates 1 at the same poin ts.

Proof. The assertion follows by induction on the number of points. For n = 1 it is obvious. Assume that the theorem holds for n arbitrary distinct points. Let p and q be the polynomials from 1I"n-l that interpolate 1 at the points Xl. ... ,Xn and Xo, ... Xn-l, respectively. Then, by an inductional hypothesis,

p(X) = I[xl. ... , xn]xn- 1 + ... ,

q(X) = I[xo, ... , Xn_l]Xn - 1 + ... It is seen that

Ln(f; x) = p(x)(x - xo) - q(x)(x - xn) Xn - Xo

for each x. Now comparing the coefficients of xn in both sides of this polynomial identity we get the wanted relation. The proof is completed.

Ch. 1, § 1.3) Divided Differences 5

Some authors use Theorem 1.2 as a definition of the divided difference. An immediate consequence of the theorem is the following recurrence relation

which yields

n

Ln(J; x) = f(xo) + L![XO, ... , Xk](X - XO) ... (x - Xk-d. (1.3.2) k=i

This is the Newton interpolation formula. It is very convenient for practical use since there is an extremely simple scheme, based on the recurrence relation (1.3.1), for the numerical evaluation of the divided differences.

Theorem 1.2 suggests a natural extension of the divided difference notion, which covers the case of multiple points.

Let Xo ~ ... ~ x N be arbitrarily given points. Suppose that

where (t, v) means that the point t is repeated v times in the sequence z. Clearly Vi + ... + Vn = N + 1.

We shall say that the polynomial p interpolates f at the points z if

for k = 1, ... , n, A = 0, ... , Vk - 1.

DEFINITION 1.3. The divided difference f[xo, ... , xNl of the function f at the points z = (xo, ... , X N) is the coefficient of xN in the polynomial p from 1f' N' which interpolates f at z.

By virtue of Theorem 1.2 this definition is equivalent to the original one in the case when all points are distinct.

As an immediate consequence of Definition 1.3 and the Taylor interpolation formula we get

f(1c) (xo) ![xo, ... ,xkl = k!

if Xo = ... = Xk. Note that a similar relation holds in the next more general situation.

For each set of points Xo ~ ... ~ xN and f E CN[xo, xNl there exists a point e E [xo, xNl such that

(1.3.3)

In order to prove this, assume that P is the polynomial from 1f' N that interpolates f at Xo, ... , x N' Then the function f - P has N + 1 zeros, counting the multiplicities. By Rolle's theorem, f(N)(e) - p(N)(e) = 0 for at least one point e in [xo, xNl. Since p(N)(e) = N!![xo,"" XN], the claim (1.3.3) is now evident.

Denote by p(z; t) the polynomial, which interpolates f at the points z.


THEOREM 1.4. The interpolating polynomial p(z; t) depends continuously on the nodes z, i.e.,

where

Proof. Let

lim IIp(z; . ) - p(y; . )1IC[a bj = 0, IIx-yll-O '

liz - yll := m!lX IXi - Yil· •

z = (xo, ... , xN ) = ((h, Vl)"'" (tn, vn»), Y=(Yo""'YN)= ((Tl,/ll), ... ,(Tm,/lm»).

Consider the interpolation problem

N

Laiu~>')(6,>.) = I(>')(ek>'), k = 1, ... , m, A = 0, ... , Ilk - 1, i=O

with nodes e = {ek>.}, where we have used the notation

Ui(t):= ti, i = 0, ... ,N.

If the determinant

D(e) := [ _ U~>')(ek>')'" ~ U~)(ek>') ] (k-1, ... ,m, A-O, ... ,/lk-1)

(1.3.4)

of the system (1.3.4) in unknowns ao, ... , aN is distinct from zero, then the unique solution p(t) = aouo(t) + ... + aNuN(t) of the interpolation problem can be written in the form

[ uo(t), ... , uN(t), ° ]

p(t) = (_l)N u~>')~u), ... , UN(~(eU); 1(>') (ek>') / D(e)· (k -l, ... ,m, A - 0, ... ,Ilk -1)

Indeed, it is clear that p E 7rN and evidently p(>')(eu) = 1(>')(6>.). Observe that the determinant D(e) is distinct from zero for e = Y since in this case it corresponds to a Hermite interpolation problem. But {Ui(t)} are smooth functions. Thus D(e) is a continuous function of e and therefore there exists an c > ° such that D(e) :j: 0 for each lie - yll ~ c. So, the polynomial p(t) is defined for each e from the domain {e: lie - yll ~ c}. Moreover, it is seen from the explicit expression of p(t) that it tends uniformly on [a, b] to the polynomial

[ uo(t), ... , uN(t), 0 ] [u~>')(Tk)"'" U(>')(Tk)]

(_l)N Ur)(Tk)"",U};~(Tk),/(>')(Tk) / ~=1, ... ,~, (k -l, ... ,m, A - O, ... ,/lk -1) A - O, ... ,/lk -1)

when e --+ y and the latter evidently coincides with p(y; t). Now suppose that liz -yll ~ c. Since the polynomial p(z; t) interpolates 1 at z, we

conclude on the basis of Rolle's theorem that there exist points TJk>' E (Tk - C, Tk + c) such that p(>')( z; TJk>.) = 1(>')(TJu). Thus p(z; t) coincides with a polynomial p(t) of the form we just discussed with parameters e = {ek>'} = {TJk>.} =: fl. Clearly fI --+ y as z --+ y and therefore p(z; t) tends uniformly to p(y; t) as z --+ y. The theorem is proved.


COROLLARY 1.5. Let f E CN[a,b]. Then the divided difference /[xo, ... , xN] is a continuous function of {xil in the domain a ~ Xo ~ ... ~ X N ~ b.

Proof. The theorem implies the convergence of the coefficients of p(:I:; t) to the corresponding coefficients of p(y; t) and, in particular, /[xo, ... ,xN] ~ /[Yo, ... 'YN] if :I: ~ y.

It follows from Definition 1.3 of the divided difference and the presentation (1.2.2) of the Hermite interpolation polynomial that the divided difference /[:1:] of f at the points :I: = ((Xl, Ill), ... ,(Xn , lin)) is a linear functional of the form

n vk-1

f[:I:] = E E akAf<).)(Xk), (1.3.5) k=l ).=0

where akA is the coefficient of xN in the corresponding fundamental polynomial HkA(x).

Note that for k = 1, ... , n. (1.3.6)

Indeed, assuming the contrary, we get that HkA is a polynomial of degree N - 1, which has N zeros. Thus Hk). == 0 and this contradicts the condition H~;)(Xk) = 1.

The presentation (1.3.5) suggests the following equivalent definition of the divided difference.

DEFINITION 1.6. The divided difference off at the points :I: = ((X1,1I1), ... , (xn' lin)) is the unique linear functional of the form

n Vk-1

D[f;:I:] = E EakAf().)(Xk), k=l ).=0

which satisfies the conditions

{ D[f;:I:] = 0 for f(x) = xk, k = 0, ... , N - 1,

D[f;:I:] = 0 for f(x) = xN

with N := III + ... + lin - l.

(1.3.7)

There is an interesting relation between the coefficients ak)' ofthe divided difference at (Xl. V1), ... ,(xn , lin) and the decomposition of l/n(z) (n(z):= (Z-X1Yl ... (Zxny .. ) in elementary fractions. Precisely,

(1.3.8)

In order to proof (1.3.8), note first that by Theorem 1.2 and (1.1.3),


for each to < ... < tN' Here w(x) = (x - to) ... (x - tN)' Applying this formula to /(t) := l/(z - t) with any fixed z ¢ to, ... , tN' we get

Letting now to, ... ,tN tend to (Xl, 111)"'" (Xn' lin) and using the continuity of the divided difference (see Theorem 1.4), we obtain

which is just (1.3.8). Next we give several important theorems concerning divided differences.

THEOREM 1.7 (Frobenius formula). Let /(z) be analytic in a closed simply connected region G. Let r be a simple closed, rectifiable curve that lies ill G and contains the distinct points Xl, ... ,xn in its interior. Then, for each set of multiplicities 111, .'" lin,

1 J /(z) /[(X1, 111)"'" (Xn, lin)] = -2' ( ) ( ) dz.

11"1 Z - Xl VI • •• Z - Xn V"

I'

Proof. Integrating both sides of(1.3.8}, multiplied by /(z) and applying the Cauchy integral formula we get

1 J /(z) n Vk-1 1 J /(z)

211"i n(z) dz = 2: 2: akA A!211"i (z _ Xd>.+l dz I' k= 1 >.=0 I'

n vk-1

= 2: 2: akA/(>')(Xk)

k=l >.=0

The proof is completed.

THEOREM 1.8 (Steffenson's rule). Let F(x) = /(x)·g(x) and Xo, ... ,xN be arbitrary points on the real line. Then

N

F[xo, . .. , x N] = 2:/[xo, ... ,Xk]g[Xk, . .. ,X N]' k=O


Proof. We first establish the equality

{(t - ~)/(t)}[~, tl. ... , tm ] = /[tl. ... , tm ], (1.3.9)

where the left-hand side of (1.3.9) is the divided difference of the function (t -~)/(t). In order to do this, let p be the polynomial from 7I"m-1 which interpolates 1 at t1'· .. ' tm. Then clearly the polynomial q(t) := (t - ~)p(t) will interpolate the function (t - ~)/(t) at ~, tb ... , tm and the relation (1.3.9) follows from the observation that p and q have the same leading coefficient.

Now using the relation (1.3.9) we shall prove the theorem by induction on N. For N = 0, the formula is evidently true. Suppose that it holds for any N points xl. ... , xN . Since

F(x) = (I(xo) + I[xo, x](x - xo))g(x)

for each x, we have

F[xo, ... ,xN] = I(xo)g[xo, ... ,xN]

+ {/[xo, x](x - xo)g(x)}[xo, ... , xN].

An application of (1.3.9) to the second term gives

F[xo, ... , x N] = I(xo)g[xo, . .. , xN] + {/[xo, x]g(X)}[X1,· .. , x N]·

Taking into account that the divided difference of I[xo, x] at Xl. ... , Xk is I[xo, Xl, ... , Xk] (the latter follows easily by induction on k), we complete the proof using the inductional hypothesis for the product I[xo, x]g(x).

THEOREM 1.9 (Hermite-Genocchi's formula). Let 1 E eN [a, b] and Xi E [a, b], t = O, ... ,N. Then

/[xo, ... ,xN]= J I(N\toxo+ ... +tNXN)dt1 ... dtN' (1.3.10)

SN

where{SN :=(to, ... ,tN):to+ ... +tN =I, ti~O, i=O, ... ,N}.

Proof. It suffices to prove the theorem for distinct points {x;}. The general case follows then by Corollary 1.5, after going to the limit.

Denote the integral in (1.3.10) by {/xo, ... ,xN}. We have

1 1-t1 1-t1-.·.-tN_1

/{xo, ... ,xN } = J J ... J ° 0 0

+ ... + tN(xN - xo)) dtN .·· dt1

= 1 J [J(N-1) (xo + t1(X1 - xo) + ... x N -Xo

SN-1

+ (1 - t1 - ... - tN-t}(XN - xo))

- I(N-1)(xO + t1(X1 - xo) + ... + tN _ 1)(xN_1 - xo)] dtN_1 ... dt1 I{xl, ... ,xN} - I{xo, ... ,xN _ 1}

x N - Xo


Thus the expression on the right-hand side of (1.3.10) satisfies the same recurrence relation as the divided difference does. Since f{x, y} = /[x, y] for any two points, we get by induction

The theorem is proved.

THEOREM 1.10 (Popoviciu's formula). Let Xo ~ ... ~ xN and Xo < xN . Then for each point e E (xo, xN ) there exists a point 0",0 < 0" < 1, such that

for each function f.

Proof. Consider first the case when Xo < ... < xN . Set

Ao := (e - xo) ... (e - xN _ l ),

Al :=(e-xt}···(e-xN),

/3 := Ad Ao· With this choice of /3 the linear combination

is of the form N

D[f] = La;f(xi) i=O

(1.3.11)

since the coefficient of f(e) vanishes. In addition, D[!] satisfies the conditions (1.3.7). Then, by Definition 1.6, D[f] coincides with f[xo, ... , xN]. Therefore the relation (1.3.11) holds for 0" = 1/(1- /3). Since /3 = (e - xN)/(e - xo) < 0, we get 0 < 0" < 1.

Now note that /3, and consequently 0", does not depend on Xl,···, xN _ l . Applying the continuity ofthe divided difference we conclude that the relation (1.3.11) remains valid for every Xo ~ Xl ~ ... ~ XN _ l ~ x N and e E (xo, x N ). The theorem is proved.

COROLLARY 1.11. Let to ~ ... ~ tN+m and {Xk}b" be any N+l points from the set to, ... , tN+m such that

to= Xo ~ ... ~ xN = tN+m.

Then there exist constants O"k > 0, k = 0, ... , m, such that 0"0 + ... + O"m = 1 and

m

/[xo, ... , xN] = LO"k/[tk, ... , tk+N] k=O

for each sufficiently smooth function f.

Ch. I, § 1.3) Divided Differences 11

Proof. Starting from xO, ... , xN and adding new nodes 6, ... , em, one at a time, we apply the result just obtained to any previous divided difference which contains ei as a node and get the required presentation.

In order to find the coefficients ak, observe that the function

tPk(t) := { ~t - tk+d··· (t - tk+N-d

has the property

for t ~ tk+1, for t < tk+1

tPk[t;, ... , ti+N] = bk;/(tk+N - tk) for i = 0, ... , m.

Therefore ak=(tk+N-tk)tPk[XO, ... ,XN], k=O, ... ,m.

The equality ao + ... + am = 1 follows immediately from the discussed relation, applied for I(x) = xN.

Exercise 1.3.1. Let I(x) = xn. Prove that

J[X1, ... , xn] = Xl + ... + Xn·

Hint. Set w(x) := (x - xd ... (x - xn). Clearly

w(x) = xn - (Xl + ... + xn)xn- 1 + ... = I(x) - (Xl + ... + xn)P(x),

where p is a polynomial of degree n - 1 with a leading coefficient equal to 1. Then

0= W[X1' ... ,xn] = J[Xl. ... ,xn] - (Xl + ... + xn) . 1.

Exercise 1.3.2. Let F'(x) = I(x) on [a,b] and a:::;; Xl < ... < Xn :::;; b. Prove that n

'L:F [X1, ... ,Xk-1, Xk, Xk, Xk+1,· .. ,xn] = I[xl> ... ,xn]· k=l

Exercise 1.3.3. Show that for a sufficiently smooth f 8

-8 J[(X1' lI1), ... , (xn, lin)] = lIk/[(X1, 111)' ... ' (Xk,lIk + 1), ... , (xn, lin)]. Xk

Exercise 1.3.4. Let 1 E qa,bj and F'(x) = I(x) on [a,b]. Prove that for every choice of the points Xo :::;; ... :::;; Xn and e in [a, b],

1

F[xo, ... , Xn, e] = J tn J[xo(t), ... , Xn(t)] dt, o

where x(t) = e + (x - e)t, k = 0, ... , n. Hint. For Xo < ... < Xn verify that the integral 1(1) on the right-hand side is a linear functional of the form

n

1(J) = B· F(e) + 'L:Ak· F(Xk). k=O

Show that l(xk) = bk,n+1 and the assertion follows from Definition 1.6. Use the continuity of the divided difference to get the general case Xo :::;; ... :::;; xn .


Exercise 1.3.5. Show that

n m

I[(a, b), (b, m)] = La;l(j)(a) + Lb;l(i)(b), j=O j=O

where

a. = .!.. (_1)m+1 (m + n - j) (b _ a)j-n-m-1 J j! n - j , j = O, ... ,n,

bj = ~, (_1)"+1 (m + n -:- j) (a _ by-n-m-1, J. m-J

j =O, ... ,m.

Hint. Use the explicit expression ofthe interpolating polynomial P in Exercise 1.2.2 and Theorem 1.2.

§ 1.4. Birkhoff Interpolation

We continue to discuss here the problem of interpolation by algebraic polynomials, this time in a more general setting. The interpolation conditions can be neatly described with the use of special type matrices.

The matrix n ~ 1, r ~ 0,

is called an incidence matrix if it has as components the integers ° and 1 only. Denote by lEI the number of l's in E. For the sake of convenience, we shall assume that E is a normal matrix, which means that the number lEI of l's equals to the number of columns in E (i.e., lEI = r + 1).

Suppose that z = (Xl, ••• , x r ) is a set of fixed points such that Xl < ... < X r • Let I = {Ii;} be given numbers. The problem of determining a polynomial p from 7r'r

which interpolates I at (z, E), i.e., which satisfies the conditions

if ei; = 1, (1.4.1)

is known as the Birkhoff interpolation problem. We shall say briefly that p interpolates I at (z, E) if p satisfies (1.4.1) with Ii; =

1(j)(Xi). The incidence matrix E is said to be poised if the interpolation problem (1.4.1)

has a unique solution for each choice of the nodes Xl < ... < Xn and the values {Ii; }.

The complete characterization of the poised matrices is still an open problem. We are going to describe here simple sufficient conditions for the poisedness of E. The P6lya condition

n

Lei; ~ k + 1 i~k

for k = 0, ... , r, ( 1.4.2)

Ch. 1, § 1.4) Birkhoff Interpolation 13

is an important assumption in this result. Before considering the formulation of the main theorem we recall some other well-known notions from the theory of the Birkhoff interpolation.

Every sequence eij, ... , ei,j+m of I-entries in a row of E is called a block if: ei,j-l = o in the case i 'I 0 and ei,j+m+1 = 0 in the case j + m 'I r. The block is odd (respectively, even) if it contains odd (even) number of l's.

The block eij, ... , ei,j +m is said to be supported if there exist indices i l , i 2 , il, h such that i l < i < i 2 , it < i, h < j and

DEFINITION 1.12. The matrix E is said to be conservative if it does not contain supported odd blocks.

THEOREM 1.13 (Atkinson-Sharma theorem). Suppose that the normal incidence matrix E is conservative and satisfies the P6lya condition. Then E is poised.

Proof. The proof is based on the classical Rolle's theorem. It goes by induction on the number of columns of E.

The theorem evidently holds for a matrix E with one column, since this situation corresponds to the Lagrange interpolation. Assume it holds for matrices with r columns. Let Xl < ... < Xn be arbitrarily fixed points and E = {eij }f=l, 1=0' We have to show that the corresponding Birkhoff interpolation problem has a unique solution or, equivalently, that the homogeneous system of equations

for eij = 1 (1.4.3)

admits only the trivial solution P == 0 in 1rr • Assume that P E 1rr and P satisfies (1.4.3). Consider the polynomial Pl(t) := p'(t). It satisfies the conditions

(1.4.4)

Moreover, for i = 1, ... , m, (1.4.5)

where {~dr are the zeros of Pl that follow by Rolle's theorem from the conditions P(XiJ = p(Xi 2 ) = 0, for every two consecutive zeros Xii < Xi 2 of P, which are prescribed in (1.4.3). Now we construct on the basis of (1.4.4) and (1.4.5) a new system (call it S) of equations for Pl. S consists of all equations from (1.4.4) and those from (1.4.5), which had not been already incorporated in (1.4.4). If the condition Pl(~i) = 0 from (1.4.5) appears in (1.4.4) as well, then ~i must coincide with some Xk. By Rolle's theorem, Pl(t) changes its sign at ei = Xk. Then the point Xk is a zero of Pl of odd multiplicity. But Xk is prescribed in (1.4.4) as a zero of even multiplicity since the corresponding block (ekl, ... , ekl) is supported. Thus Xk is presented in (1.4.4) as a zero of Pl of multiplicity at least 1 less than the actual one. In such a case we add to S a new equation, prescribing the condition that the next (i.e., (I + l)-th) derivative of Pl vanishes too. In such a way every Rolle's zero ~i of Pl will generate an additional equation to (1.4.4). Therefore S will contain

14 Interpolation by Algebraic Polynomials [Ch. I, § 1.4

exactly r equations. Clearly the matrix E I , which corresponds to S, is conservative and satisfies the P6lya condition. Then by the iductional hypothesis PI(t) == 0, i.e., p'(t) == O. It follows from the P6lya condition that P has at least one zero. Hence p(t) == O. The proof is completed.

There are examples which show that the Atkinson-Sharma conditions are not necessary for the poisedness of E.

Exercise 1.4.1. Show that the incidence matrix

1 0 0 0 O~) 100 1 1 000

is poised although it is not conservative.

Exercise 1.4.2. Prove that the two-row normal incidence matrix E is poised if and only if it satisfies the P6lya condition. Hint. Let lEI = r and Xl < X2. Assume that E satisfies the P6lya condition. Then it follows by repeat application of Rolle's theorem (as in the proof of the AtkinsonSharma theorem) that every polynomial p from 1I'r-1 such that

if eij = 1, i= 1,2,

must vanish identically. Conversely, let E not satisfy the P6lya condition, i.e.,

k

Mk := ~)elj + e2j) = k j=O

for some k.

Let k be the smallest integer with this property. Then there is a nonzero polynomial Po E 1I'l; that satisfies the conditions

if eij = 1, j::;; k - 1, i = 1,2.

Since eil; = 0 for i = 1,2 and p~j)(x) == 0 for j > k, Po satisfies the conditions p~) (Xi) = 0 if eij = 1 and hence E is poised.

§ 1.5. Budan-Fourier Theorem

We present here a stronger version of a classical result concerning the estimation of the number of zeros of an algebraic polynomial in a given interval. The proof makes use of the following simple fact.

LEMMA 1.14. Let the nonzero algebraic polynomial f vanishes at c. Then there is an £ > 0 such that

f(c+t)!'(c+t) > 0

f(c - t)f'(c - t) < 0

for each t E (0, g),

for each t E (0, g).

Ch. 1, § 1.5] Sudan-Fourier Theorem 15

Proof. Choose e > 0 so small that I'(x) i= 0 for x E [c - e, c) and x E (c, c + e]. For every t >0,

c+t

I(c + t) = J I'(x) dx. c

Thus, by the mean value theorem, I(c + t) = I'(e)t for some e E (c, c + t). Hence, sign I(c + t) = sign I'(e) = sign I'(c + t) for t E (0, e). The other assertion follows similarly from the equality

c

I(c-t)=- J I'(x)dx.

c-t

Further we shall denote by 8-(10, ... '/r) the number of the strong sign changes in the sequence of real numbers 10, ... , Ir. The zero entries of the sequence are ignored. In contrast, 8+ (10, ... , Ir) denotes the number of the weak sign changes in 10, ... , Ir' where each zero is interpreted as either + 1 or -1 whichever makes the count largest.

Given E = {eij} i:o~ j~} we shall denote by Int E the matrix which is obtained from E by deleting the first and the last row of E, i.e., IntE:= {eij}i=l, j~~. The block f3 := (eij, ... , ei,Hk) is called an interior block of E if 1 ~ i ~ n. The block f3 is Hermitian if j = 0 and non-Hermitian otherwise.

For expediency, we abbreviate the notations

to 8-(x) and 8+(x), respectively. Denote by v(E) the number of odd non-Hermitian interior blocks of the matrix E.

THEOREM 1.15. Let I(t) = aotr + .. . +ar be an arbitrary algebraic polynomial with ao i= O. Suppose that

if eij = 1 (1.5.1)

for some points a =: Xo < Xl < ... < Xn+1 := b and an incidence matrix E - { }n+l r-l T'h - eij i=O, j=O. en

(1.5.2)

If, in addition, the matrix E is conservative, then

lEI ~r. (1.5.3)

Proof. In order to establish (1.5.2) we study the behavior of 8-(x) when X moves from a to b. Note that 8- (x) may change its value only at the zeros of I, I', ... ,/(r-l) in (a, b). Clearly Xl, .•. ,Xm belong to these critical points.

Let e be a fixed point from (a, b). Suppose that

(1.5.4)


Clearly j + k -1 :::;; r-1. Set Sjk(X) := S- (J(j)(x), ... ,j<Hk)(x )). In the case j = 0, by Lemma 1.14,

for each sufficiently small e > 0. Thus each Hermitian block (eio, ... , ei,k-1) of l's in E causes a diminution of S-(x) by k when x passes through the point e = Xi.

Now consider the case j >0, f U- 1)(e) :/; 0. Set

for small c > 0. The next claim follows from (1.5.4) and Lemma 1.14:

l(k) = k if k is even,

l(k) equals to k + 1 or k - 1 if k is odd. (1.5.5)

We conclude from these observations that

S-(a) - S-(b - e) ~ IIntEI- v(E)

for an sufficiently small c > 0. This implies (1.5.2) since limS-(b - e) = S+(b) as c -+ o.

Next, we shall improve this estimate by studying further the case (1.5.5). Clearly l( k) = k - 1 only if

(1.5.6)

or equivalently, if

Otherwise l(k) = k + 1. Thus, for odd k, all sign changes (sign consistency) in the sequence Sj-1,k+1(X)

disappear when x passes through the point e from left to right (from right to left) except the highest one, of level j + k - 1, which jumps to the lowest level j - 1. This is the main observation we use to develop further the estimation (1.5.2).

Assume that E is conservative. Let f3 := (eij, ... , ei,Hk+d be a non-Hermitian block of E which is not right supported (i.e., epq = 0 for each p > i and q < j). We shall show that:

{

f3 causes a diminution of S- (x) by at least k when x passes from Xi to b or it causes a diminution by k - 1 and induces a sign change or a zero in the sequence (J(b), ... , j<j)(b)) which is not specified by a I-entry of E.

(1.5.7)

Indeed, if S-(JU-1)(X + c), f(j)(x + e)) = ° for x = Xi and sufficiently small e > 0 then, by (1.5.6), l(k) = k+ 1 and the claim is proved. If S-(JU-1)(x +c), f(j)(x+ c)) = 1, then Sj -l,k+ 1 loses k - 1 changes near e = Xi. In this case, we study further the behavior of the remaining sign change in (JU-1)(X),/(i)(x)) on the way of x toward b. Note that the zeros of f U- 1) in (xi,b] are not prescribed by (z,E) since

Ch. I, § 1.5] Sudan-Fourier Theorem 17

f3 was supposed not to be right supported. Thus, if x meets a zero of f(;-l), S- (X) will lose at least one more sign change (in addition to those k - 1 changes caused by f3) or the sign change in (I(;-l)(x),f(;)(x» will jump to a lower level. Applying again this reasoning (if necessary) we prove (1.5.7).

Similarly one shows that each odd interior non-Hermitian block f3 := (eij,""

ei,j+k-d, which is not left supported, causes a diminution of S-(x) by k or it causes a diminution by k - 1 and induces a not specified zero or a sign consistency (i.e., S-(l(I-1)(x), f(l)(x» = 0) in the sequence (f(a), ... , f(;)(a».

Denote by scon (a) (sch (b), respectively) the number of the induced sign consistencies in S-(a) (sign changes in S+(b» by the odd non-supported blocks of E. Let gz (a) (gz (b), respectively) be the number of all sign consistencies or zeros in f(a), ... , f(r)(a) (of all sign changes or zeros in f(b), ... , f(r)(b), respectively) except those prescribed by the l's of eo (em+l' respectively). Clearly

gz (a) - scon (a) ~ 0,

gz (b) - sch(b) ~ O.

(1.5.8)

(1.5.9)

Now after the detailed study of (1.5.5) we see on the basis of (1.5.7) and its left analogy that

Since

we get

S- (a) - S+(b) ~ lInt EI- (scon (a) + sch (b».

S-(a) = r -Ieol- gz (a), S+(b) = lem+ll + gz (b),

r - [gz (a) - scon (a)] - [gz (b) - sch (b)] ~ lEI.

Then (1.5.8) and (1.5.9) implies lEI:::;; r. The theorem is proved. The following is an immediate consequence of Theorem 1.15.

(1.5.10)

COROLLARY 1.16 (Budan-Fourier theorem). Let P be an algebraic polynomial of exact degree r. Then the number Z of zeros of P in (a, b), counting the multiplicities, satisfies the inequality

Z:::;; S- (P(a), ... , p(r)(a») - S+ (P(b), ... , p(r)(b»).

The Atkinson-Sharma theorem can be derived as a corollary from Theorem 1.15, too.

The following very particular consequence of Theorem 1.15 will be needed later.

LEMMA 1.17. Let:z: = (Xl,"" xn ), Xl < ... < xn , and let E { }n r-1 = eij ,=l,j=O

be a conservative P6lya matrix with lEI = r + 1. Let enA be the last I-entry in the sequence en := (eno, ... , en,r-d. Suppose that the polynomial cp from 'Trr satisfies the conditions

cp(A)(Xn ) i 0,

cp(;)(x,) = 0 for all eij = 1, (i, j) i (A, n).

Then


Proof. Denote by E the incidence matrix which is obtained from E replacing the I-entry enA by O. Clearly cp vanishes at (z, E). Then, by virtue of (1.5.10), gz (xn) = sch (xn). Thus, all sign changes or zeros in the sequence (cp(xn), ... , cp(r}(xn)) that are not prescribed by the l's in en are induced by some odd interior non-Hermitian blocks f3 of E that are not right supported. It was shown in the proof of Theorem 1.15 that the level of these sign changes is lower than the level of the corresponding odd interior block f3. Since f3 is not right supported, the level of all sign changes counted by gz (Xn) is less than or equal to A. Thus the sequence (cp(A}(Xn), .•. , cp(r}(xn)) does not contain a sign change or a zero. The proof is completed.

Exercise 1.5.1. Derive the Decartes rule from the Budan-Fourier theorem, i.e., show that the number of positive zeros of a polynomial f(x) = aoxn + ... + an, ao f 0, is less than or equal to the number of strong sign changes in the sequence of its coefficients. Hint. Show first that there exists a number M > 0 such that the functions f(x), J'(x), ... , f(n}(x) does not vanish on (M,oo) and apply the Budan-Fourier theorem to f in (a, b) = (M, 00 ) .


The material presented in this chapter is standard text from interpolation theory. Much more can be found, for instance, in Davis [1975].

Formula (1.3.8) is proved by Tschakaloff [1938] in another fashion. The integral representation of the divided difference for analytic functions have been obtained firstly by Frobenius [1871] and rediscovered later by Benedixon [1885].

The observation (1.3.9) and the present proof of Steffenson's rule comes from Hakopian [1982b].

Theorem 1.10 is due to Popoviciu [1934] (see also Favard [1940]). Theorem 1.13 was proved by Atkinson and Sharma [1969]. A similar result was

obtained independently by Ferguson [1969]. Theorem 1.15 is a particular case of a slightly more general result (treating differ

entiable functions) shown in Bojanov [1992b].

Chapter 2

THE SPACE OF SPLINES

§ 2.1. Polynomial Spline Functions

Because of its simple structure and good approximation properties, algebraic polynomials are widely used in practice for approximating of functions. The degree of this approximation depends essentially on the degree of the polynomial and the length of the considered interval [a, b]. Since the computation operations on polynomials of high degree involve certain problems it is advisable to use polynomials oflow degree. In such a case, in order to achieve the desired accuracy we have to restrict ourselves to a small interval. For this purpose, one usually divides the original interval of consideration [a, b] into sufficiently small subintervals ([Xk' Xk+1]}k:O and then uses a low degree polynomials Pk for approximation over [Xk' Xk+1], k = 0, ... ,n. This procedure produces a piecewise polynomial approximating function s(x),

s(X) == Pk(X) on [Xk, Xk+1], k = 0, ... , n.

In the general case, the polynomial pieces {Pk(X)} are constructed independently of each other and therefore they are not supposed even to constitute a continuous function s(x) on [a, b]. This is an unacceptable way of approximation particularly for functions that describe smooth processes. In such a situation it is naturally then to require the polynomial pieces {Pk} to join smoothly at Xl, ... ,Xn , i.e., all derivatives of Pk-1 and Pk, up to a certain order, to coincide at Xk. As a result we get a smooth, piecewise polynomial function, called a spline function.

This is one of the various ways to illustrate the practical application of splines. However, the astonishing properties of spline functions and their close relation to other branches of mathematics show that the appearance of splines is motivated by the intrinsic logic of the development of mathematics itself.

DEFINITION 2.1. The function s( x) is called a spline function (or simply "a spline") of degree r with knots {xkH' if -00 =: Xo < Xl < ... < Xn < Xn+1 := 00 and

i) for each k = 0, ... , n, s(x) coincides on (Xk, xk+d with a polynomial of degree not greater than r;

ii) s(x), S' (x), ... , s(r-1) (x) are continuous functions on (-00,00).

Every algebraic polynomial is a spline function without knots. It is seen from the definition that the r-th derivative of a spline of degree r with knots {Xk}l.' is a piecewise constant function with breaks, eventually, at Xl, ... ,Xn. Conversely, the r-th primitive function of a piecewise constant function is a spline of degree r.

19

20 The Space of Splines [Ch. 2. § 2.1

We shall denote by Sr(zt, ... , zn) the class of all spline functions of degree r with knots at Z1I'" ,Zn. Clearly, for fixed {Zk}1, Sr(Z1,'" ,zn) is a linear space.

A simple example of a spline function is the so-called truncated power function:

(Z - t)+ := {~Z - W for z~t, for Z < t.

We show in the next proposition that every spline can be presented in terms of a truncated power function.

THEOREM 2.2. The function I belongs to Sr(zt, ... , zn) if and only if it may be written in the form .

r n

I(z) = Eajzi + ECk(Z - Zk)+ (2.1.1) j=O k=1

with some real coefficients {aj} and {Ck}. Moreover,

(2.1.2)

for k = 1, ... , n.

Proof. We have denoted here by l(z+O) the quantity lim{f(t): t -+ z, t > z}; I(z-0) is defined similarly.

Evidently every function I of the form (2.1.1) coincides on (Zk, Z1:+d with a polynomial of degree r and I is r - 1 times continuously differentiable on (-00,00). Thus I E Sr(Z1, ... ,zn). In order to find the coefficients C1:, observe that all terms in (2.1.1) are differentiable functions at a neighbourhood of Zk except u(z) := Ck(Z - Zk)+. Thus f(r)(zk + 0) - l(r)(Zk - 0) = U(r)(Zk + 0) -U(r)(Zk - 0) = rIck and the relation (2.1.2) follows.

Now suppose that I E Sr(Z1, ... , zn). Then I(r)(z) is a piecewise constant function with breaks at Zt, ... ,Zn. Denote by hk the jump of I(r)(z) at Zk, i.e., hk := l(r)(Zk + 0) - l(r)(Zk - 0). Clearly I(r) can be written in the form

n

j<r)(z) = a + Ehk(z - Zk)~ k=1

with some constant a. Then, taking the r-th primitive of this expression, we conclude that I has a presentation (2.1.1) with some {aj} and Ck = hk/r!, k = 1, ... , n. The theorem is proved.

Now we can determine the dimension (dim) of the space of splines.

COROLLARY 2.3. Let a < Z1 < .. , < Zn < b be arbitrary fixed points. Then the functions

l,z, ... ,zr,(z-Z1)+,'''' (z-zn)+

form a basis of the space Sr(zt, . .. , zn) in [a, b] and hence

dimSr(zt, ... , Zn) = n + r + 1.

(2.1.3)

Ch. 2, § 2.2) The Closure of the Spline Space 21

Proof. Since, by the theorem, dimSr (x1, ... , xn) ~ n+r+ 1, the only thing we need to show is that the functions (2.1.3) are linearly independent on [a, b]. Assume the contrary. Then there exists a spline (2.1.1) with nonzero coefficients which vanishes at each point x from [a, b]. But

for x < Xl

and the identity I(x) == 0 on (a, Xl) implies ao = ... = ar = O. Then I(x) = C1 (x - Xl)+, for x E(Xb X2) and therefore C1 = 0, too. Similarly, considering I on (X2, X3), ... , (Xn-1, xn), (xn, b), we get C2 = 0, ... , Cn = o. The contradiction completes the proof.

§ 2.2. The Closure of the Spline Space

Let [a, b] be a given finite interval. Denote by Srn [a, b] the set of all spline functions of degree r which have, at most, n distinct knots in (a, b). In other words

Sometimes it is more convenient to say that Srn [a, b] is the space of splines of degree r with n free knots.

Note that Srn[a, b] is a nonlinear space for n ~1. The sum of two functions from Srn[a, b] may not be from Srn[a, b]. Clearly 7rr is a subset of Srn[a, b]. Because of the nonlinearity of the spline space many easily solved problems in 7rr become extremely difficult when posed in Srn [a, b]. Another feature of the space of splines with free knots is that it is not closed. A sequence of functions from Srn[a, b] may tend to a function which is not a spline. Let us illustrate this with the following example.

Denote by Im(x) the function continuous on [-1,1]' which is linear on [0, 11m] and such that

{ -1 Im(x) := 1

for -1 ~ x ~ 0, for 1 I m ~ x ~ 1.

The functions {fm}f are splines of first degree with two knots. It is seen that Im(x) tends uniformly to sign x on any compact subset K of [-1,0) U (0, 1] when m --. 00.

Clearly 1m E S12[-1,1] for each natural m but the limit function signx is not a spline from S12[-1, 1].

Recall that the set of algebraic polynomials 7r r is closed in the following sense.

LEMMA 2.4. Let K be an arbitrary bounded subset of the real line lit Suppose that I is a bounded function on K for which there is a sequence of algebraic polynomials {Pm}f of degree r such that

for each x E I<.

Then I coincides on K with a polynomial from 7rr .

22 The Space of Splines (Ch. 2, § 2.2

Proof. If K contains no more than r + 1 points the assertion is trivial since there is a polynomial from 1I'r which interpolates I at any point of K. Suppose that K contains more than r + 1 points. Then we can select r + 1 distinct points Xo, ... , Xr from K. By the Lagrange interpolation formula

r

Pm(X) = ~Pm(xi)lni(x), i=O

Denote by P(x) the polynomial L~=o I(xi)lni(x). We have

IIPm - PIIC(K): = maxIPm(x)-P(x)1 :t:EK

r

= 11~(!(Xi) - Pi(Xi»)lni(X)11 i=O C(K) r

~ ~1/(xi) - Pi(xi)I·1I1niIlC(K)' i=O

Since Pm(Xi) -+ I(xi), the last estimation yields IIPm - PIIC(K) -+ 0 when m -+ 00

and therefore I(x), as a limit of the sequence {Pm(x)}, coincides with P(x) for x E K. The proof is completed.

Now let us return to the spline set Srn[a,b]. We shall say that the sequence of splines {fm}f from Srn[a,b] tends to the function I(x), ifthere is a constant C >0 and a finite number of points 6 < ... < en in [a, b] such that I Im(x) I~ C for each m and 1m tends uniformly to I(z) on every compact subset K C [a,b]\{el.··· ,en}' Denote by Srn the closure of Srn[a, b] with respect to the convergence we just defined.

We need the following lemmas in order to characterize the closure Srn of the spline space.

LEMMA 2.5. Let Zo < ... < Zn and n ~ r. Then the functions (x-zo)+, ... , (z-zn)+ are linearly independent on each subinterval [a,p] of(xn , 00).

Proof. Assume the contrary. Then there exist numbers ao, ... , an, at least one of them distinct from zero, such that

in [a,p].

Therefore, pU)(x) = 0 for j = O, ... ,r and each Z E[a,.B]. Let e be a fixed point from [a,p]. Denote ti := e - Zi, i = O, ... ,n. The conditions pU)(e) = 0 for j = r - n, r - n - 1, ... ,r can be written in the following equivalent form

aot~ + ... + ant~ = 0

t n - 1 + t n - 1 0 ao 0 +... an n =

ao· 1 + ... + an' 1 = 0

Since this system has a Vandermonde (hence nonzero) determinant, we get ao = ... = an = 0: a contradiction.

Ch. 2. § 2.2) The Closure of the Spline Space 23

LEMMA 2.6. Let Xo < ... < Xn and n ~ r. Then the functions

u>.(x) := (x-· )+[xo, ... ,x>.], A = 0, ... , n,

are linearly independent on each subinterval [a,,B] of(xn, 00).

Proof. The function u>.(x) is the divided difference of (x - t)+ at the points Xo, ... ,x>.. Then, according to the representation (1.3.5) of the divided difference,

>.

u>.(x) = LW>.k(X - Xk)+, k=O

where WH := l/w~(xk), k = 0, ... , A, and w>.(x) := (x - xo) ... (x - x>.). Thus the functions 1.& := (uo(x), ... , un(x)) are obtained from the functions v := «x -xo)+, ... , (x - xn)+) by the linear transform 1.& = W·v, where

W'- [::: ... ::: .. J~ ......... ~ ]. WnO Wn1 Wn2 Wnn

But clearly det W =f. O. Since, by Lemma 2.5, v are linearly independent, 1.& are linearly independent too. The proof is completed.

LEMMA 2.7. Let {Xm1,"" XmN}, m = 1,2, ... , be given sets of points and {Vk}r, n ~ m, be given integers satisfying the requirements

a < Xm1 < ... < XmN < b,

V1 + ... +vn = N,

Xmi ~ {k as m ~ 00 for i E J(k), k = 1, ... ,n,

where J(k) := {V1 + ... + Vk-1 + 1, ... ,111 + ... + Vk} and {{k}r are fixed points, 6 < ... < {n. Assume otherwise that for the function I bounded on [a, b], there exists a sequence {fm}i of splines 1m from Sr(Xm1,.'" XmN) such that

lim II/m-/IIC(K) = 0 m .... oo

for each compact I< C [a, b]/ {6, ... ,{n}. Then I is a function of the form

r n min(r,vk-1)

I(x) = Lajxi + L L ak>.(x - {k)+->'. j=O k=l >.=0

24 The Space of Splines [Ch. 2, § 2.2

Proof. Let K be an arbitrary fixed closed subinterval of (';;,';i+l). Since 1m E 7rr

on K for sufficiently large m and Im(x) tends to I(x) as m -+ 00 for x E K, we conclude on the basis of Lemma 2.4 that I(x) coincides with a polynomial from 7rr

on K and, consequently, on the whole interval (';;,';i+t). Thus, I(x) is a piecewise polynomial function with eventual knots at 6, ... ,';n. We shall show next that

I Ec(r-IIk)(';b';k+t),

if Ilk ~ r. For the sake of simplicity, we assume that ';k = ° and ';k+l = 1. Then our problem is reduced to the following: it is known that for a certain II ~ r functions of the form

II-I

gm(x) := I>m>.(X - em>.)+, >'=0

where emO < ... < em,II-I, tend uniformly on each compact subset of (0,1) to a function g and em>. -+ ° for A = 0, ... ,II - 1 as m -+ 00. We have to show that g is of the form

II-I

g(x) = La>.xr->. >.=0

on (0,1)

with some constants {a>.}. In order to do this, let us introduce the functions

1 Um>.(x):= (~) (x-· )+[emO, ... ,em>.], A = 0, ... , II - 1.

According to the property (1.3.3) of the divided difference, there exists a point 1} E (emO' em>.) such that um>.(x) = (x - 1})+->'. Therefore

lim Ium>.(x) - xr->'I = ° for x E (0,1). (2.2.1) m_oo

Let e be an arbitrary sufficiently small positive number. Then, for every sufficiently large m, say, for m > mo, the inequality em,lI-l < e holds, since em>. -+ ° as m -+ 00.

But, in view of Lemma 2.6, the functions UmO, ... , Um,lI-l are linearly independent in (e, 1] for em,lI-l < e. Therefore, for each m > mo the function gm(x) has a unique representation in the form

II-I

gm(x) = LPm>.Um>.(X). (2.2.2) >.=0

Note that the sequences of coefficients {Pm>'}:=1 are bounded. Indeed, assume that Pm := max>. I Pm>. I tends to 00 as m -+ 00. Then dividing both sides of (2.2.2) by Pm and letting m to tend to infinity, we arrive in contradiction with the linear independence of the polynomial functions {xr ->. }~:~. Thus we can choose convergent subsequences of {Pm>'}:=I' Denote their limits by {a>.}. Going to the limit in (2.2.2) and using (2.2.1) we get

The lemma is proved.

II-I

g(x) = La>.xr->.. >.=0

Now we are ready to present the main result of this section.

Ch. 2. § 2.3) Splines with Multiple Knots 25

THEOREM 2.8. The closure BrN of the spline space SrN[a, b] consists of all functions of the form

r n IIk-1

I(x} = 2)jXj + E E cu(x - 6:}~->' (2.2.3) j=O k=l >.=0

with n ~N, 1 ~ I'k ~ r, k = 1, ... , n, 1'1 + ... + I'n ~ N.

Proof. According to Lemma 2.7, every limiting point of SrN[a,b] is a function of form (2.2.3). The converse is also easily seen. Assume that I is given by (2.2.3). We shall show that I EBrN, i.e., there are points {e;Jf=l in [a, b] and a sequence {fm}f of functions from SrN[a, b] which approaches I uniformly on every compact subset of (-00, oo)\{el!'" ,en}. Using again the fact that

(X-ek)~->' = (1) lim (x-.)~[ekO, ... ,eu], r ekj-+ek A j=O •...• >.

it is clear that one can choose as {1m} the functions

r . n Ilk -1 1 ?=ajxJ + E Ecu (r)(X-.}~[ekO, ... ,eu], J=O k=l >.=0 A

which obviously belong to SrN [a, b]. The proof is completed.

§ 2.3. Splines with Multiple Knots

It was shown in Theorem 2.8 that the boundary ofSrN consists of piecewise polynomial functions but they are not necessarily from c(r-1)[a, b]. The order of smoothness of I at the knots may be of order less than r -1. It is customary to call functions of this type splines with multiple knots, although the presence of break points in low-order derivatives is a discrepancy which offends the smooth nature of the spline functions.

DEFINITION 2.9. The function s(x) is said to be a spline function of degree r with knots Xl, •.. , Xn of multiplicities 111, ... , lin, respectively, if -00 := Xo < Xl < ... < Xn < Xn+1 := 00 and:

r; i} for each k = 0, ... , n, s(x) coincides on (Xk, xk+d with a polynomial of degree

.. ) c(r-vk)( ) fc k 1 11 sE Xk-1,Xk+1 or = , ... ,n. We shall denote by Sr«X1,1It}, . .. ,(Xn ,lIn» the set of all these splines. Let N = 111 + ... + lin and

(t 1 , ..• , t N) == (Xl, lid, ... , (x n, lin») =: :z:

be the sequence of points t1 ~ ... ~ tN, where each knot Xk is repeated Ilk times, k = 1, ... , n. Occasionally, it will be more convenient to say that a function f from Sr«Xl!1I1}, ... ,(Xn,lIn}) is a spline with knots tb ... ,tN and to use the notation Sr(h, ... ,tN), or simply Sr(:Z:}, for the corresponding class of splines.

26 The Space of Splines [Ch. 2. § 2.3

THEOREM 2.10. The function f(x) belongs to the class Sr«Xl, vt), ... , (xn, vn)) if and only if it can be written in the form

r n Vk-l

f(x) = 2:>j,xi + L L Ck>.(x - Xk)+->', (2.3.1) j=O k=l >.=0

where aj and Ck>. are real constants. Moreover,

(2.3.2)

Proof. Clearly every function of the form (2.3.1) belongs to Sr«XI, vI), ... , (xn, vn)). Now suppose that f E Sr«XI,VI), ... ,(Xn,vn)) and let {Pk}~ be the polynomial components of f on the intervals (Xk, Xk+1), k = 0, ... , n. It follows from the smoothness of f at x k that there exist constants {Ck>.} such that

Vk- l

Pk(X) = Pk-I(X) + L Ck>.(x - Xkr->'· >.=0

More precisely,

(2.3.3)

Applying repeatedly this recurrence relation between Pk and Pk- l , we get

m Vk-l

f(x) == Pm(x) = Po(x) + L L Ck>.(X - Xkr->' k=l >.=0

for each m E 0, ... , n and x E(Xm , xm+I). This is simply another way of expressing (2.3.1).

Formula (2.3.2) follows from the observation (2.3.3). The theorem is proved.

COROLLARY 2.11. The functions

are linearly independent on (-00,00) and they constitute a basis in Sr«XI, VI)' ... , (xn , vn )).

Now the assertion follows from the fact that every function from Sr«XI, vt), ... , (xn, vn)) has a unique representation of the form (2.3.1).

Exercise 2.2.1. Show that the functions IXk - tl, k = 1, ... , n, are linearly independent in [a, b], if a < Xl < ... < Xn < b.

Ch. 2. § 2.3] Splines with Multiple Knots 27

Exercise 2.2.2. Let a =: Xo < Xl < ... < xn < Xn+1 := b. Prove that every spline s from 8 1 (Xl, ... , xn) has a unique representation of the form

n

s(x) = LCklxk - tl. k=O

Find the coefficients Ck in terms of {S(Xk)}~~~'

Hint. Use the relation S'(Xk + 0) - S'(Xk - 0) = 2Ck to derive the expressions

for k = 1, ... , n. Then show that


The notion spline function appeared for the first time in a paper of Schoenberg [1946]. The name comes from a mechanical device used to draw a smooth curve through given points.

Splines with multiple knots were introduced in Curry and Schoenberg [1947]. The books by Schoenberg [1973] and Schumaker [1981] can be consulted for comprehensive notes on spline story.

Theorems treating the closure of spline space are proved in Barrar and Loeb [1970], and Tikhomirov [1976].

The early developments in the theory of univariate spline functions and their applications are covered by the books of Ahlberg, Nilson, and Walsh [1967], Stechkin and Subbotin [1976], and Ciesielski [1976]. The steadily growing interest in splines during the last 30 years has resulted in the foundation of a new field in mathematics of intensive research. There exist now several books reflecting different stages in the development of this interesting subject (Schoenberg [1973], Bohmer [1974], Karlin, Micchelli, Pinkus, and Schoenberg [1976], de Boor [1978], Zavialov, Kvasov, and Miroshnichenko [1980], Schumaker [1981], Korneichuk [1984], Chui [1988], and Nurnberger [1989]).

Chapter 3

B-SPLINES

We introduce here another basis for the spline space Sr(Zl,. " ,zn) which consists of functions that have a finite support (Le., which vanish outside a certain finite interval). The new basis functions possess some remarkable properties which make them a widely used tool in calculating with splines as well as in other theoretical studies.

Before introducing the main material, we recall an auxiliary result from classical analysis.

§ 3.1. Peano's Kernel

We use the standard notation W;[a, b] for the Sobolev classes

with

W;[a,b] := {I ECr-1[a,b] : I(r-l) abs. cont., II/(r)lIp < oo}

11111, ,= {i 1 I(z) I' dZ} 'I, 11/1100 := supvrai I I(z) I .

ze[a,6]

for 1 ~ p < 00,

The well-known Taylor's formula may be written in the form:

r-l/(k)( ) 1 /6 I(z) = ,, __ a (z _ a)k + (z - t)+-l lr)(t) dt

L...J k! (r - I)! k=O a

for each 1 EW[[a,b] and z E[a,b]. This identity can be proved by induction on r, using integration by parts.

Taylor's formula with such integral form remainder provides the basis of the next representation of linear functionals.

THEOREM 3.1 (Peano's theorem). Let L(f) be an arbitrary linear functional defined in W[[a, b] such that the function K(t) := L[(z - t)+-l] is integrable over [a, b]. Suppose that L(p) = 0 for each polynomial p E 1I"r-l. Then

for each 1 EW[[a,b].

6

L(f) = 1 / K(t)/(r)(t) dt (r - I)!

a

28

(3.1.1)

Ch. 3, § 3.2) Definition of B-Splines 29

Proof. Using the representation of f by the Taylor formula and the assumption that the linear functional L annihilates the polynomials from 11" r-l, we get

b

L(f) = (r ~ i)! L [j (x - t)+-l f(r)(t) dt] . (J

Now the theorem follows from the observation that

b b

L[j(X-t)+-lf(r)(t)dt] = j L[(x-t)+-lf(r)(t)]dt (J (J

for the functionals L of the type stipulated. It is seen that the theorem can be applied to functionals of the following general

form b r-l n

L(f) = j [?=aj(x)fU)(x)] dx + ~)d(Ak)(Xk)' (J )=0 k=l

(3.1.2)

where {aj(x)} are integrable functions in [a, b], {bk} are given numbers and {Ad are integers, 1 ~ Ak ~ r - 1, k = 1, ... , n.

Exercise 3.1.1. Find the Peano kernel K(t) for the error functional

b

R(f):= j f(t)dt- b~a[f(a)+4f(a;b) +f(b)] (J

of the Simpson quadrature formula and show that K(t) ~ 0 on [a, b].

§ 3.2. Definition of B-Splines

Let Xo ~ ... ~ Xr be arbitrary points in [a, b] such that Xo < Xr. It is seen from (1.3.5) that J[xo, ... , xr] is a linear functional of form governed by (3.1.2). In addition, it follows from Definition 1.3 that J[xo, . .. , xr] = 0 for each polynomial f E 1I"r_l' Thus we may apply the Peano's theorem to the divided difference functional and obtain

b

J[xo,·.·,xr] = (r~ I)! j B(xo, ... ,xr;t)f(r)(t)dt (J

for each f EW[[a, b], where

B(xo, ... , xr;t) := (- -t)+-l[xO, ... ,xr].

30 B-Splines [Ch. 3, § 3.2

As an immediate consequence of the fact that J[xo, ... , Xr] = 1 for J(t) = xr, we get

b J B(xo, ... , xr;t) dt = 1/r. (3.2.1)

a

Assume, for the sake of convenience, that

Then, according to (1.3.5), the divided difference of (x - t)+-1 can be written in the form

n vk-l

B(xo, ... , xr; t) = I: I: ak)..(tk - t)+.>.-I. (3.2.2) k=1 .>.=0

It is seen from this expression that B(xo, ... , xr; t) is a spline function of degree r-l with knots tl, ... , tn of multiplicities VI, .•. , Vn , respectively.

DEFINITION 3.2. The spline function B(xo, ... , xr; t) is called a B-spJine of degree r - 1 with knots xo, ... , xr .

Next we show that B(xo, ... , xr; t) has a finite support.

THEOREM 3.3. Let Xo ~ ... ~ Xr and Xo < Xr. Then

B(xo, ... ,xr;t)=O

B(xo, ... ,xr;t) > 0

for t < Xo and t > xr;

for Xo < t < xr .

Proof. Since (tk -t)~ = 0 for all t > tk, it follows from the presentation (3.2.2) that B(xo, ... ,xr;t) = 0 on (xr,oo). Further, ift < Xo, we have (tk -t)~ = (tk -t)i for k = 1, ... , n and therefore B(xo, ... , xr; t) coincides with the divided difference at r + 1 points of the polynomial q(x) := (x - ty-l, which is of degree r - 1. Thus, B(xo, ... , xr; t) = 0 for t E (-00, xo).

It remains to show that B(xo, ... , xr; t) is positive on (xo, xr). Let t be an arbitrarily fixed point in (xo, xr). We shall use the fact that B(xo, ... ,

xr;t) is the coefficient of xr in the polynomial p(x) from 11"r, which interpolates the function B(x) := (x - t)+-l at the points xo, ... , xr. Clearly p(x) is not identically zero and does not coincide with the polynomial (x - ty-l on (xo,xr). Then the function p(x) - B(x) may have only isolated zeros in (xo, xr)' Since p(x) - B(x) vanishes at Xo, ... , Xr , Rolle's theorem yields that p(r-l)(x) - B(r-l)(x) has:

(i) at least two sign changes in (xo, xr), if the sequence XO, ... , Xr does not contain a point of multiplicity r;

(ii) at least one sign change in (xo, xr) and a zero at Xo (xr , respectively) if Xo = ... = Xr-l (Xl = ... = xr, respectively). But B(r-l)(x) is a nondecreasing step function with a jump at t. Thus the conclusions (i) and (ii) could take place only if the linear function p(r-l)(x) is strictly increasing, i.e., if p(x) has a positive leading coefficient B(xo, ... , X r; t). The theorem is proved.

Ch. 3, § 3.2) Definition of B-Splines 31

It can be seen that (by continuity) B(xo, ... , Xr; t) vanishes at the end-points Xo and X r , ifthe corresponding multiplicities Vo and Vr are less than r. Further, it follows from the accepted definition of the truncated power function that (x - t)~ = 1 for x ~ t and hence the value of the spline s(x) at the break points t is defined as a right-hand-side limit, i.e.,

s(t) = s(t + 0) := lim (s(x) : x -+ t, x> t).

Thus B(xo, ... ,xr;t) = 0 even ifvr = r, while B(xo, ... ,xr;t) > 0 ifvo = r. Given the sequence (finite or infinite) of points {Xi}, such that

... ~ Xi ~ Xi+! ~ ...

and Xi < xi+r for all i, we shall denote by Bi,r-1(t) (or simply by Bi(t» the B-spline

Bi,r-1(t) = (._t)~-l[Xi, ... ,Xi+r].

At times it will be more convenient to use the so-called normalized B-splines

The following explains the reason for such normalization. Consider the sum of all Ni,r-1(t) at a certain fixed point t. Suppose that Xi <

t < Xj+1. Then i

L:Ni(t) = L: Ni(t), i=i+1-r

since Ni(t):= Ni,r-1(t) = 0 for t fI. [Xi,Xi+r]. Applying now the recurrence relation (see (1.3.1»

we get

i L:Ni(t) = L: {(. _t)+-l [Xi+1, ... , Xi+r] - (. _t)+-l [Xi, ... , Xi+r-d} i i=i+1-r

= (. _t)+-l [Xj+l, ... , Xj+r] - (- _t)+-l [Xi+1-r,"" Xi]

= (. _ty-1 [Xj+1, ... , xj+r].

But the last expression is the divided difference of a polynomial of degree r -1. Thus it equals the leading coefficient of this polynomial, i.e., equals to 1. Therefore

for each t. Non-negative functions on (-00,00) with such a property are said to constitute a partition of unity. They are used in statistics and approximation theory.

32 B-Splines [Ch. 3, § 3.2

Exercise 3.2.1. Let Mr(t) := rB(xo, ... , xr; t) be the B-spline of degree r - 1 with knots Xlc = -r/2 + k, k = 0, ... , r, normalized by the condition

00

/ Mr(t) dt = 1. -00

Prove that:

-00

00

b) Mr(x) = / Mr_m(x - t)Mm(t) dt; -00

c) M;(x) = Mr- 1 (x +~) - Mr-1(X - ~). Hint. Let 6/(x) be the central difference operator of step 1 (Le., 6/(x) := /(x + 1/2) - f(x - 1/2)) and 61c+1/(x) .- 6lc 6/(x), k = 0,1, .... Show first that for /(x) = ei~f

( t)r. and hence 6r /( x) = 2i sin '2 . el~t .

Then a) follows from the relation

00

6r /(0) = / Mr(x)/(r)(x) dx, -00

taking into account that /(r)(x) = (itteid. In order to prove b), apply Fubini's theorem to verify that

00 00 00 00

/ (/ 91(X - t)g2(t) dt) ei~>'dx = / g1(e)ei>.{ de / g2(")ei>.,, d" -00 -00 -00 -00

for any two integrable functions g1 and g2' Therefore, by the result of a),

/00 ( /00 ) . >. (sin >"/2) r Mr-m(x - t)Mm(t) dt e'~ dx = >../2

-00 -00

00

= / Mr(x)ei~>'dx -00

for each real >... This yields b).

Ch. 3, § 3.3] B-Spline Basis

In the particular case m = 1, equality b) implies

00

Mr(z) = J Mr- 1(z - t)M1(t) dt -00

1/2 11:+1/2

= J Mr- 1 (z - t) dt = f Mr- 1 (t) dt. -1/2 11:-1/2

Then c) follows by differentiation of the last identity.

§ 3.3. B-Spline Basis

In the case r = 1, the B-splines are the familiar "roof functions". Precisely,

Ni,l(t) = (Zi+2 - Zi)(- -t)+[Zi, Zi+1,Zi+2]

= { (Zi+2 - t)/(Zi+2 - Zi+d

(t - Zi)/(Zi+1 - Zi)

for Zi+1 ~ t ~ Zi+2,

for Zi ~ t ~ Zi+1.

33

It is easily seen that every continuous piecewise linear function s(t) with knots at Z1, ... , Zm-1 may be written in the form

m

s(t) = LS(Zi)Ni.1(t). i=O

In other words, every spline from S1(Zt, ... , Zm-1) may be presented as a linear combination of the B-splines B i •1(t), i = 0, ... , m. We shall prove here a similar property for splines of any degree.

Let us start with an important lemma.

LEMMA 3.4. Let Z1 ~ ... ~ Zr and / E Sr_1(Z1, ... ,Zr). IT /(t) = 0 for all t ¢ (ZtoZr], then /(z) is identically zero on (-00,00).

Proof. Let (Z1, ... , zr) == «tt, V1), ... , (tn, vn». Clearly V1 + ... + Vn = r. In view of Theorem 2.10, / can be written in the form

n I'i-1 (j)

/(t) = p(t) + L Laij (z - t)+-1) 1-. i=1 j=O II:-t.

with some p E 1I"r-1 and real constants {aij}. The assumption that /(t) = 0 for each t > Zr yields p == O. Therefore

34 8-Splines [Ch. 3, § 3.3

Let us choose arbitrary points 6 < ... < er < Xl and define the polynomials Pk(X) := (x - ekY-t, k = 1, ... , r. According to Lemma 2.5, {pdt form a basis in 'II'r-l. On the other hand, the assumption f(ek) = 0 means that

n IIj-1

L(Pk) := L Laijp~)(tj) = 0 (3.3.1) i=l j=O

for k = 1, ... ,r. Therefore L(q) = 0 for each q E 'II'r-l. In particular L(qij) = 0, where the polynomial % is defined by the interpolation conditions

q~?(t!,)=6i!,6j>' for J.t=l, ... ,n, A=O, ... ,v!'-l.

But L(qij) = aij' Therefore aij = 0 for all i,j and hence f == O. The proof is completed.

COROLLARY 3.5. The B-spline of degree r - 1 with knots {Xi}(i has a minimal support among the splines from Sr-1(XO, ... ,xr ).

Indeed, assume that there is a nonzero spline f from the set Sr-1(XO, .. ' ,xr ), which is distinct from zero on a subinterval [a, b] of (xo, x r ) and vanishes outside [a, b]. Then Lemma 3.4 implies f == 0: a contradiction.

Moreover, as it is shown below, B(xo, ... , Xr ; t) is the unique (up to multiplication by a constant) spline in Sr-1(XO, ... , xr ) with minimal support.

COROLLARY 3.6. Let Xo ~ ... ~ Xr , Xo < Xr , and f E Sr-l (xo, ... , x r ) be a spline that vanishes identically outside the interval [xo, Xr]. Then there is a constant c such that

f(t) = c· B(xo, ... ,xr;t) on (-00,00).

Proof. Let (xo, ... ,xr ) == ((tl,Vt), ... ,(tn,vn + 1». Note that here the number of knots is one more than in Lemma 3.4. Then, proceeding in the same fashion as in the lemma, we see that the assumption f(t) = 0 for t = 6, ... ,er implies

n "i-l

L Laijp~)(tj) = -an,""p~",,)(tn) (3.3.2) i=l j=O

for k = 1, ... ,r. For fixed an ,lI" this is a linear system with respect to {aij}, which has the same determinant P as in (3.3.1). Showing that the homogeneous system (3.3.1) admits only the zero solution we actually proved in the lemma that P :f. O. Suppose that an ,lI" = O. Then (3.3.2) implies aij = 0 for i = 1, ... , n, j = 0, ... , vi-1 and therefore f == O. So, the corollary holds in this case with c = O. Consider now the case an ,lI" :f. O. We may divide (3.3.2) by an ,lI" and get a system of equations in unknowns {aij/an,II"}' Since P:f. 0, this system has a unique solution {Il'ij}, which does not depend on f. Therefore, the coefficients of the spline f satisfy the relations

Since B(zo, ... ,Xr ; t) is one of these splines f, we conclude that any f differs from B( Xo, ... , Zr; t) by a constant multiplier. The proof is completed.

After this brief characterization of B-splines, let us return to the main question. Let Z = {zi}f+r be a given knot sequence (i.e., a nondecreasing sequence of points

such that Zi < Zi+r) and let {Bi,r-l(tnf:l be the B-splines associated with z.

Ch. 3, § 3.3] B-Spline Basis 35

LEMMA 3.7. The functions Bl,r-l (t), ... , BN,r-dt) are linearly independent on (-00,00).

Proof. We apply induction on N. For N = 1 the lemma is evidently true. Suppose that it holds for any set of r+ N -1 knots. Let z = {xi}f+r = ((tl, VI), ... , (tn, vn)) with t1 < ... < tn and VI + ... + Vn = r+ N. Assume that there is a linear dependence between the B-splines {Bi,r-l(t)}f:l' i.e., that

N

I(t) := LaiBi,r-l(t) = 0 i=l

for all t from (-00,00), but at least one ai is distinct from zero. The function 1 is a polynomial of degree r - 1 on (tn-I, t n ) and

I(t) = aNBN(t) + ... + aN_V+lBN_V+l(t)

= Cl (tn - tr- l + ... + cv(tn - tr-V

for t E(tn - 1 , tn), where Vn is abbreviated to V and {Cj} are some coefficients. Since tn is a knot of multiplicity V only of BN(t), Cv = aN·'Y, where 'Y is the coefficient of (tntYf.-v in the divided difference expression of BN(t). According to the observation (1.3.6), 'Y ::p O. Then the identity I(t) == 0 on (tn-l,tn), implies Cl = ... = Cv = 0 and hence aN = O. Now 1 is a linear combination of N - 1 B-splines and applying the inductional hypothesis we get al = ... = aN _ l = 0, which contradicts the assumption about {ail. The proof is completed.

THEOREM 3.8. Let a < Xr+l ~ ... ~ Xm < b be fixed points such that Xi < Xi+r for all admissible i. Choose arbitrary 2r additional points Xl ~ ... ~ Xr ~ a and b ~ xm+1 ~ ... ~ Xm+r and define Bi(t) := B(Xi, ... , Xi+r; t). The B-splines Bl(t), ... ,Bm(t) constitute a basis for Sr-l(Xr+1""'xm) on [a,b].

Proof. The functions Bl , . .. , Bm are linearly independent in [a, b]. Indeed, assume the contrary. Then there exist numbers {ai}]" with at least one ai distinct from zero, such that

m

s(t) := LaiBi(t) = 0 in [a, b]. i=l

Since s is a piecewise polynomial function, it vanishes actually in the larger interval (x r , xm +1)' Denote by 1 the function that coincides with s on (-00, xr ) and vanishes on (xr , 00). Evidently 1 is a spline of degree r-l with r knots and such that I(t) = 0 for all t f/. [Xl, x r ]. Then, by Lemma 3.4, 1 == 0 on (-00,00) and, consequently, s == 0 on (-00, X,.). Similarly one shows that s == 0 on (xm +1'oo). Thus s(t) = 0 for all t E(-OO,oo). Then Lemma 3.7 implies ai = 0 for i = 1, ... ,m, a contradiction. So, we proved that B l , . .. , Bm are linearly independent in [a, b]. Since the diminution of S"-1 (X,.+1" .. , xm) equals m (see Corollary 2.11), the theorem is proved.

36 B-Splines (Ch. 3. § 3.4

§ 3.4. Recurrence Relations

One of the first tasks one faces when calculating with splines is the evaluation of a spline at given points. The presence of recurrence relations between the B-splines makes it possible to build simple and fast algorithms for this and other numerical operations.

3.4.1. The basic recurrence relation. Let z = {Xi} be a given non decreasing sequence of points such that Xi < Xi+r and let {Bi,r-d be the corresponding Bsplines associated with z. Set

!(X):=X-t, g(X) := (x - t)+-2,

where t is a real parameter. Clearly (x - t)+-l = !(x)· g(x) and therefore

Bi,r-l(t) = (f. g)[Xi, ... , Xi+r].

Note that the divided difference of! at more than 2 points is equal to zero since! is a linear function. Then, by Steffenson's rule (Theorem 1.8),

r

(f. g)[Xi, ... , Xi+r] = L:![Xi' ... ' Xi+k]· g[Xi+lo, ... , Xi+r] 10=0

= !(Xi) ·g[Xi, ... ,Xi+r] + ![Xi, Xi+l] .g[Xi+l, ... ,Xi+r]

-!( .)g[Xi+1, ... ,Xi+r]-g[Xi, ... ,Xi+r-l] 1 [. . ] - X. + . 9 X.+l, ... , x.+r xi+r - Xi

={ Xi-t +1}9[Xi+1 , .•• ,Xi+r]- Xi-t g[Xi, ... ,Xi+r-l] ~~-~ ~~-~

Xi+r - t [ ] t - Xi = gXi+l,··.,Xi+r + g[Xi, ... ,Xi+r-l]. Xi+r - Xi Xi+r - Xi

So, we proved the recurrence relation

(3.4.1)

Remark that the coefficients in this relation are positive linear polynomials of t in (Xi,Xi+r) and their sum equals to 1. Thus, the value of Bi,r-l at every point t from (Xi, Xi+r) is a convex linear combination of Bi+1,r-2(t) and Bi,r-2(t). This observation, together with the fact that

B. (t) = { 1/(Xi+1 - Xi) ',0 0

for t E [Xi, Xi+1), for t rt [Xi, Xi+1)

(3.4.2)

may be used to give an inductive proof of the positivity of Bi,r(t) on (Xi, Xi+r).

Ch. 3, § 3.4) Recurrence Relations 37

Another useful application of the recurrence relation (3.4.1) is the following simple scheme for the computation of the value of the B-splines at a fixed point.

The components of the first column in this table are given in (3.4.2) and the components of any next column are computed on the basis of the preceding one, by (3.4.1).

Finally, note that the normalized B-splines satisfy the relation

and

Ni,O(t) = {~ for t E [Xi, Xi+1), for t rt. [Xi,Xi+1).

(3.4.3)

Exercise 3.4.1. Let {Xi}~+r be a given knot sequence such that Xl ::;; ... ::;; Xr < a < Xr+1 ::;; ... ::;; Xn < b < Xn+1 ::;; ... ::;; Xn+r. Show that for each X andt in [a, b],

n

(t - xt-1 = 2:IPi,r(t)Ni,r-1(X), i=l

where . (t).- {(t - Xi+1) ... (t - xi+r-d II', r .-r, 1

for r> 1, for r = 1.

Hint. Apply induction on r. The identity is obvious for r = 1. Assume that r >1. Then, by the recurrence relation (3.4.3),

n

(1'r := 2:IPi,r(t)Ni,r-1(X) i=l

38 B-Splines [Ch. 3, § 3.4

Taking into account that NO,r-2(X) = Nn +1,r-2(X) = 0 for x E [a, b], get further

_ ~. (t) {(t - xi)(xHr-1 - x) + (t - Xi+r-1)(X - Xi)} AT. () Ur - L..J!.p.,r-1 lV',r-2 X i=1 Xi+r-1 - Xi

n

= (t - X)L<Pi,r-1(t)Ni,r-2(X) = (t - X)Ur_1, i=1

and the induction is completed.

Exercise 3.4.2. Making the same assumptions as in the previous exercise show-that for each 1 ~ m ~ r,

where

n

xm - 1 = L O i,rNi,r-1(X) i=1

. = (_1)m-1 (m - I)! (r-m)(O) o.,r (r-1)! !.p.,r ,

on [a, b],

i= 1, ... ,n.

Hint. This representation ofthe basic polynomial functions follows from the identity in Exercise 3.4.1 if one differentiates r - m times with respect to t and put t = O.

3.4.2. Differentiation of splines. It follows from Theorem 3.8 that every spline s( t) of degree r - 1 may be written in the form s( t) = E OiNi,r-1 (t), as a linear combination of appropriately chosen B-splines. We derive here a simple formula for the derivative of N i ,r-1 and show how to present s'(t) in terms of normalized B-splines of degree r - 2.

By the recurrence relation for the divided difference, d

(XHr - Xi) dtBi,r-1(t) =

Therefore

= :t(·_t)+-1[Xi+1,,,,,Xi+r]- ~(._t)+-1[Xi,,,.'XHr_d = (r - 1){ - ( . -t)+-2[XHi>"" Xi+r] + ( . -t)+-2[Xi,.'" XHr-d}

= (r - 1) { - BH1 ,r-2( t) + Bi,r-2( t)}.

or equivalently,

(3.4.4)

dd Ni,r-1(t) = (r - 1){ - 1 Ni+1,r-2(t) + 1 Ni,r-2(t)}. t Xi+r - XH1 Xi+r-1 - Xi

Now using this relation we get

d "" "" 0i - Oi-1 -d L.J 0 iNi,r-1(t) = (r -1) L..J Ni,r-2(t). t . . Xi+r-1 - Xi • •

Roughly speaking, in order to find the derivative of a spline s(t) = E OiNi,r-1 (t) one just "differentiates" the coefficients {Oi} and decreases the order of the normalized B-splines by 1.

Ch. 3, § 3.4) Recurrence Relations 39

Exercise 3.4.3. Let s(t) = E~l OiBi,r-l(t) on [a,b], where Zl ~ ... ~ Zr < a < Zr+l ~ ... ~ Zm < b < zm+l < ... < zm+r. Show that

Hint. The function :I:

S(z):= J s(t) dt -00

is a spline of degree r with knots Zr+l, ... , Zm in [a, b]. Then, by Theorem 3.8.,

m

S(z) = EaiBi,r(Z) i=O

with some coefficients {ail and fixed Zo < Zl, zr+m+l > zr+m . Use differentiation formula (3.4.4) to get

S'(z) = r t ( ai - ai-l ) Bi,r-l(Z) i=l Zi+r+l - Zi Zi+r - Zi-l

~ ~ () + BO,r-l(Z) - Bm+1,r-l Z . Zr+l - Zo Zm+r+l - Zm

Since 00 = 0 and the B-spline representation of S'(z) is unique, we see that ao = 0 and

°i = r(Zi+r:: - Zi - Zi+:~~i-J, i= 1, ... ,m.

It remains to observe that

i= 1, ... ,m,

and this yields the desired relation.

3.4.3. Tschakaloff's formula. Many of the properties of B-splines can be derived from an interesting contour integral representation, which we give below.

Let (zo, ... ,zr) == «tt, lid, . .. ,(tn, lin», tl < ... < tn . Set for simplicity u(t) := B(zo, ... ,Zr; t)/(r - 1)! According to (1.3.5) and (1.3.8),

n ".-1 (tk _ t)~~-l u(t)=L Eau (r-A-1)!'

k=l ~=o

where the coefficients {ak~} are the same as in the expression

40 B-Splines

Here O(z) = (z - tt}"l . " (z - tn)"". For fixed t, consider the function

(z - tt- 1

<p(z) = (r _ 1)!O(z)'

[Ch. 3, § 3.4

Denote by r", the residue of <p with respect to z = t", . By the residue theorem,

r", = 2~i f <p(z) dz, Cle

where G", is a simply closed rectifiable curve in the plane, isolating t", from the other poles {t;} of <po Then

1 f (z - tt- 1 n "j-1 ap.A! d r" = - '" '" z 211'; (r -1)! ~ L.J (z - t·)A+1

C le J=1 A=O 1

~"~1 a'A A! f (z-W- 1

= ~ L.J (r -1)!' 211'i (z _ t.)A+1 dz J=1 A=O C le J

and, by Cauchy's integral theorem,

Now it is seen that for t E[tm - b t m ]

where r m is any simply closed rectifiable curve which isolates the points tm, tm+1' ... , tn from the other poles t1, ... ,tm- 1 (t1,"" tm- 1 lie outside the domain bounded by r m). Therefore,

1 f (z-W- 1

B(xo, ... , xr ; t) = 211'; (r _ 1)!O(z) dz (3.4.5)

r ...

which is Tschakaloff's contour integral representation formula.

Ch. 3, § 3.5) Variation Diminishing Property 41

N ext we prove the following differentiation rule

(3.4.6)

and call it Tschakalo/f's formula. In order to do this, note that

d Bi,r-dt) 1 {( ) I () ( ) ( )} -d ( ) -1 = ( ) %i+r - t Bi r-l t + r - 1 Bi,r-l t . t %i+r - t r %i+r - t r ,

Now inserting the expressions for B: r-1(t) and Bi,r-1(t) from (3.4.4) and (3.4.1), we obtain the required relation. '

§ 3.5. Variation Diminishing Property

Let [a,b] be a fixed interval. We say that the function J(%) has a sign change at the point e if for each e > 0 there exist points tl E (e - e,e), t2 E (e,e + e) such that J(tt}J(t2) < O. Denote by S-(f) the number of sign changes of J in (-00,00). As we mentioned already, S-(O"}, ... O"N) (or simply S-(o» denotes the number of strong sign changes in the sequence of real numbers 0 = (0"1, ... , O"N). We shall reveal here an interesting relation between the number of sign changes of the spline function J(t) = E O"iBi,r-l (t) and the number of sign changes in the sequence of its coefficients {O"i}. Our proof is based on the following important property of B-splines which has various other applications.

LEMMA 3.9. Let %0 ::;; ... ::;; %r be fixed points and %0 < %r. Then for each e E (%0, %r) there exists a number 0", 0 < 0" < 1, such that

B(t) = O"Bo(t) + (1 - O")Bl(t)

for every t, where B(t) = (. -t)+-1[%0, ... ,%r],

Bo(t) = ( . _t)+-1 [%1, ... ,%r,e]'

B1(t) = (._t)+-1[%0, ... ,%r_1,e].

Proof. The assertion is an immediate consequence from the corresponding property of the divided difference, described in Theorem 1.10.

In what follows, we shall use the notion refinement of a knot sequence.

DEFINITION 3.10. The sequence y = {Yih~1' Y1 ::;; ... ::;; YM' is a refinement of :I: = {%i}f:1' %1 ::;; ... ::;; % N' if there is a way to get y from :I: by adding new points e to :1:, such that %1 < e < %N .

The next proposition can be derived from Lemma 3.9. It follows also directly from Corollary 1.11.

42 B-Splines [Ch. 3, § 3.5

COROLLARY 3.11. Let Xo ::::; ... ::::; Xr be fixed points. Suppose that y = {Yi}f+r is an arbitrary refinement ofz such that Yi < Yi+r for all admissible i. Let {B;(t)}f be B-splines of degree r - 1 for the sequence y. Then there exist positive numbers {adf such that a1 + ... + aN = 1 and

N

B(xo, ... ,xr;t) = LaiBi(t) i=1

for all t.

Now we can present the variation diminishing property of B-spline sequences.

THEOREM 3.12. Let z = {Xi}~+N be an arbitrary knot sequence and let {Bi(t)}f be B-splines of degree r - 1 corresponding to z. Then

N

S- (~aiBi(t») ::::; S-(at, ... , aN) 1=1

for each choice of the numbers at, ... , aN'

Proof. Set N

/(t) := LaiBi(t). i=l

(3.5.1)

Suppose that e, Xl < e < Xr+N, is a new point added to z. Denote by aO = {anf +1 the coefficients of the function /, represented as a linear combination of the new Bsplines {Bnf+ l , corresponding to the refinement z Ue, It follows from Lemma 3.9 that there exist non-negative numbers ai, b; such that

Therefore

Using this fact, we get

S-(a~, ... ,a~+1) ::::; S-(ao, a~, al,··· ,aN,a~+1,aN+1)

= S-(at, a 2, ••• ,aN), (3.5.2)

where ao := 0 and aN+1 := O. So, the adding of new knots does not increase the number of sign changes in the sequence of coefficients.

Suppose now that tl < ... < tn are arbitrarily fixed points in (Xl, Xr+N). Let

Adding additional points to z we can get a sufficiently fine refinement y of z, such that the B-spline sequence {Bn, corresponding to y, satisfies the condition:

implies supp B~ C (tj - g, tj + g)

Ch. 3, § 3.5} Variation Diminishing Property 43

for each i and j. Then /(tj) = La?B?(tj),

i

where the summation is expanded over only those i for which BP(tj) :F O. Since the condition /(tj) > 0 implies af > 0 at least for one of these i, it is clear that

S- {/(tl),'" ,j(tn» ~ S-(o:o).

Hence, in view of (3.5.2),

The theorem is proved.


The definition and the fundamental properties of B-splines with equispaced knots are given in Schoenberg [1946]. There, these functions are referred to as "basic spline curves". The name was shortened later to "B-splines" in Schoenberg [1967]. The linear independence and other properties of B-splines listed in this section were known already to Curry and Schoenberg [1947, 1966]. Much earlier the Bulgarian mathematician Lubomir Tschakaloff arrived at the functions B( Zo, ... ,Zr; t) by studying divided differences with multiple nodes (see Tschakaloff [1938]). He proved the recurrence relation (3.4.6) and then derived, by induction, the positivity of B;(t) on (Zi' Zi+r). His proof is based on the contour integral representation of B-splines, a formula, which was rediscovered later by Meinardus [1974] and exploited further by many others (see Schempp [1982]). We find it useful to include here the original proof of the recurrence relation (3.4.6), given by Tschakaloff.

PROPOSITION (Tschakaloff [1938]). Let Zo ~ '" ~ Zr, Zo < Zr and

1 _ u(t) := (r _ I)! ( . -t)+ l[zO,"" zr],

Ul(t) := (r ~ 2)! ( . -t)+-2[zO, ... ,Zr-l].

Then, for each t E(-oo, 00),

d u(t) Ul(t) dt (zr - W- 1 = (Z,. - W .

Proof. Let (zo, ... ,zr) == «tt,vt}, ... ,(tn,vn», tl < '" < tn. The case t fI. [zo,Z,.] is trivial, since u(t) = Ul(t) = O. Let t E [zo, Z,.]. We have

d u(t) _ u'(t)(zr - t)"-1 + (r -1)u(t)(zr - t)"-l dt (zr - W- 1 - (zr - t)2r-2

u'(t) 1 u(t) = (Z,.-W-1 +(r- )(Zr-W'

44 B-Splines [Ch. 3, § 3.5

Assume that tm-l ~ t ~ tm. Then, making use of (3.4.5), we get

!!.. u(t) = 1 1! (t - xr)(z - W- 2 + (z - W- 1 dz dt (x r - W-1 (x r - W 211"i (r - 2)!n(z)

r",

= 1 1 !(Z-tt-2(z-xr)dZ= Ul(t) . (xr - W 211"i (r - 2)!n(z) (xr - ty

r",

The proof is completed. There are various ways to show that B;(t) > 0 on (Xi,Xi+r). The present proof of

Theorem 3.3 is from Bojanov [1990b]. The note that the normalized B-splines form a partition of unity is due to Marsden and Schoenberg [1966]. The B-spline basis for the space Sr-l(Xl, ... ,xm) was constructed by Curry and Schoenberg [1966]. The differentiation formula (3.4.4) is given in de Boor [1972]' The variation diminishing property of B-spline expansions was discovered by Karlin [1968]. A simpler proof and improvement can be seen in de Boor [1976a]. The proof we give here (Theorem 3.12) follows that of de Boor and De Vore [1985]. The identity in Exercise 3.4.1 is due to Marsden [1970]. Another interesting proof can be seen in Barry, Dyn, Goldman, and Micchelli [1991].

The survey paper of de Boor [1976b] and the book by Schumaker [1981] contain many further properties and applications of B-splines including some of the exercises in this chapter.

The integral representation (3.1.1) of the linear functional that annihilates algebraic polynomials from 1I"r was given by Peano [1913]. See also Birkhoff [1906].

Chapter 4

INTERPOLATION BY SPLINE FUNCTIONS

Even an elementary study of the interpolation problem

S(ti) = f;, i = O, ... ,n+ 1,

in the simplest linear case, i.e., when s E 8 1(X1, ... ,xn), shows that the solvability of the corresponding system depends entirely on the mutual location of the interpolation nodes t = {ti}~+l and the spline knots :z: = {xi}i. For example, in the case ti = Xi, i = 1, ... , n, the problem has a unique solution: the piecewise linear function with vertices at (ti' Ii), i = 0, ... , n + 1. On the other hand, in the case where three or more interpolation nodes are situated between two consecutive xi's, the problem becomes unresolvable. We shall give here a complete characterization of the Hermite interpolation problem by spline functions with multiple knots. The B-spline representation of s leads us to the study of the corresponding collocation matrix {Bi(tj n.

We start with a brief discussion of the total positivity of matrices and kernels, a question which is closely related to the interpolation problem.

§ 4.1. Total Positivity

Suppose that the function K (x, t) is defined on X x T, where X and T are given subsets of the real line ~.

DEFINITION 4.1. We say that K(x,t) is a totally positive kernel (TP-kernel) on X x T if

(4.1.1)

for each choice of the points Xl < ... < Xn and t1 < ... < tn in X, T, respectively, and each natural number n. If the sign in (4.1.1) is strictly positive, we say that K(x, t) is strictly totally positive (STP).

Recall that the set of functions <P1(t), ... , <Pn(t) forms a Tchebycheff system on T if and only if

det {<pi(tj)}~_l ~-1 > ° t_ ,1-

for each t1 < ... < tn in T. Thus, it follows from Definition 4.1 that {K(Xl, t), ... , K(xn' tn is a Tchebycheff system on T for each choice of the points Xl < ... < Xn in X, provided K is a STP-kernel in X x T. Therefore, Lagrange's interpolation

45

46 Interpolation by Spline Functions (Ch. 4, § 4.1

problem in T by linear combinations a1K(zl, t) + ... + anK(zn, t) has always a unique solution.

The TP-kernels possess other interesting properties. We mention one of them below.

THEOREM 4.2. Let K(z, t) be a continuous STP-kernel in [a, b) x [c, d) and let f be a continuous function which has exactly N sign changes in [e, d). Then the function

d

F(z):= J K(z, t)f(t) dt e

has no more than N distinct zeros in [a, b).

Proof. Assume the contrary. Then there exist N +1 distinct points Zl < ... < zN+1 in [a, b) such that

d

F(z,):= J K(z"t)f(t)dt = 0 for i = 1, ... ,N + 1. ( 4.1.2)

e

On the other hand, since f(t) has N sign changes in [e,d), there exist points {~s}f, e =: ~o < ~i < ... < eN < ~N+1 := d, such that J(~,) = 0 for i = 1, ... , Nand

(4.1.3)

for k = 0, ... , N and appropriate F: E {-I, I}. Remark here that f(t) is not identically zero on (~k'~k+l) and, consequently, f(t)::f; 0 on a subinterval of (~k'~k+l)'

Introduce the function

Expanding the determinant along the elements of the last column we see that p(t) is a linear combination of the functions {K(z" tnf:"tl. Then the equalities (4.1.2) imply

d J p(t)f(t) dt = O. (4.1.4)

e

On the other hand, note that p(~i) = 0, i = 1, ... , N, since for t = ~i the last column of the determinant coincides with the i-th one. In addition, it follows from the strict total positivity of K (z, t) that p( t) ::f; 0 if t ::f; ~, for i = 1, ... , N. Moreover, ift E(~k'~k+l) and ~(t) is the determinant which is obtained fromp(t) by placing the last column on position k + 1, then ~(t) > O. Therefore, p(t) changes sign alternatively in (~k'~k+l)' k = 0, ... ,N. Precisely, signp(t) = (_I)N-k for ~k <

Ch. 4. § 4.1] Total Positivity 47

t < e1: 1. Then, in view of (1.4.3), e( _l)N p(t)f(t) ~ 0 on [c, d] with strict inequality on certain subintervals. Therefore

d

J p(t)f(t) dt ~O, e

which contradicts (4.1.4). The theorem is proved. Given the matrix A = {aij }~lJ~l and integers 1 ~ i1 < ... < in ~ N, 1 ~ it <

... < im ~ M, we shall denote by

A (~1o ... '~n ) 310··· ,)m

the sub matrix {a>.,,: ~ = i1, ... , in, JJ = i1, ... ,im}. DEFINITION 4.3. The matrix A = {aij}~lJ~l is said to be totally positive (TP-matrix), if

det A ( ~1' ... ,i.n ) ~ 0 31,··· ,3n

(4.1.5)

for each choice of the indices 1 ~ i1 < ... < in ~ N, 1 ~ it < ... < in ~ M and the number n, 1 ~ n ~ min (N ,M).If the sign in (4.1.5) is strictly positive, A is called strictly totally positive (STP) matrix.

It follows from the definition that the TP-kernel K(z, t) generates TP-matrices {K(Zi' tj nf:1.f=1.

The next theorem shows that the totally positive matrices have a variation diminishing property in a certain sense.

THEOREM 4.4. Let A = {aij }~dl=l be a totally positive matrix and let z= (zl' ... ,

zn) be a given sequence of real numbers. Set b = (bl, ... , bN ) := Az, m:= S-(z). Assume that for each choice of the elements bio, ... ,bim (1 ~ io < ... < im ~ N) satisfying the condition bi"_l . bi" < 0, k = 1, ... , m, there exist numbers Zjo' ... , Zjm (1 ~ jo < ... < jm ~ n), which are distinct from zero and change sign alternatively, such that

det A (~o, ... ,i.m) > O. 30,··· ,3m

(4.1.6)

Then S-(Az) ~ S-(z).

Proof. Since S-(z) = m, the set z = {Zl' ... ' zn} may be divided into m+1 groups

m

z=U{Zj:jEJ1:}, 1:=0

where J1: consists of consecutive integers, such that each group contains at least one nonzero number and (-l)1: zj ~ 0 for every j EJ1:, k = O, ... ,m. We may assume here without loss of generality that all elements in the first group {Zj: j EJo} are non-negative.

48 Interpolation by Spline Functions [Ch. 4. § 4.1

Suppose that the theorem is not true and hence S-(b) ~ m + 1. Then there are indices io, ... , im+l such that the numbers {bi: i = io, ... , im+tl change sign alternatively. It follows from the definition of b that

m

L(-I)k L aijlxjl- bi = 0, i = io, ... ,im +1.

k=O jEJ~

These equalities can be considered as a system of m+ 2 linear equations with respect to the coefficients of the interior sums and that of bi. The system has the solution {I, -1,1, ... , (_I)m, -I}. Then its matrix, denote it by B, is singular. So, we have

det B=O. (4.1.7)

On the other hand, expanding det B along the elements of the last column, we get

m+l det B = L(-I)m+l+kbi~·det Ak

k=O m+l

= c L(-I)m+llbi~l·det Ak, k=O

(4.1.8)

where c := (_I)k. sign bio and Ak is obtained from B removing the last column and row k. Precisely,

Ak = { ,L aijlxjl: i = io,··· ,ik_l'ik+!,··· ,im+l' p = 0, ... ,m}. JEJp

Using elementary properties of determinants we find

det Ak = L '" L IXjol .. ·lxjJ joEJo jmEJm

X det A (io, ... , ik:-l' ik+1.' ••• , im+l ) Jo,··· ,)m

It is seen from this expression that det Am+l :I ° because at least one of the terms

IXjol ... Ixjm I det A (~o, ... , i.m ) Jo,··· ,)m

in det Am+! is strictly positive, according to the assumption (4.1.6). This, combined with (4.1.8), implies det B:I 0, which is contradictory to (4.1.7). Therefore S-(b) ~ S-(z). The theorem is proved.

COROLLARY 4.5. Suppose that A is a STP-matrix with n columns. Then, for each

The assertion follows immediately from the preceding theorem since the condition (4.1.6) is obviously fulfilled, because of the strict total positivity of the matrix A.

Ch. 4, § 4.2] Hermite Interpolation 4P

§ 4.2. Hermite Interpolation

The study of the spline interpolation problem is equivalently related to the investigation of the non-singularity of the collocation matrix {Bi(tj n. We make here the following important stipulations:

I. If tp-1 < tp = ... = tp+II _1 < tP+II for some p, writing {Bi(tj nj=1,7=1 we shall mean the matrix, in which the rows corresponding to tp = ... = t p+II - 1 are interpreted as

... ,

... , .............................

B (II-1) (t ) B(II-1)(t ) 1 P' ... , n P

II. The value of every piecewise polynomial function s and its derivatives at a point e is defined as a right-hand-side limit, i.e.,

for j = 0, 1, ....

THEOREM 4.6. Let :Il= (z1"" ,zN+r) be a given knot sequence (i.e., Z1 ~ ... ~ zN+r and Zi < Zi+r for i = 1, ... , N) and {Bi(tnt' be B-splines of degree r - 1, defined bY:ll. Suppose that n ~ N and t1 ~ ... ~ tn is an arbitrary system ofpoints such that ti < ti+r for all admissible i. Then

(4.2.1)

for each choice of the integers 1 ~ i1 < ... < in ~ N. Moreover, .6. > 0 if and only if

k =1, ... ,n. (4.2.2)

Proof. The idea is to present {Bi;(tkn~=1.r=1 as a sum with positive coefficients of triangular matrices. In order to this we use the decomposition described in Lemma 3.9 and a refinement of :Il.

Adding new additional points to the original set of knots :Il, we may get a sufficiently dense refinement y, such that every interpolation node tk lies at a knot of multiplicity r from the refinement y and the B-splines {Bi}, corresponding to y, satisfy the condition

For fixed k, let Y>. = '" = Y>'+r-1 be the knots from y which coincide with tk. Then B>.(t) := Br- 1 (Y>., ... ,Y>'+r-1, Y>'+r j t) and clearly

(4.2.3)


Note otherwise that

for j = 1, ... , r - 1. ( 4.2.4)

To verify this inequality observe that YJ. is a knot of iiJ.+j of multiplicity exactly equal to r- j. Thus YJ. is a zero of iiJ.+j(t) of multiplicity j (exactly) and consequently

for YJ. ~ t ~ YJ. + £,

where £ is sufficiently small and p is a polynomial, such that p(yJ.) :/: O. Actually

p(YJ.) > 0, since the B-spline is positive on its support. Then iiV2j(tk) = j!p(yJ.) > 0 and the claim is proved. Let v be the multiplicity of the interpolation node tk' Then, taking into account the relations (4.2.3) and (4.2.4), we see that the table {ii;n(tk): j = 0, ... , v-I, i = 1,2, ... } may be illustrated as follows:

j\i 1 2 3 A A+l A+V+ 1

0 0 0 0 0 + 0 0 0 0 0

1 0 0 0 0 z + 0 0 0 0

2 0 0 0 0 z z + 0 0 0

3 0 0 0 0 z z z 0 0 0

v-I 0 0 0 0 z z z z + 0

Fig. 4.1.

where the positive quantities are marked by + and those having an unspecified sign by z. Thus all elements of this table, except a triangle of size v, are zero.

Let Jk denotes the set of all indices m for which the support of iim is contained in the support of Bik • Clearly Jk is a sequence of consecutive integers. Denote their number by IJkl. According to Lemma 3.9,

Bik(t) = L am(k)iim(t), mEJk

with some positive coefficients am(k). Then each column of the matrix

{B, (t )}n n 'j k k=1,j=1

can be presented as a linear combination of IJkl consecutive columns of the matrix {iim(tk): k = 1, ... , n, m = 1,2, ... }. Using an elementary property of determinants, we present ~ as a sum of determinants. Precisely

~ = L a(M) . ~(M), (4.2.5) M

Ch. 4. § 4.2) Hermite Interpolation 51

where M = (ml, ... , m n ), mA: runs over he,

and a(M) = aml (1). am2 (2) .. ... am .. (n). Remark that a(M) > O. The most important observation in the proof of this theorem is that the sum in (4.2.5) is expanded over only those M for which ml < ... < mn • To show this let us consider the construction of the decomposition (4.2.5) of A, step by step, adding one new knot at a time, in order to get y from z. Suppose that after the i-th step we have a decomposition Ai of A which has the stated properties. Now let us add the next new point e. Each of the B-splines on the i-th step, which contains e in its support will be presented (by Lemma 3.9) as a convex combination of two consecutive B-splines, corresponding to the (i + 1)-st step. Thus some ofthe determinant A(M) in Ai will be written in a new form, namely, one or more consecutive columns of A(M) will be presented as sums of two consecutive columns, corresponding to the new B-splines (those on step i + 1). By an elementary property of determinants,

(4.2.6)

where Mq = (mql' ... ,mqn ), q = 1, ... ,p, are a set of indices, corresponding to the situation on step i + 1. By the assumptions, the indices M = (ml,"" mn ) satisfy the inequalities ml < .. , < mn • Then mql ~ ... ~ mqn for all q. If there is at least one equality between the elements of Mq, then clearly A(Mq) = 0, since the determinant A(Mq) would have two equal columns. Therefore, the sum (4.2.6), and, consequently, the decomposition Ai+l contains determinants with strictly increasing parameters mql, ... , m qn . Moreover, Ai+! is based on all such admissible increasing sequences (since Ai does). To complete the induction, we just note that for i = 0 (i.e., for the presentation (4.2.1» the claim was obviously true.

Now let us return to Fig. 4.1. It illustrates v consecutive rows, corresponding to the v-tuple interpolation node tA: , of a table B. It is seen that A(M) from (4.2.5) is the determinant of the matrix built up of the columns of B with numbers ml, ... , mn .

Therefore

A = 2: a(M)· det B ( 1, ... , n ), AI ml,·· .,mn

(4.2.7)

where M extends over the set of all (ml,"" mn ) such that

(4.2.8)

Let (tl,"" t n ) = «Tl' vt}, ... , (TI£' Vl£» and let Tp = YAp for p = 1, ... ,J-l. Then it is seen from Fig. 4.1, that

det B ( 1, ... , n ) # 0 ml,···,mn

if and only if

52 Interpolation by Spline Functions [Ch. 4, § 4.2

Moreover, Ll( M*) is a determinant of a triangular matrix and, in view of (4.2.3) and (4.2.4), Ll(M*) > O. Now assume that Bi,,(tk) = 0 for some k, i.e., that (4.2.2) does not hold. Then mi ~ Jk and hence M* is not in the set (4.2.8). This yields Ll = O. On the other hand, iftk E SUppBi", i.e., if Bi,,(tk):f; 0, then for sufficiently dense refinement y, the support of B::.a ,lies in the support of Bi and therefore mi. E Jk. " " ~ Thus the assumption (4.2.2) and (4.2.7) imply

The theorem is proved. We proved already the variation diminishing property ofthe matrix {Bi(tj n. Note

that the same result could be derived on the basis of Theorem 4.4 as a consequence of the total positivity property, just established.

Next we give the theorem which characterizes the Hermite interpolation problem for spline functions.

THEOREM 4.7. Let a < {I ~ ... ~ {n-r < b and a ~ t1 ~ ... ~ tn ~ b be given points, such that no more than r consecutive points from the ordered set {{d u {td coincide. Assume that (t1," ., tn) = « T1, VI), ... , (TIJ' vIJ». Then, the interpolation problem

k=l, ... ,I', j=0, ... ,vk-1, (4.2.9)

by splines s from S1'-1({1'''' ,{n-1')' has a unique solution for any fixed Uk;} if and only if

{i-r < ti < ei

for all i for which the inequality is meaningful.

Proof. Choose arbitrary additional points Xl ~ ... ~ X1' ~ a, b ~ Xn+1 ~ ... ~ Xn+1' and define the B-spline sequence Bi(t) := B(Xi' ... , xi+rj t), i = 1, ... ,n, for the knots {Xi}?~;' where Xi = {i-r for i = r + 1, ... , n. According to Theorem 3.6, {Bd1 form a basis in Sr-1({1"" ,{n-1')' Then the interpolation problem (4.2.9) may be written as

n

E niB}j) (Tk) = fkj i=l

and the assertion follows from Theorem 4.6.


We shall consider here functions of the form

1'-1 s(t) = Eht>· + E Cij(Xij - t)~-j-1, (4.3.1)

k=O ei;=l

where the real numbers {h} and {Cij} are free parameters while the knots :I: = (Xl, .•. , xm ), Xl < ... < Xm , and the incidence matrix E = {ei; }~lJ~"l are fixed.

Ch. 4. § 4.3) Sir1chofT Interpolation 53

Clearly the set (4.3.1) coincides with the class of functions f defined on (-00,00) and such that:

i)fE1rr_1 on (Xi,Xi+1) for i=0, ... ,m(xo:=-00,Xm +1:=00);

ii) f U) (Xi + 0) = f U) (Xi - 0), if ei,r-j-1 = O.

We call such functions splines of degree r-1 with knots at (z, E). A natural extension of the ordinary B-splines will be introduced, which allow us

to construct a convenient basis for the space of splines (4.3.1) and to characterize a certain general interpolation problem.

4.3.1. B-splines with Birkhoff's knots. The classical Newton interpolation problem led to the notion of divided difference, defined as the leading coefficient in the corresponding interpolating polynomial. Then the B-spline was introduced as the Peano kernel of the divided difference functional, i.e., as the divided difference of the truncated power kernel. This concept allows various meaningful extensions of B-splines, starting from different interpolation processes.

We discuss here divided differences with respect to a given pair (z, E) of points z = {Xi}l' and a matrix E, associated with the Birkhoff interpolation problem. Writing (z, E) we assume that the number m of points in the set z is equal to the number of rows ei := (eiQ, ... , ei,r-1) in E.

The results are derived under certain restrictions on E. They are well known in the theory of Birkhoff interpolation and some of them, as P6lya condition and conservative matrix have been already used in Chapter l.

We say that the matrix E = {eij }~l.J;Ol satisfies the strong P61ya condition (SP-condition) if:

m '" L: L: eij > k + 1 for k= O, ... ,r- 1. i=l j=O

DEFINITION 4.8. The pair (z, E) is regular (respectively, s-regular) if:

i) E is conservative;

ii) E satisfies the P6lya condition (respectively, SP-condition).

DEFINITION 4.9. Let (z, E) be a regular pair and lEI = r+l. The linear functional

D[(z,E);f)] := L: aij/U)(xi)

satisfying the conditions

{ D[(z,E);/] =0 D[(z,E);/] = 1

eij=l

for f(x) = x"', k=0, ... ,r-1, for I(x) = xr

is called the divided difference of 1 at (z, E).

(4.3.2)

(4.3.3)

S4 Interpolation by Spline Functions [Ch. 4, § 4.3

Note that (4.3.3) is a linear system with respect to {aij} whose matrix is the transpose of a matrix, corresponding to the Birkoff interpolation problem defined by (z, E). Then, by the Atkinson-Sharma theorem, this system has a unique solution. Thus (4.3.3) defines D[(z, E); J] uniquely.

Let P«z, E), I; t) denotes the polynomial of degree r which interpolates I(t) at (z, E), i.e., which satisfies the conditions

if eij = 1.

If P«z, E), I; t) = axr + ... , then D[(z, E); J] = D[(z, E); P) = a and therefore

{ D[(z, E); I] coincides with the coefficient of xr in the polynomial P E 1rr , interpolating I at (z, E).

This implies the following property.

( 4.3.4)

Suppose that the matrix E is conservative and satisfies the P6lya condition. Let I ECr[a, b], where r = IEI- 1, a ~ Xl < ... < xm ~ b. Then

D [(z, E); I] = I(r)(f.)/r! (4.3.5)

for some point f. E [a, b). Indeed, consider the difference

g(t) :=/(t)-P(z,E),/;t).

Clearly g vanishes at (z, E). Then by Rolle's theorem (applied as in the proof of Atkinson-Sharma theorem), g(r)(t) has at least one zero f. in [a, b). Thus

I(r)(f.) = p(r)(f.) = r!D[(z, E); I]

and the proof is completed.

LEMMA 4.10. Suppose that the pair (z, E) is s-regular,

z=(XI, ... ,xm ), E = {eij }~l ~-=-Ol and lEI = r + 1. 1- ,}_

Then

if i = 1 or i = m. (4.3.6)

Moreover,

(4.3.7)

where A is the order of the highest derivative of I at Xm , appearing in the expression D[(z,E);J].

Ch. 4, § 4.3] BirkhofT Interpolation 55

Proof. Assume that aij = 0 for some eij = 1 with i = 1 or i = m. By the AtkinsonSharma theorem there is a polynomial cp E 1rr , such that

(4.3.8)

Clearly aij = D[( z, E)j cp] = O. Then, in view of (4.3.4), cp is a polynomial of degree r - 1. But cp satisfies the homogeneous linear system of equations (4.3.8) for (k,JJ) '" (i, j). This system has a non-singular matrix, since (z, E) was supposed to be s-regular. Then cp == 0: a contradiction with cp(j)(Zi) = 1.

Next we prove (4.3.7). Let cp be the polynomial from (4.3.8), defined for i = m

and I' = A. We have

Therefore signam~ = signcp(r) (zm).

On the other hand, by Lemma 1.17,

This implies signcp(r) (zm) = signcp(~)(zm) = 1

and (4.3.9) completes the proof.

DEFINITION 4.11. For regular (z, E), with IEI= r+1 the function

B(z,E)jt) := D[(z,E)j(.-t)+-1]

is said to be a B-spline of degree r - 1 with knots (z,E).

(4.3.9)

We first show that this new, more general, definition holds the basic characteristic properties of ordinary B-splines.

LEMMA 4.12. Let the pair (z, E) be s-regular and lEI = r + 1. Then

B[(z,E)jt] = 0

B[(z, E)jt] > 0

(4.3.10)

(4.3.11)

Proof. The proof is similar to that in the ordinary case (see Theorem 3.3). The equality (4.3.10) follows from the fact that the function O(z) := (z - t)+-1 is equal to zero in (Z1, zm) for t > Zm and O(z) coincides on (Z1. zm) with the polynomial (z - ty-1 for each fixed t < Z1. To show (4.3.11) we use the observation that B[(z, E)j t] is the coefficient of zr in the polynomial p from 1rr, which interpolates o at (z, E). Since p(z) - O(z) vanishes at (z, E), we get, by Rolle's theorem and the s-regularity assumption that p(r-1)(z) - 0(r-1)(z) must have at least two sign changes in (Z1I zm). This is possible only if p(r-1) is an increasing linear function, i.e., if p has a positive leading coefficient B[(z, E)j t]. This completes the proof.


Therefore B-splines have a finite support. More precisely, supp B[(x, E); t] = [Xl, Xm], with the convention that B[(x, E); xm] = 0 and B[(x, E); Xl] =F 0 if Xl is a knot of multiplicity r.

We demonstrated in Lemma 3.9 a certain convex decomposition of B-splines with Hermitian knots, which was used in the proof of the total positivity and the variation diminishing property. Now we are going to extend this result. In order to make the presentation clearer we first describe some operations on the pair (x, E).

Suppose that Xo := -00 < { < 00 =: X m +! and k is an integer, 0 ~ k ~ r - 1. To add a new knot ({, k) to (x, E) means

(i) if Xi < { < Xi+! :

to insert the point {into x, i.e., to define a new sequence (Xb"" Xi-be, Xi,"" xm) and to insert the row e{ = (611:0, ... , 611:n) in E at the corresponding position to {;

(ii) if { = Xi :

to set ei1l: = 1. We interpret similarly the inverse operation "to remove a knot" from (x, E). Finally, we call the knots (Xb A) and (xm, p) the first, and respectively, the last

knot of (x, E) if eV. = 1, elj = 0 for j > A,

emjJ = 1, emj = 0 for j > p.

THEOREM 4.13. Let (e, k) be a new knot for (x, E) and lEI = r+ 1. Let (x, E)o and (x, Eh be the new pairs obtained from (x, E) by removing the first and, respectively, the last knot of(x,E) and adding (e,k). Suppose that (x,E),(x,E)o and (x,Eh are s-regular and that (e, k) is not an end point of (x, E) U (e, k). Then there exists a constant a such that

B[(z, E); t] = aB[(z, E)o;t] + (1 - a)B [(x, Eh; t] ( 4.3.12)

for each t. Moreover, O<a<1. (4.3.13)

Proof. Let emjJ = 1 be the last 1-entry in em. Denote, for simplicity, by A and A(e) the coefficient amjJ in D[(z, E); f] and D[(x, E)o;f], respectively. According to Lemma 4.10,

A =F 0, (4.3.14)

Now observe that the functional

() D[(x,E);f] -aD[(x,E)o;f] D f : = ---->.'----'--'--"------"-'----'----"-

1-a

satisfies the conditions in Definition 4.9 and has the form mentioned there, corresponding to (x, Eh for a = A/A({). Therefore D(f) = D[(x, Eh;f]. Thus we have proved the relation

D[(x,E);f] =aD[(x,E)o;f] +(l-a)D[(x,Eh;f] (4.3.15)

Ch. 4, § 4.3) BirkhofT Interpolation 57

with a = A/A(~). The equality (4.3.12) follows now immediately from Definition 4.11.

It remains to show (4.3.13). In order to do this, let us note that a > 0 ifthere is a point to such that B[(z,E)o;to] =I 0 and to f/. suppB[(z,Eh;t]. Indeed, in this case it follows from (4.3.12) and Lemma 4.12 that

B[(z,E);to) = aB[(z,E)o;to)

and hence a> O. Similarly, if there is a point to such that

B[(z,Eh;to]=l0 and tof/.suppB[(z,E)o;t],

then 1- a> O. Thus it remains to prove (4.3.13) in the case

suppB[(z,E);t) =suppB[(z,E)o;t) =suppB[(z,Eh;t).

In order to do this observe that

( 4.3.16)

The proof is standard. Assume that A = A(~). Then the linear functional d(f) := D[(z,E);J] - D[(z,E)o;f] annihilates the polynomials from 'Trr . Since (z,E)l is an s-regular pair, the latter implies that all coefficients of d(f) must be zero. In particular avo = 0, where (Xl. A) is the first knot of (z, E). This contradicts Lemma 4.10.

N ow define (zc, Ec) for each c > 0 in the following way: we replace the I-entries ev. and em/J by 0 in E and add new rows eo and em+l, corresponding to the new additional points Xl - c and Xm + c, with entries eOj = 8j>. and emj = 8jm, j = 0, ... , r - 1.

Clearly, the corresponding coefficients Ac and Ac(~) are continuous functions of c. In addition, in view of the result already proved,

Letting c -+ 0 we get 0 ~ A/A(~) ~ 1. But according to (4.3.14) and (4.3.16), we actually have 0 < A/A(~) < 1. The proof is completed.

A repeat use of the relation (4.3.18) yields a representation of the B-spline B[(z, E); t] as a linear combination with positive coefficients of B-splines on a refined grid. Because of the importance of this consequence we formulate it precisely below.

Given an integer r >0 and a pair (z, E) such that Xl < ... < X m ,

{ } m r-l E = eij ·-1 ·-0 ' 1- ,J- lEI = N + r,

we define the "( r + 1 )-partition" of (z, E) in the following way. Let us order the elements of E row by row, i.e., in the manner elO, ... , el,r-l, e20,·.·, e2,r-l, ... , em,r-l


and number the I-entries in this sequence from 1 to N +r. Let ep ' ep+l"'" e be the rows of E which contain r + 1 consecutive 1 's starting from the z-th one. Suppose that the first row ep (respectively, the last row eq ) contains n1 (respectively, n2)

I-entries of this (r + I)-sample. We denote by E; the matrix {ep"'" eq } in which alII-entries in the sequence e p = (epo ,"" ep ,r_1) (respectively, in e q ) except the first n1 (respectively, n2) are replaced by O. Finally, define Z; := (xp, ... ,xq), i.e., Z; denotes the subset of those consecutive points from z which correspond to the rows of E;.

We shall say that the (r + I)-partition {(z;, E;)} of (z, E) is s-regular if each (z;, E;) is s-regular.

COROLLARY 4.14. Let the (r+l)-partition {(z;, E;)}f of(z, E) be s-regular. Then there exist positive constants {ad such that a1+ ... + aN = 1 and

N

B[(z,E)jt] = La;B[(z;,E;)jt]. (4.3.17) ;=1

The assertion follows by repeated use of Theorem 4.13. The next lemma shows that B[(z, E)j t] has a minimal support.

LEMMA 4.15. Let the pair (z,E) be regular. Suppose that E = {eij}~l.J;~ and lEI:::; r. Then each spline f of degree r - 1 with knots at (z, E) such that f(t) = 0 for every t ~[X1' xm] is identically zero.

Proof. Indeed, let f be such a spline. Then

" {( )r-1 }(j) I f(t) = p(t) + L..J a;j x; - t + ~=~i e;j=l

for some p E 7rr and {aij}. It follows from the condition f(t) = 0 for t > Xm that p == O. Now let t1 < ... < tr < Xl be arbitrary points. Set Pk(X) := (x - tky-1. The polynomials {Pk}1 form a basis in 7rr -1 on (X1,X m ). On the other hand, the condition f(tk) = 0 yields

L(Pk) := L a;jp~)(x;) = 0, k = 1, ... ,r. eij=l

Therefore, L(q) = 0 for each q E 7rr -1 and, consequently, a;j = 0 for all eij = 1 since (z, E) is regular. The proof is completed.

We next use Lemma 4.15 to prove the following.

THEOREM 4.16. Suppose that {(z;, E;)}f is an s-regular (r + I)-partition of some pair (z, E) with IEI= r + N. Then for each J C {I, ... , N} the functions Bi := B[(z;, E;)j.], i E J, are linearly independent over any subinterval (Xk' xk+d C K(J), where

K (J) := n supp B;. ;eJ

Ch. 4, § 4.3) Birkhoff Interpolation 59

Proof. There is nothing to prove if J«J) is empty. Suppose that (xk' xk+1) C K(J) for some k. Let us assume that

f(t) := L a.B.(t) = 0 'EJ

for each t E (xk' x k+1). Since J has no more than r elements, it follows from Lemma 4.15 that f(t) == 0 on (-00,00). This implies a. = 0 for all i EJ. Indeed, let io be the largest i from J for which a. :I 0 and let z. = (Xj,"" xn). If en>. is the last I-entry in the last row of E., then

( )r->.-l" f(t) = a.oc>. Xn - t + + L.,..(t) ,

where c>. is the corresponding coefficient of f{>'}(x n ) in D[(z.o, E.o); f] and E(t) contains the other terms of the form a'j (Xi - t)~-j -1, i ~ n, in the representation (4.3.1) of the spline f. Since f == 0 on (Xn-1, xn) and the polynomial functions {(Xn - tr- j - 1 : j EJo} are linearly independent for any Jo C {O, 1, ... , r - I}, we get ai' c>. = O. But in view of Lemma 4.10, c>. :I O. Hence aio = 0: a contradiction. Thus a. = 0 for each i EJ. The proof is completed.

Now we can introduce a B-spline basis in the set Sr-1 (z, E) of all functions s(t) of the form (4.3.1).

THEOREM 4.17. Let (z,E) be a given pair of points Xl < ... < Xm and incidence matrix E with lEI = N. Let a, b, e 1, ... ,er, er+N+1, ... , 6r+N be arbitrary points such that

Denote bye the sequence (6, ... , er , Xl' ... ,Xm, er+1' ... ,e2r)' Let {e(ej)} be some incidence vectors of dimension r, each of them containing only one 1-entry. Denote by E the matrix consisting of the rows {e(ed,··· ,e(er),e1,'" ,em,e(er+d,·.· ,e(6r)}. Suppose that {(zi,Ei)E+N is an s-regular (r + I)-partition of (e,E). Then the functions {B[(Zi, Ei); t]}~+N constitute a basis in Sr-1(z,E) over [a,b].

Proof. Clearly, the linear space Sr-1 (z, E) offunctions (4.3.1) is of a dimension not greater than lEI + r, i.e.,

(4.3.18)

Consider the B-spline sequence Bi(t) = B[(Zi' Ei ); t], i = 1, ... , r + N. It follows from Theorem 4.16 that the functions {BiE+N are linearly independent on [a,b]. Indeed, assuming that

r+N L aiBi(t) == 0 on [a, b], .=1

we conclude, on the basis of Theorem 4.16, that a1 = ... = ar+N = O. Therefore, in view of (4.3.18), dimSr _1 (z, E) = r + N and the functions B1 , ... , Br+N form a basis in Sr-1(z,E) on [a,b]. The proof is completed.


4.3.2. Sign changes of a spline function. All further properties of B-splines with Birkhoff knots follow in the same way as they have been proved already in the ordinary case. We omit here the details.

The pair (y, F) is said to be a refinement of the s-regular pair (:1:, E) if there is a way to get (y, F) from (:1:, E) by adding new knots (e, k) to (:1:, E), one knot at a time, step by step, such that the (r + I)-partition of the new pair, obtained after each step is still s-regular.

THEOREM 4.18. Let (:1:, E) be a given pair of distinct points :I: = (Xl, . .. ,xm ) and a matrix E = {eij }~l.J~l with lEI = r + N. Suppose that the (r + 1 )-partition {(:l:i' Ei)}f of (:I:, E) is s-regular. Then

(4.3.19)

Proof. Note first that if a pair (:1:, E) admits an s-regular partition, we can get a sufficiently fine (i.e., with a sufficiently dense set of knots) refinement adding additional knots. Indeed, this is clear if E does not contain odd non-Hermitian blocks. If E contains such blocks, they must lie out of the rows ej, j E [.it , jk], where [.it, jk] is the largest interval with the property eh,o = ejk,O = 1, .it ~ jk. Then we start to add knots in the nearest to [.it, jk] row ej, containing an odd block until this block becomes Hermitian. After several such operations the transformed pair (:1:, E) will not contain any odd non-Hermitian block and we could continue to add new knots without violating the s-regularity of the (r + I)-partition.

The proof of (4.3.l9) goes further as in Theorem 3.12.

4.3.3. Main interpolation theorem. We first prove the total positivity of the collocation matrix for Hermite interpolation by splines with Birkhoff knots.

THEOREM 4.19. Let:l:= (xl"" ,xm ), E = {eij}~l,r;ol and lEI = r + N. Suppose that the (r+l)-partition {:l:i,Ei}f of (:I:, E) is s-regular. Then

(4.3.20)

for each choice of the integers 1 ~ i1 < ... < in ~ N, 1 ~ n ~N, and the points T1 ~ •.• ~ Tn (with Tk < Tk+r). Moreover, A > 0 if and only if

k = 1, ... ,no (4.3.21)

Proof. Adding new knots we obtain a refinement (y, F) of (:1:, E) such that each point Tk lies at a knot of multiplicity r. Then the sequence {Bd corresponding to (y, F) will satisfy the condition

Ch. 4, § 4.3) BirkhofT Interpolation 61

for each i. Moreover, for any fixed k, Bi(T"J #; 0 only for one i. We continue and complete the proof as in Theorem 4.6.

Note that the proposition we just proved implies the corresponding interpolation result. Precisely, for fixed (:1:, E) with lEI = r + Nand T1 ~ ... ~ TN (Tj < TH ,., all j) the Hermite interpolation at Tl, ••• , TN by splines s(t) of degree r - 1 with Birkhoff's knots (:I:, E) has a unique solution if and only if B[(:l:i, Ei)j Ti] #; 0 for i = 1, ... ,N.

Next we consider the "dual" interpolation problem.

THEOREM 4.20. Let:l: = (zo,Zl, ... ,Zm+1), a = Zo < ... < Zm+1 = b, E = {eij}?!:ii~';ol and integers {lIi}~ be given such that N = 111+ ... + lin, 1 ~ IIi ~ r, i = 1, ... , n, and lEI = N + r. Assume that (:I:,E) has an s-regular (r + 1)partition {(:l:i, EiHi". Then the interpolation problem

if eij = 1 (4.3.22)

by splines s of degree r - 1 with knots e1, ... , en of multiplicities "1, . .. , lin, respectively, has a unique solution for each given data {lij} if and only if

for i=l, ... ,N,

where (Tl, ... , TN) == «e1, 111)' ... ' (en, lin».

Proof. Let us present the spline s(t) in the form

,.-1 n 1110 -1

s(t) = L aj(t - a)j + L L al:>. (t - e"J~->'-1, j=O k=l >.=0

(4.3.23)

(4.3.24)

where {aj} and {ak>.} are real coefficients. Denote by V = V[(:I:,E), (e, v)] the matrix of the system (4.3.22) in unknowns

We have to show that det V [(:1:, E), (e, v)] #; 0 (4.3.25)

if and only if the knots (:1:, E) and the interpolation nodes e satisfy the interlacing condition (4.3.23).

Clearly, the matrix V[(:I:,E),(e,v)] consists of the rows

Wij:= {l,(z-a), ... ,(z-a)"-l, K(Z,e1), ... ,K(1I1-1)(z,e1),

... , K (z,en), ... , K(II,,-l) (Z,en)} (j) 13:=';;'

where (i, j) runs over the indices of the anI-entries eij in the sequence eoo, ... , eO,,.-l,

... , em +1,O, ... , em +1,,.-1 and

K(z, t) := (z - t)+-1,


In order to prove (4.3.25) we shall perform some elementary transformations in V, writing at the row of the number r + k (k = 1, ... , N) the linear combination

L CijWij,

ei;=l

where the sum is expanded over the I-entries of Ek, and {Cij} are the coefficients of the divided difference

D[(Zk,Ek);/] = L Cij/(j)(Xi). ( 4.3.26) ei;=l

Denote by O:k the coefficient of the highest derivative at the last point of the set zk

which participate in the expression (4.3.26) of D[(zk' Ek );/]. According to Lemma 4.10,

(4.3.27)

Denote by Vo the matrix obtained from V by the described transformation of the rows r + 1, ... , r + N. Clearly

det V = 0:. det Yo, 1 (4.3.28) 0::= ----

and the (r + k)-th row v~+k of Vo is of the form

V~+k := {Dk[I], ... , Dd(x - ar-1], Dk [K (x,6)], ... , Dk [K(Vk -1)(X,en)]} , where Dk := D[(zk, Ek);·]. Using the property Dk[J] = 0 for each / E '1rr -1 and the definition of B-splines we see that

V~+k = {O, ... , 0, Bk (6), ... ,Biv ,,-l) (en)}. Denote, for simplicity, by

the matrix consisting of the rows

(j) (t .) (j) (t .) U1 " ••• ,Un "

ordered according to the position of the I-entries eij in the sequence of consecutive rows of the incidence matrix E = {eij}. Then, expanding the determinant of Yo along the first r rows, we get by the Laplace formula

det V = 0:. det A . det {Bk (Tj) } :=1,;=1' (4.3.29)

where A= [{I, x-a, ... ,(x-ar-1}(j)lx=xi]

eij = 1, eij E Eo

and Eo is obtained from E1 replacing the last I-entry (i.e., the last 1 in the last row of E1) by O. Since {Ed{V, and in particular El, were assumed to be s-regular, Eo is regular. Then, by the Atkinson-Sharma theorem, det A f; O. Therefore, it follows from (4.3.29) and Theorem 4.19 that det V[(z, E), (e, v)] f; 0 if and only if the nodes e satisfy (4.3.23). The proof is completed.

Ch. 4. § 4.4) Total Positivity of the Truncated P(MIer Kernel 63

COROLLARY 4.21. Let ((Xl,Jll), ... ,(xm,Jlm» == (tl, ... ,tr+N), Xl < ... < Xm, and (Tl, ... ,TN ) == ((6,vd"'.,(~n,vn»' ~l < '" < ~n' be given such that N = Vl+ ... + Vn, 1 ~ Vi ~ r, i = 1, ... , n, and 1 ~ Jlj ~ r, j = 1, ... , m, r + N = Jll + ... + Jlm· Then the interpolation problem

SU)(Xi)=!;j for i=I, ... ,m, j=O, ... ,Jli-I

by splines s of degree r - 1 with knots 6, ... '~n of multiplicities VI, ... , lin, respec-tively, has a unique solution for each given data {/ij}, if and only if

ti < Ti < ti+r, i = 1, .. . ,N.

(ti < Ti is interpreted as equality in case ti = ... = ti+r-d. This proposition is a particular Hermitian case of Theorem 4.20.

§ 4.4. Total Positivity of the Truncated Power Kernel

The truncated power function K (x,~) := (x - ~rt 1 plays a fundamental role in the theory of spline functions. Many applications of splines are based on the total positivity of the kernel K (x,~), i.e., on the fact that the determinant of the collocation matrix {K (Xi, ~i)} does not change sign when the points from the ordered sets {Xi} and {~j} move on the real line. This property was actually proved in Theorem 4.20. Next we shall determine the sign of det V[(:e, E), (~, II)].

First, let us recall the definition of the coalescence c[a, b] of two incidence rows a = (aI, ... , an) and b = (b l , ... , bn) with a total number of I-entries less than or equal to n.

Suppose that the incidence matrix consisting of the rows a and b satisfy the P6lya condition. The coalescence of a and b is a single row c[a, b] with components defined by the procedure:

(i) add a + b to obtain (el, ... , en), where ek = ak + bk , k = 1, ... , n;

(ii) if ek > 1, set ek := ek - 1, ekH := ekH + 1 for k = 1, ... , n - 1;

(iii) repeat (ii) if ek > 1 for some k = 1, ... , n - 1;

(iv) set c[a, b] := (el, ... , en). The following will be needed.

LEMMA 4.22. Let (y, G) be a given regular pair of points y = (Yo, Yl, ... , Yk), a = Yo < Yl < ... < Yk, and an incidence matrix G={gij }f=o,j:1 with IGI = r. Let

A = [{I, X - a, ... ,(x - ay-l }(i)I"=lIi] . gij = 1, gij E G

Then there is a positive integer (T, depending only on G, such that

(-1)" det A> 0

for each a < Yl < ... < Yk. Moreover, if G is quasi-Hermitian with

gOj = {01 for j ~ i l ,···, ip , otherwIse,

then (T = i l + ... + ip - p(p - 1)/2.


Proof. Since (y, G) is a regular pair, by the Atkinson-Sharma theorem the interpolation problem

if gij = 1

has a unique solution. Thus, det A =I 0 for each a < Yl < ... < Yk. In order to find the sign of, det A, note that for fixed Yo, ... , Yk-l det A is a polynomial function of x := Yk - Yk-l. Denote this function by Ak(X). By Taylor's formula

Ak(X) = EA~)(O)xj Ii!· j

(4.4.1)

Let A~>')(O) be the first non-zero coefficient in (4.4.1). It is not difficult to see that

A~>')(O) is equal (up to a positive integer factor) to the determinant of the matrix

[{1,(x-a), ... ,(x-at - 1}(j)I_ .]

Ak-l := X-!l. ,

9ij = 1, 9ij E Gk-l

where Gk-l = {U;j} ::::,j~-Ol is obtained from G by the coalescence of the last two rows gk-l and gk. Then

signAk(x) = signA~>')(O) = sign det A k- l

for sufficiently small x >0. Considering now det A k- l as a function Ak_l(X) of x := Yk-l - Yk-2 and, applying the same reasoning, we get Ak-2 and so on. Finally, we come to the relation

sign det A = sign Ak (x) = sign det Ao,

where Ao is a Taylor type matrix,

Ao = [{ 1, x - a,: .. ,5x - a).r-l } (j) IX=4] , 3 = 30, ... , 3r-l

with (jo, ... , ir-d being a certain permutation of (0, ... , r - 1). Thus

sign det A = (-1)",

where q is the number of pairwise interchanges needed to obtain the natural order (0, .. , r - 1) from (jo, ... ,ir-l).

If G is quasi-Hermitian, it is easy to verify that

(jo, ... , ir-l) = (i1, ... , ip , kl, ... , kr _ p ),

where kl, ... , kr _ p are the positions of the O's in (gOO, gOl, ... , gO,r-t). Clearly ip -

(p - 1) interchanges are needed to order the sequence (ip , kl , ... , kr _ p ). Thus

q = [ip - (p - 1)] + [iP-l - (p - 2)] + ... + [i2 - 1] + i l

= i 1 + ... + ip - p(p - 1)/2.

The lemma is proved.

Ch. 4, § 4.4) Total Positivity of the Truncated PCNfIer Kernel

THEOREM 4.23. Let:z: = (xo, ... ,xm+d, a = Xo < ... < Xm+1 = b,

E { }m r-l = eij i=O,j=O'

65

1 ~ IIi ~ r, i = 1, ... , n, N = 111 + ... + lin and lEI = N + r. Let (:z:, E) admit an s-regular (r + I)-partition. Then there is a positive integer u, depending on E only, such that

(_1)<7 det V[(:z:,E),(e,v)] ~ 0

for each 6 < ... < en. The strict inequality holds if and only if

i = 1, ... ,N,

where (T1,"" TN) == «e1, lid,···, (en, lin)). If, in addition, E is quasi-Hermitian with

for j = i1 , ••• , ip '

otherwise,

then u= i1 + ... + ip - p(p - 1)/2.

Proof. In view of (4.3.29),

det V [(:Z:, E), (e, v)] # 0

sign det V[(:z:,E),(e, v)] = sign det A.

Then the result follows from Lemma 4.22.

COROLLARY 4.24. Let k l , ... , kr - p be the positions of 0 in (eoo,eo1,"" eo,r-d and

where IE is obtained from E by replacing the first p I-entries in the sequence

eOo,···, eO,r-1,···, em+l,O,"" em +1,r-1

by O. Then, under the assumptions of Theorem 4.23, det K ~ 0 for each 6 < ... < en. The strict inequality holds if and only if

i = 1, ... ,N.


Proof. By the Laplace formula

where s = 1 + ... + p + (il + 1) + ... + (ip + 1) = u and the assertion follows from Theorem 4.23.


Total positivity is introduced and studied in detail by Karlin [1968]. Determinants that appear in Lagrange spline interpolation using the truncated power basis can be discovered in an early paper of Krein and Finkelstein [1939]. The complete characterization of the spline interpolation problem for simple knots and nodes is given by Schoenberg and Whitney [1949, 1955]. The general Hermitian interpolation by splines with multiple knots was studied in Karlin and Ziegler [1966]. The related result concerning the number of zeros of a spline function can be seen, for example, in Schumaker [1976]. For a Hermite interpolation with boundary conditions, see Melkman [1974, 1977], and Karlin [1971], Karlin and Pinkus [1976]. Corollary 4.24 is due to Karlin [1971]. The total positivity of the collocation matrix {Bi(tj)} was proved by de Boor [1976]. The present proof of Theorem 4.6 follows that of de Boor and De Vore [1985]. B-splines with Birkhoff knots were introduced (see also the pioneering paper of Birkhoff [1906]) and applied to interpolation in Bojanov [1988]. Section 4.3 is based on this paper. The material of Section 4.4 is taken from Bojanov [1990b]. For more about Birkhoff interpolation by polynomials and splines see, the book by Lorentz, Jetter, and Riemmenschneider [1983].

Chapter 5

NATURAL SPLINE FUNCTIONS

§ 5.1. Interpolation by Natural Spline Functions

5.1.1. Definition. The spline function s(x) of odd degree 2r - 1 with knots Xl, ... ,Xn is said to be a natural spline function, if the restriction of s over (-00, Xl)

and (Xn, 00) is a polynomial from 1I"r-l. We shall denote by N2r- l (Xt, ... ,xn) the class of natural spline functions of degree 2r - 1 with knots {x I<}~. It is clear that N2r- l (Xl, ... , xn) consists of all splines s from S2r-l(Xl, ... , xn), which satisfy the boundary conditions:

s(j)(xt} = s(j)(Xn) = 0, j = r, ... , 2r - 1. (5.1.1)

The next lemma gives a characterization of natural splines.

LEMMA 5.1. The function s( x) is a natural spline of degree 2r-l with knots Xl, ... , Xn if and only if it can be written in tIle form

n

s(X) = p(x) + LCI«x - XI<)!'"-l, (5.1.2) 1<=1

where p E 1I"r-l and the coefficients {Cl<} satisfy tIle conditions

for j = 0, ... , r - 1. (5.1.3)

Proof. Suppose that s is a function of the form (5.1.2) with coefficients satisfying (5.1.3). Then s is a spline function of degree 2r - 1 with knots Xl, ... , Xn (see Theorem 2.10). In addition, s(x) := p(x) on (-oo,xt) and, by the assumption, p E 1I"r-l . It remains to verify that s(x) coincides with an algebraic polynomial from 1I"r-l on (xn, 00). In order to do this, suppose that X ~ Xn. Then

n

s(X) = p(x) + L CI«x - XI<)2r-l

1<=1

n 2r-l . (2r _ 1) . . = p(X) + LCI< L (-1)3 . x 2r - l -'X1

1<=1 j=O J

2r-l .(2r-l) . n . =p(x)+ L(-I)' . X2r-l-'LCI<X1.

j=O J 1<=1

67

68 N~tur~1 Spline Functions [Ch. 5, § 5.1

But

for j = 0, ... , r - 1,

according to assumption (5.1.3). This shows that 8 is a polynomial of degree r - 1 in (zn' 00).

Let us prove the converse assertion. Assume that 8 belongs to N2r- 1(Zl, ... , zn). Then 8 E S2r-1(Zl, ... , zn) and hence 8(Z) has a representation ofthe form (5.1.2) with some p E 1I"2r-1. But 8(Z) == p(z) on (-00, Zl). Since 8 is a natural spline, p must be actually a polynomial of degree r-1. Further, it is seen from the expression of 8 we just derived, that the coefficients {Ck} must satisfy (5.1.3), provided 8 is assumed to be a polynomial of degree r - 1 on (zn, 00). The lemma is proved.

Remark that 8(r) (z) = 0 outside the interval (Z1' zn) for every natural spline from N2r- 1(Zl, ... , zn).

5.1.2. Interpolation. It will be noticed that the problem we consider below is a very particular case of a general spline interpolation problem, studied already in the previous chapter (see Theorem 4.20). The next simple direct treatment, together with the fascinating extremal property of the natural spline interpolation, is actually the spark that gave rise to spline theory.

LEMMA 5.2. Suppose that! E W[[a, b]. Let a ::;; Z1 < ... < Zn ::;; b and

Then b n J !(r)(t)s(r)(t) dt = (-1r(2r - 1)! L Ck!(Zk),

a k=1

where {cd are the coefficients in the representation (5.1.2) of s.

Proof. We integrate b

JU) := J !(r)(t)s(r)(t) dt

a

repeatedly by parts and get

(5.1.4)

r-1 b JU) = L(_lt-A- 1j<A)(t)8(2r-A-1)(t)l: + (-lr-1 J !'(t)8(2r-2)(t)dt.

A=1 a

Since s(r)(t) == 0 outside (Zl, zn), the sum vanishes and thus

b

JU) = (-lr-1 J !'(t)s(2r-2)(t)dt.

a

Ch. 5, § 5.1] Interpolation by Natural Spline Functions 69

Integrating again by parts and using the fact that s(2r)(t) == 0 on each subinterval (X1:, X1:+d, we obtain

n-l "'''+1

1(1) = (-lr- l L: J !'(t)s(2r-2)(t)dt

1:=1 "'''

n-l

= (-lr- l L: [!(X1:+t}s(2r-2)(Xl:+1 - 0) - !(X1:)S<2r-2)(X1: + 0)]. 1:=1

Now taking into account that s(2r-1)(X1 - 0) = s(2r-1)(xn + 0) = 0 we rewrite the last sum as

n-l

1(1) = (-lr L: !(X1: )[s(2r-l)(X1: + 0) - S<2r-l)(X1: - 0)]. 1:=1

Then using the relations

k = 1, ... ,n,

(see 2.1.2), we get the wanted expression for 1(1). The proof is completed. Now we are ready to prove the interpolation theorem.

THEOREM 5.3. Let Xl < ... < Xn be given points on the real line and 1 ~ r ~ n. Then for every choice of the real numbers {Y1:}1' there exists a unique natural spline function s(x) from N2r- 1 (Xl, ... , xn), which satisfies the interpolation conditions

for k = 1, ... , n.

Proof. We have to find coefficients {aj}~-1 and {cd!, such that

r-l n

L: ajx: + L: C1:(Xi - X1:)!'"-1 = Yi, i = 1, .. . ,n. j=O 1:=1

These equations, together with the conditions

for j = 0, ... , r - 1,

which must hold for the coefficients {C1: 11 of every natural spline, form a system of n + r linear equations in n + r unknowns (aj and C1:). The system would have a unique solution for any fixed {y;} if and only if the corresponding homogeneous system admits only the trivial zero solution: aj = C1: = 0 for all j and k. Assume that the homogeneous system has a nonzero solution ao, ... , a r -1, Cl, .•. , cn • Denote

70 Natural Spline Functions [Ch. 5, § 5.1

by So(t) the natural spline function from N2r- l (Xl, ... , xn) with these coefficients. Consider the integral

b

,(So):= J [S~r)(t)]2 dt. a

Applying Lemma 5.2 with / = So and s = So, we get

n

,(So) = (-lr(2r - I)! L Cd(Xk). k=l

But, according to the assumption on SO'/(Xk) = SO(Xk) = 0 and therefore ,(So) = O. Then S~r)(t) == 0 on [a, b) and, consequently, So is an algebraic polynomial of degree r - 1 in [a, b). Since SO(Xk) = 0 for k = 1, ... , nand r ~ n, we conclude that So(t) == O. Now it follows from the linear independence ofthe functions 1, X, ••• , x r - l ,

{( ) 2r-l}n h - - - - - - 0 Th t d' t' X - Xk + k=l t at ao - ... - ar-l - Cl - ... - Cn -. e con ra IC Ion proves the theorem.

Let us sketch another proof of Theorem 5.3 which gives a method for the construction of the natural spline interpolant.

Let a ~ Xl < ... < Xn ~ b. Denote by {Bi(t)}, B-splines of degree r-l, associated with {Xi}~' i.e.,

i = 1, .. . ,n.

If s E N2r - l (XI, ... , xn), then S<r) E Sr-l (Xl, ... , xn) and, in addition, s(r)(t) == 0 outside (Xl, xn). Since Bl , ... ,Bn- r form a basis in this subspace of Sr-l (Xl, ... , xn), s(r) may be written as a linear combination of {B;}~-r, namely,

n-r

s(r)(t) = L akBk(t). k=l

Therefore, by Taylor's formula, every natural spline function s E N2r- l (Xl, ... ,xn) can be uniquely described in the form

(5.1.5)

with some p E 7rr -l and real coefficients {ad. Introduce the notations

6i := (r - 1)!/[xi,"" Xi+r], i = 1, ... , n - r,

where / is any function satisfying /(Xi) = Yi, i = 1, ... , n. Clearly the interpolation conditions

i = 1, ... ,n,

Ch. 5, § 5.1J Interpolation by Natural Spline Functions

are equivalent to the system of equations

S(Zj) = Yj

(r - l)!s[Zj, ... , Zi+r] = 6j,

for i = 1, ... ,r, i = 1, ... , n - r.

Using (5.1.5) we may rewrite the second type equations as

i=l, ... ,n-r.

Bence {ak}i-r satisfy the linear system

i= 1, ... ,n-r.

71

(5.1.6)

Since {Bj}i- r are linearly independent on [a, b], the system has a nonzero determinant. Thus the coefficients a1, ... , an - r of the interpolating spline S are defined uniquely as a solution of (5.1.6). Having {a" li-r, one can find p by Lagrange's interpolation formula from the conditions s(Zj) = Yj, i = 1, ... , r. The construction of S is completed.

Another way of finding the parameters {aj 1 of the interpolating natural spline is based on the dual relation between the system (5.1.6) and the minimization problem

(5.1.7)

over all real a1, ... , an - r obeying the restriction

n-r

La"6,, = 1. "=1

The next exercise reveals this relation.

Exercise 5.1.1. Show that the extremal problem (5.1.7) has a unique solution. Denote it by (a1, ... , an - r ). Prove that

k = 1, ... ,n-r,

is the solution of the linear system (5.1.6).


Hint. c) is a differentiable function in the domain

Since c) > 0 on A and c)1/2 is a convex function, c)(alo"" a n - r ) attains its minimal value at a unique point (a1, ... , a n - r ) in A. By the method of Lagrange multipliers,

i=I, ... ,n-r,

at (a1, ... , an-r). Performing the differentiation, one gets

n-r 6

2 L al: J Bi (t)Bl: (t) dt = >t6i, i = 1, ... ,n- r. l:=1 /J

Thus {2al:/ >t }~:r satisfy (5.1.6). To find >t, add up the equations of the last system, multiplied by ai, respectively, and make use of

n-r

Lai6i = 1. i=1

5.1.3. Holladay's tbeorelll. We show here an interesting extremal property of the natural spline interpolant. The following simple fact will be used.

LEMMA 5.4. Let It and 12 be arbitrary orthogonal functions from L2[a, b], i.e.,

6 J It(t)l2(t)dt = O. /J

Then 6 6 J n(t)dt ~ J (It(t) + l2(t»2dt

/J /J

and the equality is attained if and only if l2(t) == 0 a.e. on [a, b].

Proof. The assertion follows immediately from the equality

and the orthogonality of It and 12. Given the points z = (Zlo ... , zn), a ~ Z1 < ... < Zn ~ b, and the values y =

(Yl. ... , Yn), we denote by W(z, y) the class of all functions f E W2[a, b] for which f(zl:) = Yl:, k = 1, ... , n.

Ch. 5, § 5.1) Interpolation by Natural Spline Functions 73

THEOREM 5.5 (Holladay's theorem). Let 1 ~ r ~ n and let the interpolation data {Xl:, Yl:}r be given. Let s be tIle unique natural spline from N2r- l (Xt, ... ,xn) which interpolates {Xl:, Yl:n . Then, for each IE W(z, y),

b b J [s(r)(t)]2dt ~ J [J(r)(t)]2dt .

4 4

The equality is attained if and only if 1== s a.e. on [a, h).

Proof. Let I be arbitrary function from the class W(z, y). Since I and s take the same value at Xl: for k = 1, ... , n, Lemma 5.2 yields

b J s(r)(t) [J(r)(t) - s(r)(t)] dt = 0,

4

i.e., the functions s(r) and 1(1') - s(r) are orthogonal. Then, by Lemma 5.4, b b b J [s(r)(t)]2dt ~ J ([/(r)(t) - s(r)(t)] + s(r)(t»)2dt = J [J<r)(t)]2dt .

444

Moreover, the equality holds if and only if I(r)(t) - s(r)(t) == 0. Since I(xl:) = s(xl:) for k = 1, ... , nand r ~ n, the last identity implies I(t) == s(t). The theorem is proved.

Exercise 5.1.2. Let 1 ~ r ~ n, a = Xl < ... < Xn = b and IE W2[a, h). Let sf be the unique spline from N 2r- 1(X1, ... ,Xn) for which sf(Xl:) = I(xl:), k = 1, ... ,n. Prove that

b b J [/(r)(t) - s)")(t)] 2 dt ~ J [J<")(t) - s(")(t)] 2 dt

4 4

for each s E N2r- l (Xl, ... ,xn). The equality is attained ifand only if s(r)(t) == S})(t) on [a, h).

Hint. Apply Lemma 5.2 to I - sf and show that J<r)(t) - S})(t) is orthogonal to

g(")(t) for each g E N2,.-1 (Xl, ... ,xn). In particular, b J [J(")(t) - s)r)(t)] [S})(t) - s(r)(t)]dt = 0.

4

Then Lemma 5.4 implies the required result. It follows from the general interpolation result in Chapter 4 (see, for example,

Corollary 4.21) that given I E Cr-l[a, b) and a < Xl < ... < Xn < h, there exists a unique spline s from 8 2,.-1 (Xl, ... , xn) which satisfies the interpolation conditions

s(j)(a) = l(j)(a), j = 0, ... , r - 1,

s(j)(h) = l(j)(b), j = 0, ... , r - 1,

s(x;) = I(x;), i = 1, ... ,71.

This kind of interpolation is known as complete spline interpolation.

74 Natural Spline Functions (Ch. 5, § 5.1

Exercise 5.1.3. Prove directly the existence and uniqueness of the complete spline interpolant.

Hint. The problem is to show that if a spline s E S 2r-1 (Xl, ... , xn) satisfies the corresponding homogeneous system of equations then s == O. Using integration by parts show first that

b b

p(s) := J [s(r)(t)]2dt = (-It-1 J sC2r- I)(t)s'(t)dt

a a

for each s E S2r-I(XI, ... , xn), provided sU)(a) = sU)(b) == 0 for j = 1, ... , r - l. If, in addition, s(x;) = 0, i = 0, ... , n + 1 (with Xo = a, Xn+1 = b), then, clearly, p(s) = 0 and hence s(r)(t) == O. This yields s == O.

Exercise 5.1.4. Given 10 E Cr-l[a, b], denote by F the set of all functions 1 from W[[a, b] for which

IU)(a) = I~j)(a),

IU)(b) = I~j)(b), I(x;) = lo(x;),

j = O, ... ,r-l,

j=O, ... ,r-l,

i = 1, ... ,n.

Let s be the unique spline from S2r-I(XI, ... , xn) n F. Prove that

b b J [s(r)(t)]2 dt < J [!(r)(t)]2 dt

a a

for each 1 E F, different from s.

Hint. This is Holladay's theorem for complete spline interpolation. Use the method applied already in the case of natural spline interpolation (see the proof of Theorem 5.5).

Exercise 5.1.5. Let (x;, y;), i = 1, ... , n, be given, a :::; Xl < ... < Xn :::; band 1 :::; r:::; n. Denote 11e := {f E Wna, b]: I/(x;) - y;1 :::; c, i = 1, ... , n}. Show that, for every fixed c ~ 0, the extremal problem

b

inf { J [!(r) (t)] 2 dt: 1 E 11e} a

has a unique solution. Moreover, this solution is a natural spline from the class N2r- I (Xb ... , xn).

Hint. The existence and the uniqueness of the solution follows from the observation that 11e is a compact, convex set, and the Minkowski inequality.

Ch. 5, § 5.2] Best ApprOlCimation of Linear Functionals 75

Let f. be the extremal function to our problem. Then, by Holladay's theorem,

b b J [s<r)(t)]2dt ~ J [J$r)(t)]2dt, (J (J

where s is the natural spline from N2r- 1 (Xl> .•• , x n ), which interpolates the function f. at Xl> ... , X n . Then, from the uniqueness, s == f •.

§ 5.2. Best Approximation of Linear Functionals

We are going to show that the natural spline interpolation scheme have best approximation properties, in a certain sense. In order to this, we introduce first the necessary notations and prove a general result about the best recovery of a given linear functional on the basis of partial information.

Let H be a given linear space. Suppose that Land L1 , ••• ,Ln are linear functionals defined on H. Assume that the values L1(f), ... , Ln(f) are known or easily available for each f E H. We shall consider the problem of approximation of the functional L(f) in H on the basis of the information T(f) := (L1(f), ... , Ln(f)) only. In other words, we have to construct a method S, which applies to each f E H and assigns an approximation value S(f) to L(f) using T(f) only. For example, any function S(tl' ... ,tn ) of n variables generates such a method in the following way:

L(f) ~ S(L1(f), ... , Ln(f)) =: S(f) for each f E H.

Of course, there are infinitely many such methods. Our goal will be to construct an extremely "good" one.

The error Rs of the method S in H is defined as

Rs := sup{IL(f) - S(f)I: fEn},

where n is a fixed subset of H (usually n is the unit ball in H).

DEFINITION 5.6. The method S· is called the best method of recovery of the functional L on the basis of the information T(f) if

Rs* = R(T) := inf Rs. s

The infimum here is expanded over all admissible methods, i.e., over all functions S of n variables.

The next lemma, based on a simple geometric interpretation, shows that under certain fairly general restrictions on n there is a linear best method of recovery.

LEMMA 5.7 (Smolyak's lemma). Let the linear functionals L(f), L1(f), ... , Ln(f) be defined in the linear space H. Suppose that n is a convex and centrally symmetric body in H. Let

sup{L(f): f E no} < 00, (5.2.1 )


where k=l, ... ,n}.

Then there exist numbers A l , ... ,An' such that

Proof. Let us recall first that n is a body in H if it has a nonempty interior. The central symmetry means that fEn implies - fEn.

Denote by Y the set of all points (Yo, ... , Yn) of jRn+1, such that Yo = L(I), Yle = LIe(l), k = 1, ... , n, for some fEn. It follows from the assumptions in the lemma that Yis a convex, centrally symmetric body in jRn+l. Set

Ao := sup{yo: (Yo, 0, ... ,0) E Y}.

The assumption (5.2.1) yields Ao < 00. Let

n

co(yo - Ao) + LCIcYIe = 0 Ie=l

be the equation of a hyperplane which is a tangent to Yat the point (Ao, 0, ... ,0). Clearly, Co i= 0 (since Yis a body in jRn+1). We may assume without loss of generality that Co > O. Because of the symmetry, the hyperplane

n

co(yo + Ao) + L CleYIe = 0 Ie=l

will be a tangent to Yat (-Ao, 0, ... ,0). Then Ywill be situated between these two tangent hyperplanes (see Fig. 5.1.). Therefore, the coordinates Yo, ... , Yn of every point from Ywill satisfy the inequalities

which become

n

co(yo - Ao) + L CIcYIe ~ 0, Ie=l

n

co(yo + Ao) + LCIcYIe ~ 0, Ie=l

n

Yo + L(cle/co)YIe ~ Ao, Ie=l

n

Yo + L(cle/co)YIe ~ -Ao. 1e=1

Ch. 5, § 5.2) Best ApprC/lCimation of Linear Functionals 77

L(f)

Fig. 5.1.

Set Ak = -ck/co, k = 1, ... , n. Then the latter inequalities can be written as

IYO - tAkYkl ~ Ao. k=l

Recalling now the meaning of the coordinates {ydo, we get

for each lEn. This shows that the error R of the linear approximation method

n

LU) ~ EAkLkU) k=l

equals to Ao. Hence R(T) ~ Ao. We shall show that R(T) = Ao. Indeed, let e be a sufficiently small positive number and let Ie be the element from no that corresponds to the point (Ao-e, 0, ... ,0), i.e., for which LUe) = Ao-e, LkUe) = 0, k = 1, ... , n. It follows from the symmetry of n that -Ie E no. But for every method S that uses only the information TU)

Then

LUe) = Ao - e ~ SUe),

L(-Ie) = -(Ao -e) ~ S(-Ie) = SUe).

LUe) - SUe) = Ao - e - SUe), L(-Ie) - S(-Ie) = -Ao + e - SUe).


Therefore, at least one of the values L(fe) or L( - !e) will be approximated by the method S with an error which is greater or equal to Ao - c. Then R(T) ~ Ao - c and since c was arbitrarily chosen, R(T) ~ Ao = R. This shows that the linear method of recovery

n

L(f) ~ LAJ:LJ:(f) J:=1

is the best one. The lemma is proved. The next proposition is an important consequence from the proof of Smolyak's

lemma.

COROLLARY 5.8. Under the same assumptions as in the lemma,

R(T) = sup {L(f): ! E n, LJ:(f) = 0, k = 1, ... , n}.

The assertion follows from the equality R(T) = Ao and the definition of Ao.

§ 5.3. Extremal Property of the Natural Spline Interpolation

Consider the problem of the best approximation of linear functionals in the space WHa, b] on the basis of the information !(X1),'" ,!(xn ), where {Xi}]' are arbitrarily fixed distinct points in [a, b]. Let B be the unit ball in W2[a, b], i.e.,

The error of approximation of a given linear functional J(f) in W2[a, b] by a method S: J(f) ~ S(f) is defined as max {IJ(f) - S(f)I: ! E B}. Clearly B is a convex symmetric body in W2[a, b]. Then, by Smolyak's lemma, there is the linear best method of recovery of J(f), i.e., an approximation formula of the form

n

J(f) ~ LAJ:!(xJ:) (5.3.1 ) J:=1

of minimal error in W2[a,b]. We show here how the coefficients {Ad of the best method can be found.

First, we prove an auxiliary theorem which is of independent interest as well.

THEOREM 5.9. Let J be an arbitrary linear functional defined in C 2r-2[a, b]. Let 1 ~ r ~ n and a ~ Xl < ... < Xn ~ b are given points. Then there exists a unique system of numbers {AJ:}~ such that

n

J(s) = LAJ:s(xJ:) J:=1

Ch. 5, § 5.3) Extremal Property of the Natural Spline Interpolation 79

Proof. Denote by tPk(Z) the unique natural spline from the class N2r-l(Zl' ... ' zn), which satisfies the interpolation conditions

i = 1, ... ,n.

Note that n

S == L:S(Zk)tPk k=l

because both sides are natural splines that take the same values at Z1. ... , Zn. Then

n

J(s) = L:S(Zk)J(tPk) k=l

and therefore Ak := J(tPk), Ie = 1, ... , n, is a solution of our problem. Suppose that {akn is any other solution. Then

n

Ai = J(tPi) = L:aktPi(Zk) = ai· k=l

Thus {Ak} is the unique solution. The theorem is proved. Denote by sJ the spline from N2r- l (zl' ... ' zn) which interpolates the function 1

at the points Z1. ... , Zn. Since

n

sJ(z) = L:/(Zk)tPk(Z), k=l

the functional we constructed in the previous theorem may be written as J(sJ). Therefore the approximation scheme J(f) ~ J(sJ) is the unique method of the form (5.3.1) which is exact for each 1 E N 2r-l (z1' ... , zn). Next, we show an interesting extremal property of this method.

Recall that a class of linear functionals was described in Peano's theorem (Theorem 3.1). For the sake of convenience we shall refer to it (or even to the more specific one, given in (3.1.2» as Peano's class.

THEOREM 5.10. Let J(f) be an arbitrary linear functional from Peano's class, defined in W2[a, b], and {zk}i be fixed distinct points in [a, b]. Suppose that 1 ~ r ~ n. Then the approximation scheme

J(f) ~ J(sJ) (5.3.2)

is the best method of recovery of the functional J on the basis of the information I(Z1), ... , I(zn) in the class W2[a, b].


Proof. By virtue of Smolyak's lemma it suffices to prove that (5.3.2) is the best of the linear methods. Moreover, we can restrict ourselves to the linear methods that are exact for all polynomials of degree r - 1. This observation comes from the fact that every linear method J(I) ~ S(I) which is not exact in 11",.-1 has an unbounded error in W2[a, b]. Indeed, suppose that J(p) - S(p) = c i= 0 for some p E 11",.-1.

Since Cp E W2[a, b] for every constant C and J(Cp) - S(Cp) = Cc, we see that the error of the method S in W2[a, b] may reach an arbitrarily large number. On the other hand, there are linear methods, such as, for example, the one based on the Lagrange interpolation at Xl, ••• , X n , which have a finite error. Therefore, the best linear method must be exact for all polynomials from 11",.-1. So, in order to prove the theorem we need to show that an arbitrary linear method J(I) ~ L1(1) which is exact for f E 11",._1 has an error in W2[a, b] greater than the error of the method J(I) ~ Lo(l) := J(sJ).

Introduce the functionals

Ri(l) := J(I) - Li(l), R2(1) := Lo(l) - L1 (I).

i = 0,1,

All ofthem are linear functionals from Peano's class and they annihilate the algebraic polynomials from 1I"r-1. Then by the Peano theorem,

b

R;(I) = J Ki(t)j<")(t) dt, i = 0, 1,2,

a

for each f E W2[a, b], where

( (X - t)+-l) Ki(t) := Ri (r _ I)! .

By the Cauchy-Bouniakovski inequality,

( b) 1/2 ( b ) 1/2 Ri(l) ~ J Kl(t) dt . J [J(r)(t)] 2 dt ,

a a

i = 0,1,2.

Therefore

where the supremum is attained for the function

The theorem will be proved if we show that IIKII2 ~ IIKtib and the equality holds only for K(t) = K1(t) on [a, b].

Consider the function K2(t). Clearly K2(t) = 0 for t ~ Xn , since K2(t) is a linear combination of the truncated power functions (Xk - t)+-l, k = 1, ... , n. Further, K2(t) = 0 for t ~ Xl, since, for such t, K2(t) = R2(p), where p(x) is the algebraic

Ch. 5. § 5.3) Extremal Property of the Natural Spline Interpolation 81

polynomial (x - W- 1 j(r - I)!. ( Recall that R2 annihilates 7rr -d. Thus, K2(t) is a spline function of degree r - 1 with knots {Xi}!, which vanishes outside (Xl. xn ).

Therefore, there exists a function G(t) E N2r- 1(X1, ... ,xn ), such that G(r)(t) == K2(t). Note that

b b

q:= J Ko(t)K2(t) dt = J Ko(t)G(r)(t) dt = Ro(G) a a

and therefore q = 0 since the method (5.3.2) is exact for all natural splines from the class N2r-1(X1, ... ,Xn ). Now, the orthogonality of Ko and K2 and Lemma 5.4 give

b b

J K5(t) dt ~ J (I{o(t) + K2(t)Fdt. a a

But R2 = Lo-L1 = (J -Lt}-(J -Lo) = R1-RO. Therefore, K2(t) = K1(t)-Ko(t). Then K2(t) + Ko(t) = K1(t) and

b b

J K5(t) dt ~ J K;(t) dt. a a

Again by Lemma 5.4, the equality is attained if and only if K2(t) == 0, i.e., K1(t) == Ko(t). The proof is completed.


For the early development of natural spline interpolation see Quade and Collatz [1938], Holladay [1957], de Boor [1963], and Schoenberg [1964]. The paper by Greville [1969] gives a fundamental account of this subject. The extremal property of the natural spline interpolant described in Theorem 5.5 was discovered by Holladay [1957] in the particular case r = 2. Then it was extended to the present form by de Boor [1963] and Schoenberg [1964]. A similar problem of minimization the Lp-norm of /(r)(t) over all functions from W;[a,b] that interpolate {(Xi,Yi)}l was considered by de Boor [1976c]. Further extensions covering BirkhofT interpolation, and interpolation by generalized splines, were given by Ahlberg and Nilson [1966], and Schoenberg [1968].

Lemma 5.7 was proved by S. A. Smolyak [1965] in his dissertation (Moscow State University, 1965). The paper of Bakhvalov [1971], which contains the proof, is more easily available. The approximation methods that minimize the L2-norm of the remainder, expressed by Peano's theorem, are known as best in tlle sense of Sard (see Sard [1963]). Schoenberg [1964] revealed the remarkable property of the natural spline interpolant showing that the best approximation method in the sense of Sard to a linear functional L(f) is obtained by operating with L on the unique natural spline Sf E N2r- 1(X1, ... , xn) which interpolates / at the same nodes Xl,"" Xn.

Chapter 6

PERFECT SPLINES

A perfect spline of degree r with knots 6, ... ,em is any expression of the form

pet) = taiti-1 +, (tr + 2 ~(-I)k(t - ek)+ ),

where {ad'i and, are real numbers. The characterizing property of these splines is that lP(r)(t)1 = constant for every

ton(-oo,oo).

§ 6.1. Favard's Interpolation Problem

The perfect splines came into Approximation Theory as solutions of interesting extremal problems in the Sobolev space W~[a, b]. We shall consider here the so-called Favard interpolation problem of the characterization ofthe function f from W~[a, b], which satisfies some preassigned interpolation conditions and minimizes IIf(r) 1100. The next simple example is devoted to a particular case.

LOUBOUTIN'S PROBLEM. Show that

inf{llf(r)lloo: f EW~[-I,I], f(-I) = 0, f(l) = 1,

f(j)( -1) = f(j)(I) = 0, j = 1, ... , r - I} = (r - 1)!2r - 2

and the infimum is attained only for the function

1

1/J(X) = (_lr-12r-2 j(x - t)+-l signUr_1(t)dt,

-1

where Ur -1(t) is the Tchebycheff polynomial of the second kind, i.e.,

U. () __ 1_ sin (r arccos t) r-1 t - 2r-1 (1 _ t2)1/2

Solution. First we shall show that the function 1/J satisfies the required interpolation conditions. In order to this, recall that the Tchebycheff polynomial Ur - 1 (t) is completely characterized by the following orthogonality property:

1

j p(t)signUr_1(t)dt = ° -1

82

for each p E 1l"r-2. (6.1.1)

Ch. 6, § 6.1] Favard's Interpolation Problem

Note otherwise that

and therefore

1 ,.. ,../r

1 I sin (r arccos t) I 1· 1 . (1 _ t2)1/2 dt = Ism r(}ld(} = r Ism r(}ld(} -1 0 0

1

,..

= 1 I sin (}Id(} = 2 o

1 !Ur-l (t)1 dt = 2r~2· -1

83

(6.1.2)

It follows immediately from the definition of 1/; ( x) that 1/;(j) ( -1) = 0 for j = 0, ... , r - 1. Further,

1

1/;(j)(I) = (_lr-12r- 2 (r ~ ~ ~)~)! 1(1-q-j-lsignUr_1(t) dt -1

and by (6.1.1), 1/;(j)(I) = 0 for j = 1, ... , r-1. Finally, using again the orthogonality of sign Ur -l(t) to 'lrr-l and (6.1.2), we get

1 1

1/;(1) = (_lr-12r- 2 1(1- tr-1sign Ur- 1(t) dt = 2r- 2 J !Ur-l(t)1 dt = 1.

-1 -1

Now, we shall show that 1/; is the unique extremal function for our problem. Indeed, assume that there is a function f E W~[-I, 1] which satisfies the required interpolation conditions and

IIf(r)lloo ~ c:= 111/;(r) 1100 = (r_l)!2r- 2 •

Consider the difference g(t) := 1/;(t) - f(t). Clearly

g(j)( -1) = g(j)(I) = 0 for j = 0, ... , r - 1.

(6.1.3)

Then, by Rolle's theorem, g(r)(t) will change its sign at least r times in (-1,1), provided f is not identically equal to 1/;. But signg<r)(t) = sign1/;(r)(t) on (-1,1), because of the assumption (6.1.3). It remains to note that

sign1/;(r)(t) = (-lr-lsignUr_l(t),

which shows that g(r)(t) can have at most r - 1 sign changes. The contradiction completes the proof.

It is easy to see that 1/;'( t) is a B-spline of degree r - 1 with knots at the extremal points 11k = cos k'lr Ir, k = 0, 1, ... ,r, of the Tchebycheff polynomial of the first kind Tr(t) := cos(r arccos t). Since 1/;'(t) is, in addition, a perfect spline, it is termed a perfect B-spline.

84 Perfect Splines [Ch. 6, § 6.1

Exercise 6.1.1. Show that

Hint. Take into account that the result was proved already for [a, b] = [-1,1] (in Louboutin's problem) and apply a linear transformation.

Now let us consider Favard's interpolation problem under more general, Birkhoff interpolation, conditions. In order to simplify the proof we shall use an important result from topology, known as Borsuk's antipodality theorem (see Borsuk [1933]).

THEOREM 6.1 (Borsuk's theorem). Let X m +l be a given normed linear space of dimension m+ 1. Let Sm be the unit sphere in Xm +l , i.e., Sm := {:e E X m+1: 1I:e1l = I}. Suppose that tjJ(:e) is a continuous, odd (i.e., tjJ(-:e) = -tjJ(:e)) mapping of Sm into]Rm. Then there exists a point :e* E Sm such that 4>(:e*) = 0.

Before considering to the formulation of the Favard's interpolation problem we list below certain requirements which will be imposed on the incidence matrix E = { } n r-l eij i=l,j=O· C1. The number of I-entries in E is equal to N := r + m + 1, m ~o. C2. E is conservative. C3. There exist components eikjk = 1, k = 0, ... ,r-I, of E, such that the system

of linear equations

{al + a2x + ... + arxr- l }(jk) L=";k = Yikik, k=O, ... ,r-I,

with respect to {aill has a nonzero determinant. We shall denote this determinant by det w.

With any given E = {eij }~=l,j:~, :e = (Xl, ... , Xn), a ~ Xl < ... < Xn ~ b, and y = {Yij}, we associate the set of functions

F(E,x,y):= {! EW~[a,b]: fU)(Xi) = Yij if eij = I}.

THEOREM 6.2. Suppose that the incidence matrix E = {eij}i=l,j:~ satisfies the conditions CI-C3. Then for each :e and y the set F(E,:e,y) contains a perfect spline P(x) of degree r with at most m knots. Moreover,

(6.1.4)

Proof. The extremal property (6.1.4) is an immediate consequence from the construction of P. Indeed, let P(x) be a perfect spline from F(E,:e,y) with no more than m knots. Assume that there is a function f EF(E,:e,y) such that

(6.1.5)

Then the function g(t) := P(t) - f(t) satisfies the conditions 9U)(Xi) = ° for each eij = 1. Apply Rolle's theorem consecutively to g(t), g'(t), ... , g(r-l)(t). Since E was

Ch. 6, § 6.11 Favard's Interpolation Problem 85

supposed to satisfy the conditions CI-C3, Rolle's theorem will imply the existence of at least N - r = m + 1 sign changes of gCr)(t). On the other hand, it follows from assumption (6.1.5) that signg(r)(t) = signp(r)(t) and, evidently, p(r)(t) changes its sign only at the knots 6, ... ,em of P, i.e., at most m times. The contradiction proves (6.1.4).

So, it remains to show the existence of a perfect spline P with the desired properties. Without loss of generality, we may assume that [a, b] = [0,1]. Let

8m := {h = (ho, ... , hm ): Ihol + ... + Ihml = I}.

We define below a mapping ¢(h) of the sphere 8m into ~m. Set

k= 1, ... ,m.

Clearly ° ~ 6 ~ ... ~ em ~ 1. Introduce the function O(h;t), defined on (-00,00) in the following way:

O(h;t):= signh"

where eo := -00, em+1 := +00. For fixed h E 8m, let us consider the system of equations

(6.1.6)

for k = 0, ... ,r, with respect to the unknowns c, 0"1, ... ,O"r. Here i", i" are defined as in the condition C3 for k = 0, ... , r - 1, while ei.j. is some other I-entry of E. Denote by D the matrix of the system (6.1.6). Let

r

q(x) = l)tixi-1 "=1

be the unique solution of the system, given in C3. Adding to the first column of D all others multiplied by iiI, ... , iir , respectively, we see that

(6.1.7)

Assume that det D = ° for each choice of the element ei.j. = 1 of E. Then (6.1.7) and the condition C3 would imply

for each eij = 1.

Therefore the polynomial q is the desired solution of the interpolation problem in this trivial case.


Now suppose that there exists an element ei.j. = 1 for which det D :I O. Then the system (6.1.6) will have a unique solution c(h), lXo(h), ... , lXr_l(h). Note that each of these functions is odd and continuous in 8 m • Order the quantities

for eij = 1, eij E E, (i, j) :I (i", jk), k = 0, ... , r, in some order and denote them by IjJl(h), ... , IjJm(h). Define the mapping 1jJ: 8 m -+ ~m in the following way: ljJ(h) = (ljJl(h), ... , IjJm(h)). It transforms continuously 8 m in ~m and ljJ(h) = -1jJ( -h), i.e., IjJ is odd. Then, by Borsuk's theorem, there exists a point h* E 8 m such that ljJ(h*) = O. We claim that c(h*) :I O. Indeed, otherwise the perfect spline

would satisfy the conditions gU)(1:i) = 0 for each eij = 1 from E. This follows from (6.1.6) for (i, j) = (ik, jk), k = 0, ... , r, and from the equalities IjJl(h*) = ... = IjJm(h*) = 0, for the others (i,j). Now applying Rolle's theorem consecutively for g, g', . .. , g(r-l), we conclude that g(r)(t) must have at least N -r = m+1 sign changes. But g(r)(t) = O(h*;t) and O(h*;t) changes its sign at most m times, according to the definition of (J. The contradiction shows that c(h*) :I O. Then the function g(t)/c(h*) is the desired interpolating perfect spline. The proof is completed.

We formulate separately an important particular case of this theorem, which corresponds to the Hermite interpolation.

Let {Vkn be preassigned integer numbers satisfying the requirements: 1 ~ Vk ~ r, k = 1, ... , n, N := VI + ... + Vn ~ r. Given z = (1:1, ... ,1:n), a ~ 1:1 < ... < 1:n ~ b, and y= {Yij}?=l.j~(/, we denote by F(z,y) the set

{J E W::'[a,b]: j<j)(1:i) = Yij for i = 1, ... , n, j = 0, ... , Vi - I}.

COROLLARY 6.3. The set F(z, y) contains a perfect spline P of degree r with no more than N - r-1 knots. Moreover,

The incidence matrix E that corresponds to the Hermite interpolation problem with m nodes of multiplicities Vb . .. , Vn, respectively, satisfies the conditions C1 -C3. Then the assertion follows from Theorem 6.2.

Next we derive another consequence of Theorem 6.2, which can be recognized as the fundamental theorem of algebra for perfect splines.

THEOREM 6.4. Let rand N, 0 ~ r ~N, be given natural numbers and

Ch. 6, § 6.1] Favard'$ Interpolation Problem 87

be arbitrary fixed points, 1 :::;; VA: :::;; r, k = 1, ... , n, V1 + ... + Vn = N. Then there exists a unique (up to multiplication by -1) perfect spline <P of degree r with at most N -r knots, which satisfies the conditions:

a) 1I<p(r)lIoo = 1, b) <p(A)(ZA:) = 0, k = 1, ... , n, A = 0, ... , VA: - 1.

This spline has exactly N -r knots.

Proof. Let Z be an arbitrary point such that Z > TN' By virtue of Corollary 6:3 there exists a perfect spline 1/J of degree r with no more than N - r knots, which satisfies the interpolation conditions

k = 1, ... ,n, A = 0, ... ,VA: - 1, (6.1.8)

Note otherwise that any such spline has a minimal norm of its r-th derivative amid all functions from W~[Tb TN], which satisfy (6.1.8). Clearly, 111/J(r)lIoo t= 0, since otherwise 1/J must coincide with a polynomial of degree r - 1 and then, because of the assumption r:::;; N, 1/J could not satisfy the interpolation conditions (6.1.8).

Consider the perfect spline <p(t) := 1/J(t)/II1/J(r)lIoo. It satisfies conditions a) and b) of the theorem.

We shall show that the spline <p has exactly N zeros, counting the multiplicities. More precisely, <p vanishes only at the points Zl," ., Zn with multiplicities 111, ..• ,lin, respectively. Indeed, assume the contrary. Then <p(t) vanishes at some point to, distinct from Zl, ... , Zn, or <p(lI k )(ZA:) = ° (<p(r)(t) changes sign at ZA: if 1IA: = r) for some k E {I, ... , n}. Applying Rolle's theorem we conclude that <p(r)(t) changes its sign at least N - r + 1 times, i.e., <p has at least N - r + 1 knots, which contradicts the definition of <po

One can see in the same fashion that <p has exactly N -r knots. Indeed, since <p(t) has N zeros, Rolle's theorem implies that <p(r)(t) has at least N - r sign changes. But, by the definition, <p has at most N - r, and therefore, exactly N - r knots. The existence part of the theorem is proved. Next we shall show the uniqueness. In order to do this, assume that there are two distinct splines <PI and <P2, satisfying a) and b). Then there is a point e t= ZA:, k = 1, ... , n, for which <PI(e) t= <P2(e). We saw already that <P1(e) t= 0, <P2(e) t= 0, since <Pi(e) vanishes only at {XA:}~' Suppose, in addition, that <P1 is distinct from -<P2. Then we may assume without loss of generality that <P1 (e) > <P2(e) > 0. Construct the function

<P2(e) g(t) := <p2(t) - <P1(e) <P1(t).

It has at least N + 1 zeros: TI, ... , TN and e. But signg(r)(t) = sign<p~)(t) for each t, since <P2(e)/<P1(e) < 1. Then g(r)(t) cannot vanish on an interval and, hence, e, T1, ... , TN are isolated zeros of g. By Rolle's theorem g<r)(t), and consequently <P2(t), will have at least N + 1 - r sign changes. This contradicts the assumption that <p2(t) has N - r knots at most. The theorem is proved.

Further, for any given system of zeros Z = «Zl' 111)"'" (Zn, lin)), we shall denote by <Pr(Z; t) the perfect spline from Theorem 6.4, which is positive for t > Xn . Clearly, -<Pr(z; t) satisfies also the requirement of the theorem.

Now we shall reveal an interesting extremal property of the perfect spline <Pr(z; t).

88 perfect Splines [Ch. 6, § 6.1

COROLLARY 6.5. Let

Then, for each e E [a, b],

sup{f(e): f E F(z)} = l<Pr(z;e)l·

Proof. Assume the contrary. Then there exists a point e in [a, b], distinct from Xl, ... ,Xn , and such that

f(e) > l<p,.(z; e)1 > O.

Let c := sign<pr(z;e). Consider the function

It vanishes at least at the points z and e. This implies by Rolle's theorem that g<")(t) has at least N + 1- r sign changes. But this is impossible, since sign g(r)(t) = sign<p~r)(z;t). The theorem is proved.

Let us note that the zeros (1'1, ... , TN) == «Xl, V1), ... ,(Xn , vn )) and the knots 6 < ... < eN-,. of the perfect spline <p,.(z; t) satisfy the so-called interlacing condition

k= 1, ... ,N, (6.1.9)

with any e-(,.-l) ~ ... ~ eo < 1'1 and TN < eN-r+1 ~ ... ~ eN' Indeed, assume that there is an index k, for which Tic ~ elc. Since TN ~ b < eN-,.+1' it is seen that k ~ N - r. Denote by t/J(t) the spline of degree r with knots elc+1,'" ,eN-r, which coincides with <Pr(z;t) on the interval (elc, b). It is clear that t/J is a perfect spline with N - r - k knots: ek+1,' .. ,eN _,.. On the other hand, by the assumption 7"Jc ~ elc, t/J vanishes at Tic, ... , TN and then, in view of Theorem 6.4, t/J has exactly m knots, where m is the number of zeros minus the degree, i.e., m = N + k - 1- r. This yield a contradiction. Therefore Tic < elc for each k.

Assume now that Tic ~ 6-,. for some k. It follows from the construction of the sequence {ed that k > r. Let t/J(t) be the perfect spline of degree r with knots 6, ... ,elc-,.-l ,which coincides with <p,.(z;t) on (a,elc-r). It has k zeros and k-r-1 knots. This contradicts Theorem 6.4. The claim (6.1.9) is proved.

Finally, let us mention another important property of <p,.(z; t).

LEMMA 6.6. The perfect spline <Pr(z;t) depends continuously on z. Moreover, the parameters 0'1, ... ,a,., e1, . " ,eN-r of <p,.(z; t) are differentiable functions of Xl, ... ,

Xn in the domain Xl < ... < X n •

Proof. The parameters 0'1, ... , O'r, 6, ... ,eN -r of <Pr (z; t) are defined by the system of N equations

Ch. 6, § 6.1] Favard's Interpolation Problem 89

for k = 1, ... , n, j = 0, ... , Vic - 1. It is seen that the Jacobian ~ of this system with respect to a1,' .. , a r , 6, ... ,{N -r coincides, up to a nonzero constant, with the determinant which corresponds to the Hermite interpolation problem by splines from Sr-1(6, ... ,(N-r) at the nodes z. Since (by (6.1.9» the knots {{i}f-r and the points z satisfy the interlacing condition, Theorem 4.20 implies that ~ :I O. Then, by the implicitfunction theorem, 01, ... , Or andet, ... ,(N-r are differentiable functions of Xl, ••• ,Xn . The proof is completed.

Exercise 6.1.2. Let z = «a,r),(b,r», a < b. Find IPr(z;t).

Hint. Using the orthogonality ofsignUr(t) to all f E 1I"r-1 show (as in Louboutin's problem, section 6.1) that the function

1

1/Jr(X) := (r ~ I)! J (x - t)+-l sign Ur(t) dt -1

satisfies the condition 1/J~j)( -1) = 1/J~j)(I) = 0, j = 0, ... , r -1. By Taylor's formula, 1/J~r)(x) = sign Ur(x). Thus 1/Jr(x) is a perfect spline of degree r with r knots in (-1,1). In view ofthe uniqueness of IPr(z; t) (see Theorem 6.4), 1/Jr(t) = IPr(z; t) on [-1,1]. Therefore, for every a < b,

( b-a)r (2 a+b) IPr(z;t)= -2- 1/Jr b-at-b-a =:1/Jr([a,b];t),

i.e., IPr(z; t) is the perfect B-spline of degree r in (a, 11).

Exercise 6.1.3. Let z = «a, r), (b, r», a < b. Find

b

I(a,b):= J IPr(z;t)dt. a

Hint. Use Exercise 6.1.2 to get

1

( b - a)r+1 J I(a, b) = -2- 1/Jr(X) dx -1

= C ~ ar+' j (hr-~)l~~' d. ),;gnU,(t)dt -1 -1

1

( b- a)r+1 J (l-ty = -2- r! signUr(t)dt

-1

1

( b - a)r+1 (_I)r J = -2- ----;:! IUr(t)1 dt

-1


and then by (6.1.2), (-It (b - a),.+l

I(a,b) = ~ 22,.

Exercise 6.1.4. Let Z = «X1,1I1), ... ,(Xn ,lIn », Xl < ... < Xn , 1 ~ IIA: ~ r, k = 1, ... , n. Suppose that

"'.+1 I J SO,.(Zjt)dtl ~ C, k = 1, ... , n-1.

"'. Show that there is a constant L (independent of Xb .•. , xn) such that Xn - Xl ~ L.

Hint. Assume the contrary. Then for every M > 0 there is a pair a := xA:(M) < xJ:+1(M) := b with b - a >M. By Corollary 6.5,

I",,.([a, b]j t)1 ~ ISO,.(Zj t)1 on [a, b].

Thus, using Exercise 6.1.3,

6 6

C ~ If SO,.(Zj t) dtl ~ If ",,.([a, b]j t) dtl = II(a, b)1 = M,.+1/22,.,

a a

which is a contradiction.

§ 6.2. Oscillating Perfect Splines

Let [a, b] be a given finite interval. For fixed "1, . .. , lin and r we denote by 1',.(111, ... , lin} the set of all perfect splines of degree r with "1 + ... + lin - r knots, which have n distinct freely chosen zeros Xl < ... < Xn in (a, b) of multiplicities 111, ... , lin,

respectively. According to Theorem 6.4, every P from 1',.(111, ... ,lin) is defined uniquely (up to a constant multiplier) by its zeros {Xi}~. The multiplicities lib .•. , lin

fix the mode of oscillation of P(t). The perfect splines from 1',.(111, ... , lin}, like the algebraic polynomials with real

zeros, appear prominently in the error analysis of interpolation schemes based on the zeros of P. Before proceeding to this subject in the next section, we give here some important results involving oscillating perfect splines.

6.2.1. Splines with preassigned integrals over subintervals. Let %1 < ... < Zm be fixed points on the real line, and 1'1, . .. , I'm, together with "1, ... , lin, be preassigned multiplicities which do not exceed r. Set N := 1'1 + .. . +I'm +111 + ... +lIn •

Given the positive numbers {eA:}? , we consider the problem of determining a perfect spline

(6.2.1)

Ch. 6, § 6.2) Oscillating Perfect Splines 91

of degree r with N - r knots, which satisfies the conditions

P().)(Zi)=O, i=l, ... ,m, ,\=O, ... ,J,li-1,

P(i)(Xk)=O, k=l, ... ,n, j=0, ... ,lIk-1,

%"+1

I J p(t)dtl=ek' k=l, ... ,n,

for some Xl < ... < Xn < Xn+l := Zl, 6 < ... < eN -r and real coefficients al, ... , ar . These conditions form a system of N + n equations

k = 1, ... ,N +n, (6.2.2)

in unknowns X = (Xl, ... , Xn , at, ... , ar , 6, ... ,eN-r). Assume that the equations in (6.2.2) are ordered in the following way

p(",,-l)(Xk) = 0, k = 1, ... , n,

p(Xk) = 0, p'(Xk) = 0, ... ,p(v,,-2)(Xk) = 0,

i 1

p(t) dt - Ckek = 0, k = 1, . .. ,n,

i = 1, ... , m, ,\ = 0, ... , J,li - 1,

where Ck is the sign of p(t) on (Xk, Xk+d, i.e., cn := (_l)l1l+'''+l'm and Ck = cn( _1)""+1+"'+"" for k = 1, ... , n-l.

Denote by J(XlJ ... , Xn , al,.'" a r , 6, ... ,eN-rj et, ... , en) (abbreviated to J(Xj el, ... , en), or simply J) the Jacobian matrix ofthe system (6.2.2) with respect to X.

LEMMA 6.7. Let {lIkn and {J,li}'r be an arbitrary system of natural numbers satisfyingtherequirements 1 ~ Ilk ~ r, 1 ~ /Ji ~ r, N:= 111+' .. +lIn +/Jl+ ... +J,lm ~ r. Then for every given set of points Zl < ... , zm and positive numbers {ek n,

detJ(Xjel, ... ,en):I 0

at the solution X of the system (6.2.2).

Proof. Suppose that X satisfies (6.2.2). Let us find the value of det J at X. Clearly J has the form

Xl .. 'Xn al···ar , 6 ···eN-r 1

Dl n

n+1

0 D

N+n


where the block marked by 0 consists of zero elements and

Notice here that the points z = «Zl' Vl), ... , (ZA' VA» and

define p uniquely. Precisely, p(t) == CPr(z, Zj t), where CPr is the perfect spline introduced in Theorem 6.4 for (Tl, ..• ,TN ) = (z,z). Thus, if Vi = r for some~, then Zi does not coincide with a knot of p(t) (see the proof of Theorem 6.4) and hence p(lIi )(Zi) is well defined.

Unfolding det J by the LaplacJ formula, along the first n columns, we get

n

det J = II p(lIi )(Zi) . det D, i=l

(6.2.3)

where the matrix D is obtained from J by deletion of the first n rows and columns. For convenience, we shall use the notations

and for Yl ~ ... ~ YN (Yi < Yi+r-l),

subject to the usual convention that for coincident y's the repeated rows are replaced by consecutive derivatives of {Ui}.

Since ~.+1 ~.+1

0:; J p(t) dt = J t;-ldtk, ~. ~.

~.+1 ~.+1

a~i J p(t) dt = (_1)k+12r J (tk - ei)+-ldtk,

we can write det D in the form

~2 ~ .. +1

det D = J ... J F(tl, ... , t A ) dtl ... dtA , (6.2.4)

:1:1 ~"

where

Ch. 6, § 6.2] Oscillating Perfect Splines 93

with

Y = (Y1, ... , YN) = {(Xl, V1 - 1), t1, ... , (xn, Vn - 1), tn, (Zl, Jl1)' ... , (zm, Jlm)}.

Observe that Xk ~ tk ~ Xk+1, k = 1, ... , n, i.e., the points yare arranged in non-decreasing order. Then, in view of the total positivity of the truncated power kernel (see Corollary 4.24), there is a u such that

on the domain of integration O. Since F is continuous in 0, we shall deduce from (6.2.4) that det D :F 0, showing that there is a point (t1,' .. , tn) in 0 at which F does not vanish. To this end, recall from (6.1.9) that the zeros (T1"'" TN) = (:1:, %) and the knots {edi"-r of the perfect spline p satisfy the interlacing condition ei-r < Ti < ei, i = 1, ... , N. But these inequalities are strict. Thus, y will satisfy the interlacing condition too for each (t~, ... , t~) very close to (Xl> ... , xn). Then, according to Corollary 4.24,

(-ltF(t~, ... ,t~) >0.

This implies det D :F O. Using the fact that p(/I;)(Xi) :F 0 (since p has exactly N zeros, counting the multiplicities), we get from (6.2.3) that det J:F O. The lemma is proved.

THEOREM 6.S. Let {Vk}r and {Jldi be arbitrary preassigned natural numbers satisfying the requirements 1 ~ Vk ~ r, 1 ~ Jli ~ r, V1+ ... +Vn+Jl1+ ... +Jlm ~ r. Let the points Zl < ... < Zm be fixed on the real line. Then for every set ofpositive numbers {ek lr there exists a unique system of points {Xk n, Xl < ... < Xn < Xn+1 := Zl, and consequently a unique perfect spline p(t) of the form (6.2.1) such that

P(>')(Zi) = 0,

pU)(Xk) = 0,

i=1, ... ,m, ).=O, ... ,Jli-1,

k = 1, ... ,n, j = O, ... ,Vk - 1,

"k+l

\ J P(t)dt\ = ek, k = 1, .. . ,n.

"k

Proof. The proof proceeds by induction on n. When n = 0, there are not integral conditions and the existence and uniqueness of the solution p(t) follows from Theorem 6.4. In this case, p(t) = <Pr(%;t) for each natural m.

Assume that the theorem holds for some n ~ 0 and every m. Suppose that {Vk}~+l are arbitrary preassigned multiplicities. Let {ek }~+1 be given positive numbers. Fix the system of zeros % = « Zl , Jlt), ... , (zm, Jlm)), Zl < ... < Zm· Let e be an arbitrary fixed point in (-00, Zl). The induction hypothesis ensures that there exists a unique system of points {Xk(enr, X1(e) < ... < xn(e) < e =: Xn+1(e), such that

"k+l(e)

\ J pe(t) dt\ = ek, k = 1, .. . ,n, (6.2.5)

"k(e)

94 Perfect Splines

where pe(t) = it'r(z(e), z(e);t) and

z(e):= (Zl(e),V1), ... ,(xn(e),vn)),

z(e):= (e,Vn+1),(Zl,l'l), ... ,(Zm,l'm)).

[Ch. 6, § 6.2

The theorem will be proved if we show that there is a unique e E (-00, Zl) such that the equality (6.2.5) holds for k = n + 1 as well with Xn +2(e) := Zl. In order to this, we begin a careful study of the function

Zl

l'(e) := I J pe(t) dtl, e

which is actually the left-hand side of (6.2.5) for k = n + 1. Consider the system of equations

k=l, ... ,N+n, (6.2.6)

that defines the parameters X = (Xl, ... , Xn, all"" a r ,6, ... ,eN-r) of pe(t) provided e < Zl < ... < Zm are fixed zeros of multiplicities Vn+1, 1'1, ... , I'm, respectively. Here N := V1 + ... + vn+1 + 1'1 + ... + I'm. According to Lemma 6.7, the Jacobian det J of (6.2.6) with respect to X is non-zero at the solution X(e) of (6.2.6) for each e E (-00, Zl). Thus, by the implicit function theorem, all components of X(e) are differentiable functions of e in (-00, zI). Moreover, the corresponding derivatives are expressions of the form A(e)/det J(e) with a certain determinant A(e) whose elements, like those of J(e), are continuous functions of e. Therefore l'(e) has a continuous derivative at each e from (-00, zd.

Next, note that, by Corollary 6.5.,

where 771, ... , 77r is any subsequence of {z(e), z(en, which contains e and Zl. This inequality implies

Zl

l'(e) ~ ~ J I(t - 77d··· (t - 77r )Idt. e

Since there is a constant L such that 77r - 771 ~ L (see Exercise 6.1.4), we conclude that l'(e) -+ 0 as e -+ Z1' Similarly, using the inequality

on [e,zd,

where 1/Jr is the perfect B-spline on [e,zd (see Exercise 6.1.2 for the definition), we get


(This integral was evaluated in Exercise 6.1.3). Therefore U(e) -+ 00 as Zl - e -+

00. Thus the integralu(e) takes all values from 0 to 00 as e traverses (-OO,Zl)'

In particular, there exists a point eo E (-oo,zt), for which U(eo) = en+!' So, X1(eO), ... , xn(eo), eo is a solution ofthe problem. The existence part ofthe theorem is proved. Next we prove the uniqueness.

Recall first that the system (6.2.6) was associated with the problem with n free (Xl < ... < xn) and m+1 fixed (e < Zl < ... < zm) zeros. Now consider the system

1e=1, ... ,N+n+1, (6.2.7)

this time with X O = (X1, ... ,Xn,e,a1, ... ,ar,e1, ... ,eN-r), which corresponds to the problem with n + 1 unknown (Xl < ... < Xn < xn+! := e) and m fixed (Zl < ... < zm) zeros. Clearly (6.2.7) is obtained from (6.2.6) adding only one additional equation

and considering e as a free parameter in f~+n+! and (6.2.6). We assume further that the equations in (6.2.7) are ordered in the following way

f2 := fie = 0, Ie = 1, ... , N + n (i.e., this is (6.2.6»,

f~+n+! := U(e) - en +1 = O.

We shall find U'(e). Set for simplicity

Clearly,

But

of~oen+! = :e I] pe(t) dtl = 0,

e since pe(e) = O. Further, by the implicit function theorem (applied to (6.2.6»,

, (e) - _ det A.1e ale - detJ' Ie = 1, ... ,N,

where A.1e is the matrix obtained from J by replacement of the (n + Ie )-th column by

(aft OfN+n)T ae""'---ae .


Therefore

'et) ___ 1_ (OI'k+n+!.6. + + ol'k+n+l.6. ) fl .. - det J Oal 1· . . oa N N'

Now, taking into account that t2 == be for k = 1, ... , N +n, it is seen that the expression in the brackets is just the expansion of the Jacobian det.fl of If,· .. , IJ. +n+l with respect to XO, along the elements of the last row (up to a sign which we do not need to worry about). Therefore

'et) = det .fl fl.. C det J

with some c = +1 or c = -1. By Lemma 6.7, lee) i= O. Since we know already that flee) is small for Zl - e close to zero and is large for large Zl - e, we conclude that flee) is a strictly decreasing function in (-00, zt}. Therefore, there exists exactly one e E (-00, Zl) for which flee) = en+l. The induction is complete. The theorem is proved.

THEOREM 6.9. Let the multiplicities {VA:}r be fixed satisfying the requirement 1 :::; VA: :::; r, k = 1, ... , n, N := VI + ... + Vn ~ r. Then, for every finite interval [a, b] and each system of positive numbers {edo, there exists a unique set ({XA:}1, c) of points a < Xl < ... < Xn < b and a constant c such that

k = 1, .. . ,n, j = 0, ... , VA: - 1,

k = O, ... ,n, (6.2.8)

where pet) is a perfect spline of the form (6.2.1). The coefficient c is a strictly increasing function of eo, ... , en in the domain eo > 0, ... , en > O.

Proof. Note first that the perfect spline pet) is uniquely defined by its zeros x = «Xl, VI), ... , (xn, /.In)). To be precise, in view of Theorem 6.4, pet) = If'r(x; t).

The existence and uniqueness of c and {XA:}r follows easily from the previous theorem. Indeed, have we proved already that for a fixed point Zl there exists a unique system of points Y = {YA:}r such that Yl < ... < Yn = Zl and

Yk+l

I J If'r(Y; t) dtl = eA:, k = 1, ... ,n-1. (6.2.9)

Yk

Since 1f'~(Y; t) has no zeros outside (Yl, Yn), If'r(Y; t) is strictly monotonic in (-00, yt) and (Yn, 00). Then there exist a unique Yo and Yn+! such that Yo < Yl, Yn+! > Yn, and (6.2.9) holds for k = 0 and k = n as well. Now the linear transformation (J: [Yo, Yn+!] -+ [a, b] (precisely, (J(t) = a + (t - Yo)(b - a)/(Yn+! - Yo)) defines the points

k = 1, ... ,n.


Further, since !lk+l "'10+1

I j <Prey; t) dtl = I j <Pr (y; 0-1 (t») dtl !lk "'10

( b-a )r+11"'jk+l I = <Pr(Z; t) dt , Yn+1 - Yo

"'10 it is clear that

c=I/( b-a )r+1 Yn+1 - Yo

and pet) = <Pr(z; t). The uniqueness of c and {xdf follows from the one-to-one correspondence defined by the linear transformation o.

It remains to show the monotone dependence of c on eo, ... , en. In order to this, consider (6.2.8) as a system of N + n + 1 equations in unknowns X = (Xl, ... , X n , c, at, ... ,ar, 6, ... ,eN -r). Suppose that the equations are ordered in the following way

k = 1, ... ,n, "'I

j pet) dt - coeo = 0,

"'10+1 p(x,:) = ... = p( Jlk- 2)(Xk) = 0, j pet) dt - Ckek = ° (k=I, ... ,n),

where Cn = 1, cit = (_IYk+l+ ... +JI .. for k = 1, ... , n - 1, Xo := a, X n +1 := b. Denote by J 1 the Jacobian matrix of this system with respect to X. It is seen that J 1 has the form

Xl·. 'Xn C a1 ... ar 6 ···eN-r 1

D1 n

n+l

° D2

N+n+l

where D1 = diag {p( Jll)(xd, ... ,p{JI .. )(xn )}. The structure of D2 is similar to that of D in the proof of Lemma 6.7. Observe that the first column of D2 is

( "'I "'2 ",,,+1) T

j pet) dt, 0, ... , j pet) dt, 0, ... , j pet) dt , ~o %1 ~ft

98 Perfect Splines (Ch. 6, § 6.2

where the integrals are on positions 1, 1 + 0"1, 1 + 0"2, ... ,1 + O"n (O"k := V1 + ... + Vk, k = 1, ... ,n, 0"0 := 0). Unfolding det D2 along the elements of the first column we get

Xk+l

det D2 = t J p(t) dt (-Irk ~k, k=O Xk

where

~k:= f··· J Fk(t1, ... ,tn) dt1· .. dtn,

It I~

and y~, ... , y~ is the ordered set of points (Xl, V1 - 1), ... , (xn, Vn - 1), tl, ... , tn. Here ti E If, i = 1, ... , n, and

By the total positivity of the truncated power kernel (see Corollary 4.24),

with some 0" independent of k. Moreover, it is seen, as in the proof of Lemma 6.7, that (-1)" Fk(t1, ... ,tn) > 0 for those t1 E If, . .. ,tn E I~, which are very close to Xl, ... , X n , respectively. Thus,

n

det D2 = LCkek( _Irk (-lrl~kl. k=O

n

det D2 = (-1 r+N L ek I~kl k=O

and consequently det J1 = det D1det D2 #= o.

Then, by the implicit function theorem, c( eo, ... , en) is a differentiable function of ek and

This shows that


provided ek > ° and hence c is a strictly monotone function of ek. The proof is complete.

6.2.2. Interpolation at the extremal points. Let 10, ... , Im+1 (m ~ r + 1) be a given sequence of numbers satisfying the requirement ("':+1 - Ik)(fk - Ik-d < 0, k = 1, ... , m. Consider the problem of determining a perfect spline P(t) of degree r with m + 1- r knots which interpolates 10, ... , 1m+! at some points a = TO < TI < ... < Tm+l = b and, in addition, remains monotone on (Tk, Tk+d for k = 1, ... , m. The last condition implies that I must have a local extremum at every interior node Tk, i.e., that P'( 7);) = ° for k = 1, ... , m. The next theorem treats a more general interpolation problem involving a block of vanishing consecutive derivatives at Tk.

THEOREM 6.10. Let {Ilk}i" be given multiplicities, 1 ~ J.lk ~ r, k = 1, ... , m, M := III + ... + J.lm, M ~ r + 1. Then for every fixed interval [a, b] and every set of numbers 10, ... , I m+1' such that

k= 1, ... ,m,

there exists a unique set of points a = TO < TI < < Tm+! = b and a unique perfect spline P(t) of degree r with M + 1 - r knots which satisfies the conditions

P(fk)=lk, k=0, ... ,m+1,

P'(Tk) = P"(Tk) = ... = p(l'k)(Tk) = 0, k= 1, ... ,m.

The quantity IIP(r) 1100 is a strictly increasing function of I/HI - Ik I, k = 0, ... , m.

Proof. Set G(t) := P'(t). The interpolation problem described in the theorem reduces to the following conditions on G:

{G(j)(7);) =0, k=1, ... ,m, j=0, ... ,J.lk-1,

I TkJ+l () I 1 ( ) ()I 1 1 (6.2.10) Tk G t dt = P Tk+l - P Tk = Ik+! - Ik , k = 0, ... , m.

By virtue of Theorem 6.9, there exists a unique perfect spline G of degree r -1 with M + 1 - r knots which satisfies (6.2.10). Then

:z:

P(t) := 10 + J G(t) dt (J

is the unique solution of the original problem. Finally, note that

and again by Theorem 6.9, c is a strictly increasing function of ek := I/HI - Ik I, k = 0, ... , m. The proof is completed.

Now let us formulate separately the following important particular case.


COROLLARY 6.11. Let {lIk}r be given multiplicities, 1 ~ Ilk ~ r, Ie = 1, ... ,n, N := 111 + ... + lin ~ r. Then, for every fixed interval [a, b] and every set of numbers ho > 0, ... , hn > 0, there exists a unique set of points a = to < Xl < t1 < X2 < ... < tn-1 < Xn < tn = b and a unique perfect spline P of degree r with N - r knots such that

P(j)(Xk) = 0, k = 1, ... ,n, j = 0, ... ,Ilk - 1,

P(tk) = (-l)<1khk' k=O, ... ,n, P'(tk) = 0, Ie = 1, ... , n - 1,

where Un := 0, Uk := Ilk+! + ... + lin, Ie = 1, ... , n - 1. Moreover, IIP(r)lIoo is a strictly increasing function of ho, ... , hn in the domain ho > 0, ... , hn > 0.

Proof. The assertion is a particular case of Theorem 6.10, corresponding to

(fo, ... ,fm+d == (ho, 0, h1' 0, ... , hn- 1, 0, hn)*,

where the asterisk • is a reminder that the value ° appears between hk and hJ:+1 only if 1IJ:+1 > 1 for k = 0, ... ,n - 1.

6.2.3. Perfect splines of least unifonn norm. The next question we study is the characterization of the perfect spline P of minimal uniform norm

IIPII := max IP(t)1 tE[a,b]

in the set {P E Pr(V1, ... , lin): IIP(r)lIoo = 1}. Let us mention first an immediate consequence of Corollary 6.11, corresponding

to the case ho = ... = hn = 1.

COROLLARY 6.12. Given [a, b], r and v = (111, . .. ,lin), there exists a unique perfect spline Tr(v; t) from Pr(lIl, ... , lin), and consequently a unique set ofpoints a =: to < t1 < ... < tn- 1 = tn:= b, such that

i = 0, .. . ,n, i = 1, ... ,n -1,

where (Tn := 0, (Tk := Ilk+! + ... + lin, Ie =F n. Moreover, IITr(v; ')11 = 1. The notation T for this equi-oscillating perfect spline has to stress the relation of

Tr(v;t) to the famous Tchebycheff polynomials Tr(t) := cos(r arccos t) (for -1 ~ t ~ 1). It is seen that Tr(v; t) == Tr(t) if 111 = ... = lin = 1, n = r and [a, b] = [-1,1]. We shall show that Tr(v; t)/IITrr) (v; ')1100 has a minimal uniform norm of all perfect splines P from Pr(lIl, ... , vn) with IIP(r)lIoo = 1. The proof is based on the monotone dependence of the coefficient c ofthe perfect spline P(t)= c· p(t), with p in form (6.2.1), on its local extremal values ho, ... , hn .

THEOREM 6.13. Let 1'(t):= Tr(v;t)/IIT~r)(v; .)1100' The inequality

111'11 < IIpli

holds for each p E Pr(1I1, ... , lin), such that IIp(r)lIoo = 1 and p =F 1'.

Ch. 6, § 6.2] Oscillating Perfect Splines 101

Proof. Suppose that there is a perfect spline P E 'P,.(lI!, ... ,vn) with IIp('')lIoo = 1, such that IIpll ~ IITII· Let cJ denote 11/(")1100 for every f from 1',. (v!, ... , lin). Clearly

On the other hand, Cp = c(ho(p), ... hn(p)),

Cr = C(hO(T), ... hn(T)),

where {hi(f)}~ denotes the nonzero local extrema (including the values at the endpoints) of the perfect spline / E 1',. (Vi, ... , vn ). It follows from the assumption IIpll ~ IITII that

hi(p) ~ hi(T), i = 0, .. . ,n. (6.2.11)

Since p :f. T, there is at least one strict inequality in (6.2.11). (Otherwise, hi(p) = hi(T), i = 0, ... , n, would imply p == T, by Corollary 6.11). But according to the same Corollary 6.11, c(ho, ... , hn) is a strictly increasing function with respect to each hk • Then the inequalities (6.2.11) imply

c(ho(p)" .. hn(p)) < c(ho( T), ... hn( T)),

i.e., 1 = cp < Cr = 1, which is a contradiction. Therefore IITII < IIpli. The theorem is proved.

Exercise 6.2.1. Denote by 'P,.N the set of all perfect splines of degree r with no more than N - r knots. Given [a, b], let TN(t) := T,.(vj t)/IIT,.(vj ')1100 for n = Nand Vi = ... = Vn = 1. Prove that

Hint. Assume the contrary. Then there is an s E 'P,.N with IIs(")lIoo = 1 such that

IIsll < IITNII. Without loss of generality we may assume that s(")(t) = T;;>(t) = 10n (a, a + c) for some small c > O. It follows from the equi-oscillation property of TN that (TN - s)(t) must have at least N isolated zeros in [a,b]. Then, by Rolle's theorem, ( TN - s)(,.) (t) has N - r sign changes in (a, b). In order to get a contradiction observe

that sign (TN - s)<")(t) = sign T;P(t) if T~P(t) :f. s(")(t). Thus, there exist N - r + 1 subintervals 10 ,"" IN_,. such that (-l)i(TN - s)(")(t) > 0 on Ii and Ii C (ei,ei+d, where 6 < ... < eN-,. are the knots of TN' eo := a, eN-,.+l := b. But this requires that the knots 711 < ... < 71m (m ~ N - r) of s must be situated in the following way

711 < 6 < 712 < 6 < ... < em-l < 71m < em < ... < eN-,..

Thus (TN - s)(t) == 0 on (em,em+l) and hence on 1m, a contradiction.

102 Perfect Splines (Ch. 6, § 6.3

§ 6.3. Optimal Recovery of Functions

The extremal problem considered here illustrates the spline interpolation as a most appropriate tool for the approximation of differentiable functions.

6.3.1. The best method of recovery. The general setting of the problem of best recovery was discussed in Section 5.2. Let us recall and specify some definitions and notations needed in the study of our particular problem.

Let z = «Xl,lIl), ... ,(Xn ,lIn» be a fixed set of points, where a ~ Xl < ... < Xn ~ b, 1 ~ Ilk ~ r, k = 1, ... , n. With any function f from W~[a, b] we associate the set of values

T(z; I) := {J(~)(Xk), k = 1, ... ,n, A = 0, ... ,Ilk - I}

and call it information about f. We are going to construct a method of approximation of differentiable functions on the basis of T(z; I), which has a minimal error in the class W~[a,b]. In order to do this, let us choose an arbitrary point t from [a,b]. Evidently L(J) := f(t) is a linear functional in W~[a, b]. Then, by Smolyak's lemma, there exist numbers {CkA(t)}~=l,~==~l such that

Here B := {J E W~[a, b]: IIf(r)lIoo ~ I} and R(z; t) is the error ofthe best method, i.e.,

R(z; t) := infsup If(t) - S(T(z; I)) I, S B

where the infimum is expanded over all functions S of N = III + ... + lin variables. Letting now t run over the whole interval [a,b], we get the functions Ck~(t) on [a,b], which define a linear method of approximation

n /I~-l

f(t) ~ L L CkA(t)j<~)(Xk). k=l ~=O

This method has a minimal error in W~ [a, b]. We call it a best method of recovery of the functions from W~[a,b] on the basis ofT(z;l). It turns out that the functions CkA(t) are splines. We show below how they could be constructed for every set of nodes z.

LEMMA 6.14. For every t E [a, b]

R(z;t) = lCf'r(z;t)l.

Proof. According to Corollary 5.8 (of Smolyak's lemma),

R(z; t) = sup{J(t) : fEB, T(z; I) =o}. On the other hand, by virtue of Corollary 6.5 the supremum above is equal to lCf'r(z; t)l. The assertion is proved.

Ch. 6, § 6.3] Optimal Recovery of Functions 103

LEMMA 6.15. Let Z = «Xl,Vl)," .,(xn,vn)) be a given set of points in [a,b], a ~ Xl < "'<Xn ~ b, 1 ~ Vic ~ r, k = 1, ... ,n, and N:=Vl+",+Vn ~ r. Let el, ... ,eN- r be the knots of the perfect spline <Pr(z; t) of degree r. Then for each function / E Cr-l[a,b] there exists a unique spline function sf(t) of degree r - 1 with knots at 6, ... , eN -r' which satisfies the interpolation conditions

k = 1, ... , n, A = 0, ... , Vic - 1. (6.3.1)

Proof. Let us point out first the fact that in the case Vic = r, XIc does not coincide with a knot of <Pr(z; t). (This was mentioned in the proof of Theorem 6.4). Thus, the interpolation conditions (6.3.1) are well defined even for A = r - 1.

Clearly, a < el < ... < eN-r < b. Choose 2r arbitrary additional points e-(r-l) < ... < eo < a, b <; eN-r+l < ... < eN, and denote by {B;(t)}f the B-splines of degree r - 1, corresponding to the sequence e-(r-l),'" ,eN, i.e.,

We know already (see Theorem 3.8) that {B;(t)}f constitute a basis in the space Sr-l(6, ... ,en-r) of splines in [a,b]. Thus the interpolation conditions (6.3.1) can be rewritten as a system of linear equations

N

L (}:;BV') (XIc) = /{>')(XIc), k=I, ... ,n, A=O, ... ,VIc-1. ;=1

Let (Tl, ... ,TN ) == «Xl.Vl), ... ,(xn,vn)). According to Theorem 4.7, this system has a unique solution for each / E Cr - l [a, b] if and only if the interpolation nodes {Ti}{" and the knots {ed~(r-l) satisfy the so-called interlacing condition

k= 1, ... ,N.

But we showed already (see (6.1.9)) that these inequalities indeed hold in our case. The lemma is proved.

LEMMA 6.16. Let Z = «Xl, VI)"'" (Xn, vn)) be a given set of points in [a, b], a ~ Xl < ... < Xn ~ b, 1 ~ Vic ~ r, k = 1, ... ,n, and N:= Vl+ ... +Vn ~ r. Let sf(t) be the spline of degree r - 1 with knots at the knots {ed{,,-r of <Pr(Z; t), which interpolates / at z. If / E W~ [a, b] and II/(r) 1100 ~ 1, then

I/(t) - sf(t)1 ~ l<Pr(z; t)1

for each t in [a, b].

Proof. Assume that there is a point to E [a, b] and a function / E W~[a, b] with 1I/(r)lIoo ~ 1, such that

I/(to) - Sf (to)1 > l<Pr(z;to)l· (6.3.2)


Clearly, to fI. {X!, ... ,xn }. Construct the function

h(t):= I(t) - 8,(t) - a<Pr(z;t),

where a = [/(to) - 8,(tO)]j<Pr(z;tO). It follows from the assumption (6.3.2) that lal > 1. The function h(t) has at least N + 1 zeros: z and to. Then, by Rolle's theorem, Mr-2)(t) will have at least N + 1 - (r - 2) = N + 3 - r distinct zeros. Since h(r-2)(t) is continuous and its derivative breaks eventually only at the points 6, ... ,eN-r, we conclude that the function h(r-1)(t) will change its sign at least N + 2 - r times. We shall show that this is impossible. Indeed, it follows from the assumptions

lal > 1, l<p~r)(z; t)1 = 1, lIJ<r)lIoo = 1,

that sign h(r)(t) = -sign [a<p~r) (z; t)]. Since a<p~r)( z; t) changes its sign alternatively when t passes through a knot of <Pr(z; t), the function h(r)(t) will have the same behaviour and therefore h(r-1)(t) is monotone in the intervals (ei,ei+d, i = 0, ... , N-r (eo = a, eN-r+1 = b). Moreover, the type of monotonicity (increasing, decreasing) changes alternatively from one interval to the other. But it is easily seen in this case that a function of such a type could have at most N + 1 - r changes of sign. The contradiction completes the proof.

THEOREM 6.17. Let z= (T1, ... , TN), Ti < Ti+r, i = 1, ... ,N-r, be given points in [a, b] and r ~ N. The spline interpolation method

I(t) ~ 8,(t) (6.3.3)

with nodes z is the best method of recovery of functions 1 from the class W~[a,b] on the basis of the information T( z; f).

Proof. The theorem follows immediately from the three lemmas we just proved. According to Lemma 6.15, the interpolating spline 8, (t) is defined for every 1 E W~ [a, b]. Further, by Lemma 6.14 and Lemma 6.16,

I/(t) - 8,(t)1 ~ l<Pr(z; t)1 = R(z; t), which shows that the method (6.3.3) is best. The proof is completed.

6.3.2. Characterization of the optimal nodes. For a given set of nodes z = «Xl, V1), ... ,(xn, vn)) the error R(z; t) of the best method of recovery of I(t) on the basis of the information T(z; t) is a nonnegative function of t on [a, b]. By Lemma 6.14, the uniform norm lI<Pr(z; ')11 of <Pr(z; t) is the exact estimation of the error R(z; t) on the whole interval [a, b]. The quantity lI<Pr(z; ')11 depends essentially on the location of the nodes Xb"" Xn in [a, b]. An appropriate choice of {xJJ~ could lead to a better approximation scheme. Hence the question of characterization of those xi, ... , x~ which minimize lI<Pr(Z; .)11 (and thus IIR(z; .)ID comes quite naturally.

Let the multiplicities V1, ... , Vn be fixed. The nodes z* = «xi, V1), ... ,(x~, vn)) with a ~ xr < ... < x~ ~ b, for which

IIR(z*;.) II = inf{IIR(z; ')11: Xl < ... < xn} are said to be optimal of type (v!,, .. , vn ). The corresponding best approximation scheme (6.2.4) for these nodes z* is termed optimal of type (Vb ... , vn ) in W~[a, b].

The next theorem gives the complete characterization of the optimal nodes.

Ch. 6, § 6.4) Smoothest Interpolant 105

THEOREM 6.18. For every choice of the multiplicities {Ilk}! such that 1 ~ Ilk ~ r, k = 1, ... , n, N := III + ... + lin ~ r, there exists a unique set of optimal nodes xi, ... ,x: of type (Ill. ... ,lin). The points xi, ... , x: are the zeros of the perfect spline from the set {P E 'Pr (1I1, ... , lin): IIP(r)lIoo = I} of the least uniform norm in [a, b].

Proof. According to Lemma 6.14, R(zj t) = l<,Or(zj t)l. Thus the theorem is another equivalent statement of the result from Theorem 6.13.

§ 6.4. Smoothest Interpolant

We are going to give here another example in which the perfect splines appear as extremal functions. Roughly speaking, this is the problem of characterizing the smoothest function from W~[a, b] of a preassigned shape.

Let [a, b] be a fixed finite interval. For any given set of values y = {Yd~ and points Z = {Xi}~, a = Xo < Xl < ... < Xn = b, define the class

F(z,y):= {J EW~[a,b]: I(Xi) = Yi for i = 0, ... ,n}.

The values {Ydii prescribe the general shape of the interpolant I, while the nodes {Xi}~ it specify further, with a certain influence mainly on the local behaviour of f.

We know already from Corollary 6.3 that F(z, y) contains a perfect spline P of degree r with at most n - r knots. Furthermore, P is the smoothest interpolant in F( z, y) in the sense that

Assume now that {Yi}ii stay fixed and consider Xl < ... < Xn-1 to be free parameters in the problem of minimizing lIt<r)Hoo over all functions from W~[a,b] that sequentially take on the given values Yo, ... , Yn' In other words, consider the problem

inf min 11/(r) II . a=zo<zl< ... <z .. =b IE F(e,tI) 00

(6.4.1)

The answer is trivial for n ~ r - 1. In this case, any polynomial of degree n - 1 that interpolates y at some z is a solution of (6.4.1). We assume further that n ~ r.

Without loss of generality we may assume also that the values y satisfy the requirement

for k = 1, ... , n - 1. (6.4.2)

Indeed, if Yk-1 < Yk < Yk+1 for some k and I(Xk-d = Yk-b I(xk+d = Yk+l, then I(x) takes the value Yk at some point Xk between Xk-1 and Xk+1 and hence the condition I(Xk) = Yk is automatically fulfilled.

Note that the infimum in (6.4.1) is actually attained for some points xi < ... < x:_1 in (a,b). In order to show this, choose some points Xl < '" < Xn-1 in (a,b) (say, the equispaced points) and denote by C the quantity min{lI/(r)lIoo: 1 E F(z, yn for this particular choice of z. C is an upper bound of (6.4.1). Let Zm = {xr}ii,


m = 1,2, ... , be a minimizing sequence for the problem (6.4.1). Assume that xr+1 - xr -+ ° as m -+ 00 for some fixed k. Let Pm be the smoothest function from F(zm, y). Because of (6.4.2), p:n has n - 1 sign changes in (a, b). Then every derivative pM)(x), for j = 1, ... , r-1, has at least one zero Tlj in (a, b). Observe now that 11P:n1l00 > (Yl:+1 - Yl:)/(xr+1 - xr) and hence IIp:nlloo -+ 00 as m -+ 00. Similarly, considering P:n(x) between T/1 and the point e for which IP:n(e)1 = IIp:nlloo, one concludes that IIP~lIoo tends to 00 as m -+ 00. Repeating this argument one comes to the conclusion that IIPJ;")lIoo is greater than C for large m. This contradicts the assumption that Zm is a minimizing sequence. Thus Zm tends to some Z· = {xD~ with a = Xo < ... < x~= b.

We shall prove that the extremal function to (6.4.1) is a perfect spline P(x) of particular form, namely, that remains strictly monotone between the nodes Xl:, Xl:+1 for each k = 0, ... ,n - 1. This, of course, implies P'(Xl:) = 0, k = 1, ... ,n - 1.

THEOREM 6.19. Given [a, b] and y = {ydo, there is a unique set of points z· = {xDo, a = Xo < ... < x~= b, and a unique perfect spline P E F(z·, y) of degree r with n - r knots for which

P is uniquely characterized by the fact that

P'(xi) = ° for k = 1, ... , n-1.

Proof. By Theorem 6.10 there exists a unique set of points z· = {xD(i and a unique perfect spline P.(t) of degree r with n - r knots such that

P.(xi) = Yi,

p~(xn = 0,

i = 0, . .. ,n, i = 1, ... , n - 1.

We shall show that p. is the required smoothest interpolant. For given z = {x"}o, a = Xo < ... < X n = b, let P(z; t) be the perfect spline from

F(z, y) with n - r knots. Recall that by Theorem 6.2,

IIp(r)(z;')lloo =inf{llt<r)lloo: JEF(z,y)}.

Since the values {Yl:}~ satisfy (6.4.2), P'(z; t) has n - 1 simple zeros in (a, b). Let us denote them by TIl < ... < TIn-I. We have

el:(z):= Ip(z;Tll:+I)-P(z;Tll:)1 ~ IYl:+1-Yl:1 =e,,{z·)

for k = 0, ... , n -1. By Theorem 6.10, IIP(r)(z; ')1100 is a strictly increasing function of eo, ... , en -1. Then

(6.4.3)

for each a = Xo < ... < x n= b. The equality is attained if and only if z = z·. Therefore z· is the unique solution of (6.4.1) and consequently p. is the unique perfect spline extremal function to this problem.

Finally, note that P'(t) has exactly n - 1 zeros in (a, b), namely, xi, ... , x~_1 . Then P.(t) is strictly monotone on [XI:-l, XI:], k = 1, ... , n. The proof is completed.

Ch. 6, § 6.4] Smoothest Interpolant 107

THEOREM 6.20. The perfect spline p. from Theorem 6.19 is the unique extremal function to (6.4.1).

Proof. Assume that I is an extremal function to the problem (6.4.1) which is distinct from p •. It follows from (6.4.3) that

for each::t: I ::t:. and 9 E F(::t:, y). Thus, if I is extremal, then necessarily IE F(::t:·, y). Suppose further that I'(xi.) I 0 for some k E {I, ... , n-l}. Then there is a point

t,., E (Xi_l,xi) such that f'(t,.,) = O. Let t:= (xo, ... ,Xi_l,t,."xi+1" .. 'x~). Then I/(xi+l) - l(t,.,)1 > IYk+l - y,.,1 and, by Theorem 6.10,

which is a contradiction. Therefore 1'(xV = 0 for each k = 1, ... ,n - l. As we assumed in the beginning, I t- p.. Then f t-p. on [xi_I' xi] for some

k E {I, ... , n}. Since I(t) coincides with P.(t) at xi_l and xi, the function f'(t)P! (t) must change sign in (xi_I' xi). It is seen that for sufficiently small g > 0 the function P~ (t) - (1- g)f'(t) would also change its sign in (xLI' xk). Note otherwise that f'(x';) = P.(x,;) = 0 for i = 1, ... , n-l and thus P!(t) - (l-g)f'(t) has at least n distinct zeros in (a,b). Then, by Rolle's theorem, p~r)(t)_(I_g)/(r)(t) must have at least n-r+l sign changes. But signp~r)(t) = sign[p~r)(t)-(I-g)/(r)(t)] almost everywhere in [a, b] and p. has n - r knots by the assumptions. The contradiction completes the proof.


The name "perfect splines" comes from a paper by Glaeser [1973]. Louboutin's problem was considered in Louboutin [1967]. Another solution was given by Schoenberg [1971]. The problem of minimizing IIf(r)lIoo over all functions that interpolate a given data was initiated by Favard [1940]. Karlin [1973] announced Corollary 6.3 and supplied the proof in Karlin [1975]. Meanwhile a short proof was given by de Boor [1974]. Theorem 6.2 was proved by Goodman [1979]. Our proof here is different. It was used by Bojanov [1990] for the study of Favard's problem in classes defined by linear differential operators. Section 6.2 is based on Bojanov [1980]. The question of optimal recovery of functions from W; [a, b] (1 ~ p ~ 00) on the basis of {f(x,.,),f'(x,.,), ... ,j(r-l)(x,.,), k = 1, ... ,n} was considered by Bojanov [1975]. Theorem 6.17 is due to Micchelli, Rivlin, and Winograd [1976] (see also Gafney and Powell [1976]). The periodic variant was studied in Bojanov [1977]. The existence and uniqueness of the optimal nodes of fixed type (Ill, ... , lin) was proved by Bojanov [1980]. Exercise 6.2.1 can be found in Tikhomirov [1969]. He proved in the same paper that the perfect spline of degree r with no more than N - r knots of least uniform norm must equi-oscillate at N + 1 points. Then Karlin [1976b] showed the uniqueness. The problem of interpolation by polynomials and splines at the extremal points was studied by Fitzgerald and Schumaker [1969] in the simple node case. More


about perfect splines can be seen in Fisher and Jerome [1975], Cavaretta [1975], Karlin, Micchelli, Pinkus, and Schoenberg [1976], and Korneichuk [1984]. Section 6.4 is based on Pinkus [1988], where the more general problem of minimizing the Lp-norm of f(r)(t), for 1 ~ p ~ 00, is considered. The uniqueness of the extremal function for p i= 00 is still an open problem. Particular cases are studied by Marin [1984] (p = 2, r = 2) and Uluchev [1991] (p = 2, r = 3).

Chapter 7

MONOSPLINES

Functions of the form t r Ir! + s(t), where s(t) is a spline of degree r - 1 are called monosplines. To be precise, a monospline of degree r with knots Xl < ... < Xn of multiplicities Vl, .•. ,Vn , respectively, is any expression of the form

with real coefficients {aj} and {CkA}, 1 ~ VI< ~ r, k = 1, ... n. The interest in monosplines comes from their close relation with quadrature formulae.

§ 7.1. Monosplines and Quadrature Formulae

Let [a, b] be a fixed finite interval. It is more convenient for us to consider the monosplines of degree r multiplied by (-1 r and presented on [a, b] in the following way

(b - tr r-l (b _ tr- j - l n Vk- l (XI< _ W-A-l

r! -?= Bj (r _ j _ I)! - L L akA (r - oX - I)! 3=0 1<=1 A=O

(7.1.1)

with some real coefficients {Bj} and {akA}. Given the subsets J l and J2 of {O, 1, ... , r - I}, we shall denote by VJl(z;~) the

class of all monosplines M of the form (7.1.1), which satisfy the boundary conditions ~(Jb J2):

Introduce the sets

MCi)(a) = 0

MCi)(b) = 0

for j E h, for j E J2.

J i := {O ~ j ~ r - 1: r - j - 1 ¢ J;}, i = 1,2,

associated with J l and J2 , and define the quadrature formulae

b

I(f) := J f(x) dx ~ Q(f) a

109

110 MonO$p/ines leh. 7, § 7.1

of the form

6 n ".-1 J /(z) dz ~ ~ A;!(j}(a) + ~ B;!(j}(b) + L: L: aJ:>./(>'}(Zk). II ;Eh iEJ2 k=1 >'=0

(7.1.2)

We say that the formula 1(f) ~ Q(f) is exact for the function p if l(p) = Q(p). There is a one-to-one correspondence between the monosplines M from (7.1.1) and

the quadrature formulae 1(1) ~ Q(I) of the form (7.1.2), which are exact for all polynomials p of degree r -1. Moreover, the pair of correspondences {M, Q} satisfies the equality

6

1(f) - Q(I) = J M(t)/(r}(t) dt (7.1.3)

for each / E W[ [a, b]. In order to show this, observe that successive integration by parts on [Zi, Zi+1] yields the relation

"'i+l

+ (-It J M(r}(t)/(t)dt.

Summing with respect to i = 0,1, ... , n (zo := a, Zn+1 := b), and taking into account that M(r}(t) = (-It on each subinterval (Zi' ZH1), we get

6 6 J M(t)/(r}(t)dt = J /(t)dt II II

r-1 r-1 - L:( -It-i - 1 M(r-i -1}(a)/(j}(a) - L:( -It-i M(r-i -1}(b)/(j}(b) i=O i=O

n r-1 - L: L:( -It->.-1 [M(r->.-1}(Zi + 0) - M(r->.-1}(Zi - O)]j<>'}(zd. k=1>.=0

But M(r->.-1}(Zi + 0) = M(r->.-1}(Zi - 0) for A = r - 1, r - 2, ... ,r - Vi and 1 ~ i ~ n. Further, it follows from (7.1.1) that

Bi = (-It-i M(r-i -1}(b),

ak>' = (-It->.-1 [M(r->.-1}(Zi + 0) - M(r->.-1}(Zi - 0)].

Thus, using the notation

Ch. 7, § 7.2] Zeros of Monosplines 111

we can rewrite the obtained equality in the following way

b b ,.-1 ,.-1 J M(t)f(")(t) dt = J f(t) dt - ~Aj/(j)(a) - ~Bjf(j)(b) a a 3=0 3=0

n v,,-l (7.1.4)

- L L a/c).t<)..)(Xk). k=l )..=0

We derived this relation for an arbitrary monospline M. Assume now that M E rot(:l:j !B). Then

Aj = 0

Bj = 0

for r - j -1 E J 1,

for r - j -1 E J 2,

and (7.1.4) reduces to the quadrature formula (7.1.2) with a remainder given by (7.1.3). Remark that the coefficients {Bj}, {a/c).} of M(t) are the same as those in the corresponding quadrature formula. So, every M from rot(:l:j!B) defines a unique quadrature formula of form (7.1.2). Since f(")(t) == 0 for f E 11",._1 the remainder of this quadrature vanishes (see (7.1.3)) and, hence, is exact for all polynomials of degree r - 1.

It remains to prove the converse: If 1(1) ~ Q(I) is a quadrature formula of form (7.1.2), which is exact for every p E 11",.-1, then the monospline M(t), defined by (7.1.1) with the same coefficients {Bj}, {a/c).} as in Q(I), satisfies the boundary conditions !B, i.e., M E rot(:l:j !B). This follows immediately from the relations

Aj = (_I)'"-j-1 M(,.-i- 1)(a),

Bj = (-I)'"-jM("-i- 1)(b).

The conditions Aj = 0 for j ¢ J 1 , Bj = 0 for j ¢ J 2 implies

which means that M E rot(:l:j !B). Note that the presentation (7.1.3) of the remainder can be derived by Peano's

theorem. The established relation between monosplines and quadrature formulae is often

used to construct quadrature formulae with certain extremal properties.

§ 7.2. Zeros of Monosplines

Let M(t) be a monospline of simplest form, namely

M(t) = (b - t)'" _ ~ ak (Xk - t)+-l rl L....J (r - 1)1

k=l

112 Monosplines [Ch. 7, § 7.2

Assume that M(e) = 0 and set <pe(x) := (x - e)+-l I(r - 1)! Observing that

b J <pe(x) dx = (b - tY Ir!, a

we rewrite the condition M(e) = 0 as

b n J <pe(x)dx = Eak<Pe(Xk). a k=l

This shows that every zero of M induces the exactness of the quadrature formula n

I(f) ~ E ak!(xk) (7.2.1) k=l

for a certain truncated power function. Thus, the more are the zeros of M, the wider is the class of exactness of (7.2.1), the "better" is the quadrature formula. This observation leads us to the problem of estimating the number of zeros of a given monospline. Before considering this question we specify first the definition of zero of a monospline.

Since M(t) coincides with a nonzero polynomial between two consecutive knots, the definition of a zero there is the same as the usual one: if e is not a knot of M(t), then we say that e is a zero of M of multiplicity k provided

Next we consider the case when e is a knot of M. Suppose that e = Xi. Denote by M+(t) (M_(t), respectively) the monospline which agrees with M(t) to the right (left) of Xi and has no knots in (-OO,Xi] ([Xi, 00), respectively). Suppose that M+ has a zero of multiplicity a at Xi and M_ has a zero of multiplicity fJ at Xi, where the usual definition is applied to M+ and M_. Then we say that e is a zero of multiplicity max(o:,fJ) provided the sign of M~-Vi) to the right of Xi is the same

as the sign of M¥-Vi) to the left of Xi. Otherwise, M has a zero of multiplicity max(a,fJ) + 1. Note that according to this definition M(t) changes its sign at e if the multiplicity is odd and does not change sign if it is even.

Denote by Z(M; (a, b)) the number of zeros of M in (a, b) counting multiplicities. For given multiplicities {Vi} of the knots of M we set

if Vi is even, if Vi is odd.

THEOREM 7.1. Let M(t) be an arbitrary monospline of degree r with knots {Xi}1,

a < Xl < ... < Xn < b, of multiplicities VI, ... ,Vn , respectively. Then n

Z(M; (a, b)) ::::; E(Vk + Uk) k=l (7.2.2)

Ch. 7, § 7.2] Zeros of Monosplines

Proof. M (t) is an algebraic polynomial of the form

{-Irtr Ir! + ...

113

on each subinterval (Xi, Xi+l), i = 0, ... , n (xo := a, Xn+1 := b). Then, by the Budan-Fourier theorem (see Corollary 1.16),

Z(M; (Xi, Xi+1)) :::; S- (M{Xi + 0), ... , M(r){Xi + 0))

- S+ (M(Xi+1 - 0), ... , M(r){Xi+1 - 0)).

If ai is the multiplicity of the zero of M at Xi, summing the last inequality over i yields

where

n

Z(M; (a, b)) :::; L: Tr{M; Xi)

k=1

+ S- (M(a), ... , M(r)(a)) - S+(M(b), ... , M(r){b)),

Tr(M; Xi) := ai + S- (M(Xi + 0), ... , M(r){Xi + 0))

- S+ (M(Xi - 0), ... , M(r){Xi - 0)).

The theorem will be proved if we show that

Tr{M; Xi) :::; IIi + Ui. (7.2.3)

Suppose that 0 :::; ai :::; r - IIi - 1. Since M(j)(x) is a continuous function at Xi for j = 0, ... , r - II; - 1, M(j){Xi - 0) = M(j){Xi + 0) = 0 for these j and hence

Tr(M; Xi) = S- (M(a;){Xi + 0), ... , M(r)(x; + 0))

- S+(M(a;)(Xi - 0), ... ,M(r)(x; - 0))

:::; S- (M(r-v;-1)(x; + 0), ... , M(r)(x; + 0))

- S+ (M(r-v;-1){Xi - 0), ... , M(r){x; - 0)).

Evidently, (7.2.4)

and this implies Tr(M;Xi):::; IIi + 1, which coincides with (7.2.3) in the case of odd IIi. If the equality is attained in (7.2.4), then

sign M(r-j) (x; + 0) = (-Ir+i, j = 0, ... , IIi + 1.

In particular, for even IIi, this implies

sign M(r-v; -1)( Xi + 0) = sign M(r-v.-1) (Xi - 0) = (-1 r+ 1

and since M(r){t) = (-It, we get

S+(M(r-v;-1)(Xi-0), ... ,M(r)(x;-0)) ~ 1.

N ow it is clear that Tr (M; Xi) :::; IIi for even IIi. This completes the proof of the estimation (7.2.3) in the case 0 :::; ai :::; r - IIi - 1. The reasoning in the other cases is similar. The theorem is proved.


§ 7.3. The Fundamental Theorem of Algebra for Monosplines

Propositions which assert the existence and uniqueness of a monospline with a preassigned maximal number of zeros are known as fundamental theorems of algebra for monosplines. In view of the comment in the beginning of the previous section, any such fundamental theorem is actually a theorem about quadrature formula of highest "degree of precision" with respect to a given spline space.

We prove below a theorem of this kind restricting ourselves to the case of monosplines with simple knots.

Let us note first an immediate consequence of Theorem 7.1. Denote by IJI the cardinality of the set J (i.e., the number of components of J).

COROLLARY 7.2. Let the boundary conditions ~ = ~(h, h) be defined by the sets h, h c {O, 1, ... , r - I}. Then for every M E 9Jl{z,~) with z = (Xl, .•• , xn) and a < Xl < ... < Xn < b,

(7.3.1)

Proof. Since M satisfies the boundary conditions ~,

s- (M{a), ... , M(r){a)) ~ r -!Jll, S+(M{b), ... , M(r){b)) ~ IJ2 1.

Then (7.3.1) follows from Theorem 7.1. The next fundamental theorem of algebra for monosplines shows that the estima

tion (7.3.1) cannot be improved. For given J 1, h and Ill, ... ,I'm we shall denote by E( h , h, Ill. ... , Ilm) the quasi

Hermitian incidence matrix {eij } i:!:t~;l with

eOj = 1

em +1,j = 1 eij = 1

eij = °

for j E J l ,

for j E J2 ,

for i = 1, ... ,m, j = 0, ... ,Ili - 1,

otherwise.

THEOREM 7.3. Let J1, h be given subsets of {O, 1, ... , r - 1} and 1 ~ I'i ~ r, i = 1, ... , m. Suppose that the matrix E(J1 , J2 , 1'1> .. . I'm) satisfies the P6lya condition. Then, for every fixed set of points a =: to < tl < ... < tm < tm+1 : = b, there exists a unique monospline M{t) of the form

t r n

M(t) = I" + p{t) + L adxk - t)~-l r.

k=l

(7.3.2)

with some p E 7rr -1 and a < Xl < ... < Xn < b, r+2n -lhl-IJ21 = III + .. '+I'm, such that

if eij = 1.

Ch. 7. § 7.3] The Fundamental Theorem of Algebra for Monosplines 115

Proof. In the case r = 1, the desired monospline M(t) is a piecewise linear function and can be easily constructed. We assume further that r >1.

The existence of M will be shown on the basis of the Borsuk antipodality theorem (see Theorem 6.1). In order to do this, introduce the sphere

with N := 2n + 1 - a - P, where

'" ._ { 1 .... - 0

p:= {~

if r -1 E Jl, otherwise,

if r - 1 E J2, otherwise.

Every h from SN determines a partition a = eo < 6 < ... < eN < eN+! = b of [a, b) in the following way:

el: =a+ (tlhjl)(b-a), J=1

Define the broken line function

l(hi t) := { (t - 61:-1-a) s~gn h21:-1 (t -e21:-1+a)Slgnh21:

k=l, ... ,N.

on (61:-2,61:-I), on (e21:-1,61:),

for all admissible k. Note here that if h}, h2 , • •• , hN+! change sign alternatively, than l(hi t) is a monospline of degree 1 with n knots at {61:-d, if a = 1, or at {e21:}, if a = O. Our goal is to show that there exists a function in the class {l(hit): hE SN}, whose (r - I)-tuple integral

" 1 J -M(hit) :=p(hit) + (r-2)! (z-t)+ 21(hi t )dt, G

with an appropriate p(hi t) E 1rr-2, is the desired monospline. To this purpose, for each h E SN we determine l(hi t) and then define p(hi t) E 1rr-2 by the interpolation conditions

for eij = 1, eij E Eo,

where Eo is any poised submatrix of E = E(Jl,J2,Pl, ... Pm) with IEol = r-1. The existence of Eo follows from the assumption that E contains a P6lya matrix. Denote the quantities

eij = 1, eij E E\Eo,


in some fixed order, by ~l(h), ... , ~N(h), respectively. Consider the mapping ~: SN -+ ]RN,

~(h) := (~l(h), ... , ~N(h».

Clearly, ~(h) is a continuous odd function on SN. Then, by the Borsuk's theorem, there exists an h* E SN such that ~(h*) = O. In other words,

for all eij = 1.

It remains to show that the function M(h*;t) is a monospline. Since M(h*;t) vanishes at (t, E), where t := (to, tl,"" tm+d, and lEI = r+2n, it follows by Rolle's theorem that M(r-l)(h*; t) has at least N = r + 2n - (r - 1) - a - f3 sign changes in (a, b). But M(r-l)(h*;t) = l(h*;t). Thus l(h*;t) changes sign alternatively in (ei,ei+d, i = 0, ... , N. This means that hi, ... , hN+1 change sign alternatively, and hence, as we mentioned already, l(h*;t) is a monospline of degree 1 with n knots. The existence of M is proved.

Next we prove the uniqueness. Suppose that there exist two monosplines Ml and M2 of form (7.3.2), which vanish at (t,E). Then sign Ml(t) = signM2(t) on (a, b), since Ml and M2 have the maximal number of zeros in (a, b). Suppose that Ml :I M2· Then there is a point to E (a, b) such that Ml (to) :I M2(tO)' Assume, without loss of generality, that 0 < M2(tO) < Ml (to). Consider the function

with ..\ = M2(to)/Ml (to). Clearly, g vanishes at (t, E) and g(t) has only isolated zeros, since Ig(r)(t)1 :I 0 almost everywhere in [a, b]. Adding the fact that g(to) = 0 we conclude on the basis of Rolle's theorem that g(r-l)(t) has at least 2n + 2 - a - f3 sign changes in (a, b). But, since 0 < ..\ < 1, M~r-l)(t) - ..\MIr-l)(t) could have at most 1 sign change between two successive knots Xi, Xi+l of M2 (i = 1, ... , n -1). Thus g(r-l)(t) may change its sign eventually: at Xl, ... ,Xn; at most once in (Xi,Xi+d, i = 1, ... ,n -1; at most once in (a,xd if a = 0; at most once at (xn, b), if f3 = O. According to this estimation g(r-l)(t) changes sign at most n + (n - 1) + (1 - a) + (1 - f3) = 2n + 1 - a - f3 times. This contradiction shows that Ml = M 2 . The proof is completed.


The relation between monosplines and quadrature formulae can be observed in the classic papers of Birkhoff [1906], Peano [1914], Tschakaloff [1938]. This relation was described and exploited further by Nikolski [1950] (see also the book by Nikolski [1979]). The name monospline was used for the first time by Schoenberg [1965].

The estimation for the number of zeros of monosplines with multiple knots was obtained by Micchelli [1972]. We follow here his proof of Theorem 7.1.

The present simple proof of the fundamental theorem of algebra (Theorem 7.3) is from Zensykbaev [1981]. A more general result concerning monosplines with multiple knots and multiple zeros was obtained by Barrar and Loeb [1980] and Zensykbaev [1989].

Chapter 8

PERIODIC SPLINES

The function Ion JR is said to be periodic with a period T (or in brief, T-periodic) if I(x + T) = I(x) for each x E JR. For the sake of convenience, we shall consider here 211"-periodic functions. In this case, one may think of I as a function defined on the unit circle.

If a periodic function I has to be approximated by a simpler function !p, it is quite reasonable to require !p to also be periodic. In this way the most characteristic property of I, its periodicity, would be preserved a priori in the approximation I ~ !po

Since the splines are widely used as an approximation tool, we find it necessary and useful to study and review some of the specific properties of periodic splines.

Given the points Z = ((Xl,VI), ... ,(xn,vn», we denote by Sr-l((Xl,Vl), ... , (x n , vn» (or abbreviated to Sr-l(Z» the set of all 211"-periodic splines of degree r - 1 with knots Xl, ... , Xn of multiplicities Vb ••• , Vn , respectively. Of course, we assume that ° < Xl < ... < Xn < 211" and 1 ~ Vk ~ r, k = 1, ... , n. The set Sr-l(Z) is a linear space. We shall see later that

§ 8.1. Basis

Clearly, Sr-l (z) is a subset of the class Sr -1 ( z) of all splines of degree r - 1 with knots z. Precisely,

Hence, every periodic spline s from Sr_l(Z) may be written in the form

r-l n vk-l

s(t) = :~:::>}:iti + L L ckj(t - Xk)~-j-l, (8.1.1) i=O k=l j=O

where the coefficients {at} and {ckj } satisfy the condition

j = O, ... ,r - 1. (8.1.2)

The converse is also true: every function (8.1.1) which satisfies (8.1.2) is a 27r-periodic spline from Sr-l(Z).

117

118 Periodic Splines (Ch. 8, § 8.1

The expansion (8.1.1), with the accompanying condition (8.1.2), is not very convenient for the presentation of periodic splines. Next, we construct a periodic basis for S"_l(Z).

8.1.1. Periodic B-splines. Let us write the points z as

where Xi is repeated exactly IIi times (i = 1, ... , n) and N := 111 + ... + lin. Assume that r ~N. Introduce additional r points eN+i := ei + 211", i = 1, ... ,r. Let

i= 1, ... ,N,

be the B-splines corresponding to the sequence {ei}i" +,.. Now we define N 211"periodic splines Bi(t):

Bi(t) := Bi(t) on [0,211")

- {Bi(t) Bi(t):= Bi(t + 211")

for i = 1, ... , N - r,

if e,. ~ t <211", if 0 ~ t ~ e,.,

for i = N - r + 1, ... ,N. In other words, Bi(t) is the 211"-periodic extension of B(ei,'" ,ei+,.;t) on the whole real line. Thus Bi E 8,._1(Z), i = 1, ... ,N.

The splines {Bi(t)}f are linearly independent on [0,211"). Indeed, assume that

N

so(t) := L O!iBi(t) = 0 on [0,211") i=1

and at least one coefficient, say O!k' is distinct from zero. Let 6 > 0 be so small that (ek,ek + 6) does not contain knots and forms the sequence {ei}f. Then the sum in the above presentation of so(t), if when considered on (ek,ek +6), reduces to r+l terms, including O!kBk. But the B-splines {Bd coincide on this subinterval with r+ 1 distinct nonperiodic B-splines corresponding to the periodic extension of {ei}f. Then it follows from the linear independence of the B-splines (in the nonperiodic case) that all coefficients of the sum are zero, which contradicts the assumption O!k =F O. The claim is proved.

We shall call {.Hi}f 211"-periodic B-splines corresponding to the knot sequence z.

THEOREM 8.1. The splines {Bdf form a basis in 8,._1(Z) on the interval [0,211").

Proof. It suffices to show that every function s from S,.-l (z) may be presented as a linear combination of the B-splines {Bi}f. In order to do this, let us continue periodically the knot sequence z to the right and to the left by r points. Denote the extended sequence by e = {e-,.+1, ... ,eN +,.}. Let {Bd~_"+1 be the B-splines

corresponding to e. Now observe that if s E 8,._1(Z) then s belongs to 8,._1(Z), too.

Ch. B, § B.l) Basis 119

But we know from Theorem 3.8 that {Bdf:-r+1 is a basis in Sr-1(Z). Then s has a unique presentation in the form

N

s(t) = L: aiBi(t) on [0,211l (8.1.3) i=-r+1

Recall that for t e[O, 211-)

l1i(t) = Bi(t) for i = 1, ... , N - r,

for i = 0, ... , r - 1. (8.1.4)

It follows from this relation that we could obtain from (8.1.3) a representation of s(t) as a linear combination of 111 (t), ... , l1N(t), provided we show that

for i =0, ... , r - 1. (8.1.5)

Let us prove (8.1.5). Evidently,

r-1 s(t) = L:a-iB-i(t)

i=O r-1

set) = L:aN_iBN-i(t) i=O

Observe that BN_i(t + 211") = B_i(t) for t e[xn - 211", xd, because of the periodicity of the knot sequence (. Then taking into account the periodicity of s as well, we get: For t e[Xn - 211", Xl],

r-1 s(t) = L: a_iB_i(t)

i=O r-1

= s(t + 211") = L: aN_iBN-i(t + 211") i=O

r-1 = L:aN_iB_i(t).

i=O

Therefore a_i = a N _ i for i = 0, ... , r - 1. This completes the proof of the theorem.

COROLLARY 8.2. The dimension of Sr-1 «Xl, lid, ... , (Xn, lin» is equal to III + ... +lIn ·

8.1.2. Representation by the Bernoulli polynomials. Now we shall construct another basis for Sr-1(Z) using the 211"-periodic extension ofthe Bernoulli polynomials. Let us recall that the characterizing property of the Bernoulli polynomial br(t) of degree r is the equality

211'

J W)(t)dt = ° for j =0, ... ,r - 1. (8.1.6)

o

120 Periodic Splines [Ch. 8, § 8.1

Usually these polynomials are normalized by the condition that they have a leading coefficient equal to 1. It follows from (8.1.6) that

b~j)(O) = W)(211"), j = 0, ... , r- 2.

Thus the 211"-periodic extension br(t) of the polynomials br(t) on the whole real line is a piecewise polynomial function of degree exactlr r, which has continuous derivatives of order j for j = 0, ... ,r - 2. In other words, br(t) is a 211"-periodic monospline of degree r.

For every natural number r, consider the functions

( ) ._ ~ cos (kt - r1l"/2) Dr t .- L.J kr

k=l

(8.1.7)

on (-00,00). In particular, for r = 1 and r = 2, we have

Dl(t) = ~=1 sinkkt = {0(1I" - t)/2 for 0< t < 211", Ai for t = 0,

D ( ) = _ ~ cos kt = _ (11" - t)2 11"2 2 t L.J k2 4 + 12

k=l

for 0:::;; t :::;; 211".

These equalities can be verified, for example, if we find the Fourier series for the polynomials in the right-hand side.

It is seen from (8.1.7) that 2,.-

J Dr(t) dt = 0. (8.1.8)

o Furthermore, since

( COS(kt - r1l"/2») , = cos(kt - (r - 1)11"/2) kr kr - 1

it follows from the definition (8.1.7) that

lY,.(t) = Dr_1(t), r = 3,4, ... , (8.1.9)

for every real t. This equality holds also for r = 2, provided t ::j: 2k1l", for all integers k. The recurrence relation (8.1.9), together with the fact that Dl is a polynomial of degree 1 on (0,211"), implies that Dr(t) coincides with an algebraic polynomial of degree r on the interval (0,211"). Further, it is clear from the equalities (8.1.8) and (8.1.9) that this polynomial satisfies the conditions

2,.-J Dy)(t)dt = ° for j = O, ... ,r-1. o

Thus, the 211"-periodic functions Dr(t) coincide on (0,211") with the Bernoulli polynomials br(t).

The functions Dr(t) play an important role in the classes of 211"-periodic differentiable functions, which is similar to the role of the truncated power functions in the corresponding nonperiodic classes. Let us recall the following representation theorem.


THEOREM 8.3. Every 27r-periodic function f from cr-1(-oo,oo) with locally absolutely continuous (r '- l)-st derivative can be written in the form

2~ 2~

f(z) = 2~ j f(t) dt + .; j f(r)(t)Dr(z - t) dt. o 0

The converse is also true: if O(t) is 27r-periodic and

then the function

is 27r-periodic,

2~

j O(t) dt = 0,

o

2~

f(z)= ao + .; j O(t)Dr(z - t) dt o

2~

ao = 217r j f(t) dt, o

f E cr-1(-oo,oo) and JCr)(t) = O(t) a.e. in [0,27r).

Proof. It follows from the periodicity of f(r)(t) and Dr(t) that

2~ 2~

j f(r)(t)Dr(z - t) dt = j f(r)(z - t)Dr(t) dt.

o 0

Then, using integration by parts, we get

2~ 2~

j JCr)(z - t)Dr(t) dt = j f(r-l)(x - t)Dr_1(t) dt = ... o 0

2~ 2~

= j!'(x-t)D1(t)dt= j!'(z_t)7r;t dt

o 0

2~

7r - t 12~ 1 j =--f(x-t) -- f(x-t)dt 2 0 2

o 2~

= 7rf(x) - ~ j f(t) dt, o

which yields the representation (8.1.10).

(8.7.10)

(8.1.11)

(8.1.12)


Now suppose that 1 is defined by (8.1.12). We need to prove only that I(r)(x) = 8(x) on [0,211"] (the rest of the claim follows immediately from the established properties of Dr). Clearly,

Since

we get further

2,.-

I(r-l)(x) = ~ j 8(t)D1(X - t) dt. o

for 0 < t < x, for x < t < 211",

z 2,.-

I(r-l)(x) = ~ j8(t)(t - (x - 11"») dt + ~ j8(t)(t - (x + 11"» dt 211" 211"

o z 2,.- 2,.- z 2,.-

= ;; j 8(t) dt + 2~ j t8(t) dt + ~ (j 8(t) dt - j 8(t) dt) o 0 0 z

and, because of (8.1.11),

2,.- z 2,.-

l(r-1)(x) = 2~ j t8(t) dt + ~ (j 8(t) dt - j 8(t) dt) . o 0 z

The last expression is a differentiable function of x and I(r)(x) = 8(x) a.e. in [0,211"]. The proof is completed.

N ow we are prepared to give the new basis for the spline space Sr-l (:I:) based on the 211"-periodic Bernoulli monosplines Dr(t).

THEOREM 8.4. The function 1 belongs to Sr-1 «Xl, V1), ... , (xn , Vn» if and only if it can be written in the form

n 11,,-1

I(x) = ao + L L ck>.Dr->.(x - Xl,)' k=1 >.=0

where ClO + C20 + ... + CnO = O.

Moreover, 2,.-

ao = 2~ j I(t) dt, o

Ck), = ~ [J(r->.-I)(xk + 0) - l(r->.-I)(Xk - 0)],

k = 1, ... ,n, A = 0, .. . ,v" -1,

and hence the presentation (8.1.13) is unique.

(8.1.13)

(8.1.14)

(8.1.15)


Proof. It follows from the mentioned properties of Dr(t) that every function I of the form (8.1.13) is a 21r-periodic piecewise polynomial function of degree not greater than r with a continuous j-th derivative in a neighbourhood of x", for j = 0, ... , r - v", - 1. So, it remains to show only that IE 1I"r_1 on (Xj' xH1), provided the coefficients {c",on satisfy (8.1.14). But for x E(Xj ,xH1),

n

j<r-1)(x) = Ec",oD1 (x - x",) "'=1

n _ '" 1r - X + x", - L.Jc",o 2

"'=1 and the last expression is a constant, because of(B.1.14). Thus I E Sr_1 (:1:).

Now let us prove the converse assertion. Suppose that IE Sr_1«X1,V1), ... , (xn, vn)). Consider the function

2..-

F(x) = ;: J O(t)D1(X - t) dt

° with O(t) := I(t) - ao. Evidently 0 satisfies (B.1.11). Then, by Theorem B.3, F'(x) = O(x). Observe that

2..-J D1(X - t) dt = O.

° Therefore 2..-

1rF(x) = J l(t)D1 (x - t) dt. o

Using the fact that I(r)(t) = 0 on (x"" x1:+1) for k = 0, ... , n (xo := 0, Xn +1 := 211"), the periodicity of I and (B.1.9), we get, integrating by parts,

""+1

1rF(x) = t J l(t)D1(x - t) dt

"'=0 ""

= - t f/(j)(t)D2+j(X _ t)I""-O "'=0 j=O "" +0

n r-1 = E l: [lei) (x", + 0) - lei) (x", - 0)] D2+j (x - x",)

"'=1 j=o

and since I has a knot of multiplicity v", at x", (i.e., I, I', ... , I(r-v" -1) are continuous in a neighbourhood of x",),

n r-l

1rF(x) = l: l: [/(r->.-l) (x", + 0) - I(r->.-l) (x", - 0)]D1+r_>. (x - x",), "'=1 j=O


and hence

F'(z) = I(z) - ao n r-1

= .!. L: L: [j<r->.-l)(Zl: + 0) - j<r->.-l)(Zl: - 0)] Dr_>.(Z - Zl:)' 1f'l:=1 j=O

which is a presentation of the desired form (8.1.13) with coefficients (8.1.15). To show that {cl:>.} satisfy the condition (8.1.14), we consider Ion any of the subintervals (Zj,ZH1) and use the assumption that 1 E 1f'r-1 there. Then

n

j<r-1)(z) = L cl:OD1 (z - zl:) = const on (Zj' ZH1) l:=1

and thus the coefficient of Z in this polynomial expression must be zero. The theorem is proved.

COROLLARY 8.5. Let 0 < Zl < ... < Zn < 21f' and 1 ::;; vl: ::;; r, k = 1, ... , n. Then the functions

j = 1, ... , n - 1,

k=l, ... ,n, ),=1, ... ,vl:-1,

form a basis in Sr-1«Z1, vd, ... , (zn' vn)).

Proof. The number of the functions is N := V1 + ... + Vn and, as seen from Theorem 8.4, every spline 1 from Sr-1(Z) can be presented uniquely as a linear combination of them.


The complete characterization of the Hermite interpolation problem of a periodic function by periodic splines is known only in the case of an odd number of knots.

Let 0 < Zr+1 ::;; ... ::;; zr+n < 21f' and 0 < t1 ::;; ... ::;; tn < 21f' be fixed points, such that Zj < Zj+r, i = r + 1, ... , n, tl: < tl:+r , k = 1, ... , n - r. For a given sufficiently smooth function 1 on [O,21f'), consider the problem of determining a spline s from Sr-1{Zr+1, ... , zr+n) satisfying the interpolation conditions

i = 1, ... , n, j = max {), : tj = ... = tj-A} (8.2.1)

(i.e., this is the Hermite interpolation, where the multiple points from the sequence t1, ... , tn are taken as multiple nodes).

Let Z1, ... , zn+r, ... be a 27r-periodic extension of the sequence Xr+1,· .. , zr+n. For r ::;; n introduce the B-splines of degree r - 1

Bj(t) := B(xj, ... , Xi+r; t)

Ch. B, § B.2] Hermite Interpolation 125

and denote by Bi(t) the 211"-periodic extension of Bi(t) on the whole real line. Clearly,

on [0,211"].

Since {B;}~ form a basis in 8,.-1 (Z,.+1,' .. , Z,.+n) on [0,211"] (see Theorem 8.1), every spline s from 8,.-1 (Z,.+1 , ... , Z,.+n) may be presented in the form

n

s(t) = L: Ci [Bi(t) + Bn+i(t)] i=l

and hence the interpolation problem (8.2.1) reduces to the study of the collocation matrix

G:= [~ly.1~.~ ~~:.1.(~~~ .. " .. " ... ~.ny.1? ~ .~~~~~~~ 1 B1(tn) + Bn+1(tn) ... Bn(tn}+B2n(tn)

under the standard stipulation that the corresponding rows are replaced by the rows of derivatives in the case of coalescent ti.

THEOREM 8.6. Let n be odd and r ~ n. Then for each choice of the sequences Z,.+l ~ ... ~ Z,.+n and t1 ~ ... ~ tn in (0,211"),

det G ~ 0.

Moreover, det G> 0, if and only if

Bp+i (ti) > 0, i = 1, ... ,n, (8.2.2)

for some integer p.

Proof. We shall use the same technique as in the nonperiodic case (see Theorem 4.6). Adding additional knots e, e + 211" to the original sequence Zl, ... , Zn+,., one pair

of knots at a time, we get a new sequence y of knots. Denote by {B?}f" the corresponding sequence of new B-splines, associated with y. Assume that the refinement y is so dense that:

a) BP(tj) f; 0 implies BP(tJJ = 0 for all other knots tJ: from the periodic extension Of{t1, ... ,tn}'

b) ti lies at a knot from y of multiplicity r, i = 1, ... , n. Set for convenience

Gi(t) := Bi(t) + Bn+i(t) = Bi(t) + Bi(t - 211"),

G?(t) := B?{t) + B?(t - 211").

By the decomposition formula of Lemma 3.9,

BJ:(t) = L: O:m(k)B~(t), mEJ.


where h. := {j: supp BJ C supp Bk} and all coefficients am (k) are positive. This implies

Gk(t) = 2: am(k)G!!,(t). mEJ ..

Then det G = 2: a(M)G(M), (8.2.3)

M

where M = (mi,"" mn ), m k runs over Jk,

and a(M) := am1 (1) ... am .. (n). It is seen in the same way as in the proof of Theorem 4.6 that the summation in (8.2.3) is expanded over only those M for which mi < ... < mn •

Next we find the sign of G(M), provided mi < ... < mn . Assume first that B~ (ti) ::f:. O. Then, for a sufficiently dense refinement y, supp B~ (t) C (0,211") for

1 ..

all k = 1, ... , n and therefore

G!!, .. (t) == B! .. (t) on [0,211"].

Then

G(M) = det [~~~ .(~~~ ........... ~~~ ~~~~ 1 B!l (tn ) . . . B! .. (tn )

and by Theorem 4.6, G(M) ~ O. The strict inequality holds if and only if B!; (ti) ::f:. 0 for i = 1, ... , n. Now note that the sum (8.2.3) is expanded over all mi < ... < mn , mk E Jk. In particular, (8.2.3) contains the term a(M')G(M'), where M' = (mL ... , m~) is chosen (if possible) in the following way:

k = 1, ... ,n.

The choice is possible if and only if the condition (8.2.2) is fulfilled. Thus the theorem will be proved if we show that G(M) ~ 0 (with strict inequality only if (8.2.2) holds) in the remaining case B~l (ti) = O.

Assume that B~l (tt) = O. If B~;(tt} = 0 for all i = 1, ... , n, then

G(M) := det {B!; (tk - 211")} ~=i~=i

and G(M) ~ 0 by Theorem 4.6. Let B~l(tt} = ... = B~ (tt) = 0 and B~ (ti) ::f:. O. Then B~ (ti) =

"-1 .. 1

B~ (ti) = 0 for all i = 1,2, ... , n. Therefore "-1

G!;(t) = B!;(t - 211") for i = 1, ... , k - 1,

G!,(t) = B!,(t) for i=k, ... ,n. J J

Ch. 8, § 8.2) Hermite Interpolation 127

Observe further that

i = 1, ... ,k - 1,

for some mn+1 < m n+2 < ... < mn+k-l. Then G(M) may be written in the form

G(M) = det [~~~:~~~~~ .. ::: .. ~~ft.~ •• ~l.~~l?~~ .. ?.l: ... '.'.' ... ~.~~.(~~~]. B!ft+l (tn ) .,. B!ft+._l (tn ) B!. (tn ) ... B!ft (tn)

Now using the assumption that n is odd, we see that the determinant will not change if we put the first column on the last position. Repeating this operation k - 1 times we get

G(M) = det

and again by Theorem 4.6, G(M) ~ O. The proof is completed.

COROLLARY 8.7. Suppose that n is odd. Let Xl, X2, ... be a 27r-periodic extension of X,.+1 ~ ... ~ X,.+n. Then, for given I, the Hermite interpolation problem (8.2.1) by splines s from 8,.-1 (x"+1 , ... , X,.+n) has a unique solution, if and only if

k = 1, ... ,n,

for some integer p. The first inequality should be interpreted as an equality in the case xp+l: = ... = xp+l:+,._l'

Exercise 8.2.1. Let 0 < Xl < ... < Xn < 27r and It, ... , In be arbitrary given numbers. Prove directly that there exists a unique spline s E 82,.-1 (Xl, ... , xn) which satisfies the interpolation conditions S(Xi) = Ii, i = 1, ... ,n.-Hint. The assertion follows immediately from the general interpolation result stated , in Corollary 8.7. To prove it directly, show that

2,.-J [s(,.)(t)] 2 dt = 0

o

for each s E 82,.-1 ( Xl, ... , X n) that satisfies the homogeneous system of equations S(Xi) = 0, i = 1, ... , n. Follow the hint in Exercise 5.1.2.

The central interpolation result in this chapter was proved for interpolating splines S with an odd number of knots. The following is an example of an uniquely solvable interpolation problem in case of even number of knots of s.

Exercise 8.2.2. Let cp be the 27r-periodic perfect spline defined in Corollary 8.9. Let {ei}f be the knots of cp in [0, 27r). Show that for each system of numbers {lij} there exists a unique spline s E 8,.-1(6, ... ,eN) which satisfies the Hermite interpolation conditions

for i = 1, ... , n, j = 0, ... , IIi - 1.

128 Periodic Splines leh. B. § 8.2

Hint. Show that the corresponding homogeneous system ~dmits only the trivial zero solution. To this aim, assume that there is a spline 8 E Sr-1(et. ... ,eN) such that 8(i)(Zj) = ° for i = 1, ... ,n, j = O, ... ,lIi -1 and 8('1) > ° for some '1 E [0,211"). Define get) := cp(t) - a8(t) with a = cp('1)/8('1). The function 9 has at least N + 1 zeros in [0,211"). Then, by Rolle's theorem g(r-2)(t) has also at least N + 1 zeros. Now observe that g(r-2)(t) is continuous and g(r)(t) changes its sign only at the knots {ei}{"" of cpo This implies that g(r-2)(t) has N zeros at most in the period, which is a contradiction.

§ 8.3. Favard's Problem

All the results from Chapter 6 concerning perfect splines on a given interval [a, b] have their periodic analogues. We discuss briefly here only the Favard's interpolation problem for periodic functions, which is the basis for further periodic modifications of the material from the previous chapter.

Denote by W~ the class of all 211"-periodic functions from We: [0, 211"], i.e.,

w~ := {f EW~[O, 211"]: 1(i)(O) = 1(i)(211"), j = 0, ... , r - I}.

When studying functions I from W~ it is important to remember that I and its derivatives are periodic functions. Thinking of I as being defined on the unit circle one observes that it has always an even number of sign changes in the period [0,211"). In particular, every periodic perfect ,..!pline has an even number of knots. Note otherwise that if a function I from W~ has 2m zeros in [0,211") Rolle's theorem implies that I' has at least 2m zeros there.

Taking into account all these remarks concerning periodic functions we are ready to sketch the proof of the periodic version of Theorem 6.2.

With any given incidence matrix E = {eij}i=1,i;~, a system of points z = {Zk}~' ° < Z1 < ... < Zn < 211", and a set of values y = {Vij}, we associate the set of functions

F(E;z,y):= {f EW~: 1(i)(Zi) = Vij for eij = I}.

THEOREM S.S. Let E = {eij}i=1,i;~ be a given incidence matrix with lEI = N and let N be odd. Suppose that E does not contain odd blocks of 1'8 of the form eij = ... = ei,j+l-l = 1, j >0, ei,j-1 = 0, ei,HI = ° if j + I < r. Then for each Y = {vij} and for each system of points z = {Zk}~, 0 < Z1 < ... < Zn < 211", the

set F(E; z, y) contains a periodic perfect spline 8 with no more than N - 1 knots. Moreover,

Proof. Suppose that 8 is a periodic perfect spline from F( E; z, y) which has no more than N -1 knots in [0,211"). Assume that II/(r)lIoo < 118(r)lIoo for some I EF(E; z, y). Then the function g(z) := 8(Z) - I(z) will vanish at (z, E) and, by Rolle's theorem, g(r)(z) would have at least N sign changes. But signg(r)(z) = sign8(r)(z) because

Ch. B, § B.3) Favard's Problem 129

Is(r)(x)1 = IIs(r)lIoo > IIf(r)lIoo. Thus s<r)(x) has at least N sign changes and this contradicts the assumption. The extremal property of the perfect spline interpolant is proved. N ext we show the existence following the same idea we used in the non periodic case.

If YiO = 0 for all eiO = 1 the interpolation problem admits the trivial solution s(x) == const. Further, we assume that there is at least one I E {I, ... , n} for which e"O = 1 and Yl,O #= O. Set

Introduce the points

and the function

,p(h; t) := sign hi

Set for convenience u(t) == 1. Consider the system of equations

2,..

ao - cYIO = -; J ,p(h; t)Dr(Xl - t) dt, o

2,..

Ie= 1, .. ,N,

aou(!')(Xm) - CYm!' = -; J ,p(h; t)Dr_!, (Xm - t) dt o

(8.3.1)

in unknowns ao and c, where (m,JJ) is any pair of the set A := {(i,j): eij = 1, (i,j) #= (l,O)}. Let D be the determinant of the system. Assume first that D = 0 for every choice of (m, JJ) E A. Then our problem admits the trivial solution s(x) = const = YIO.

Suppose now that D #= 0 for some (m, JJ) E A. Then the system (8.3.1) has a unique solution ao(h), c(h), and it depends continuously on h in SN+l. Define the mapping c): SN+l -+ ~N in the following way: c) == (c)l(h), ... ,c)N-l(h», where c)l(h), ... , c)N-2(h) are the expressions

2,..

ao(h)u(j)(Xi) - C(h)Yij -; J 'I/J(h;t)Dr_j(Xi - t) dt o

for (i,j) E A\(m,JJ) and 2,..

c)N-l(h):= J 'I/J(h;t) dt. o

The mapping c) is continuous and odd. Then by Borsuk's theorem (see Theorem 6.1), there is h" E SN+l for which

130 Periodic Splines

Note that the condition I)N_I(h) = 0 guarantees that the function

2,..

s(hj x) := ao(h) + .; J "'(hj t)Dr(x - t) dt

o

is 211"-periodic (see Theorem 8.3).

[Ch. 8, § 8.3

Clearly c(h*) i= 0, since otherwise the function s(hj t) would be a periodic perfect spline with N - 1 knots which vanishes at (z, E) (with lEI = N) and a standard argument based on Rolle's theorem leads to contradiction. Then the function s(h*j t)/c(h*) is the desired solution of Favard's problem. The proof is completed.

Note that the incidence matrix E that describes an Hermite interpolation problem satisfies all the requirements of the previous theorem. We apply this particular (Hermitian) case of Theorem 8.8 to derive the following proposition, which is known as the fundamental theorem of algebra for periodic perfect splines.

COROLLARY 8.9. Let z = «Xl, vt}, ... , (xn , vn)) be any set of fixed points such that 0 < Xl < ... < Xn < 211", 1 ~ v k ~ r for k = 1, ... , n and let N:= VI + ... + Vn

be even. Then there exists a unique (up to multiplication by -1) 211"-periodic perfect spline tp of degree r with no more than N knots which satisfies the conditions IItp(r)lIoo = 1,

i =1, .. . ,n, j =0, ... , vi - 1.

Moreover, this spline has exactly N knots.

The proof goes in the same way as that in the nonperiodic case (see Theorem 6.4). We leave it as an exercise.

In the following exercises, PrN denotes the set of all 211"-periodic perfect splines of degree r with at most N knots in [0,211").

Exercise 8.3.1. Let z = «Xl, VI), ... ,(xn , vn)) be a given set of points, 0 ~ Xl < ... < Xn < 211", 1 ~ Ilk ~ r for k = 1, .. . , n. Let N := VI + ... + Vn be even. Suppose that tp E PrN and satisfies the conditions: IItp(r)lIoo = 1 and tpl .. = o. The latter

condition means that tp(i)(Xi) = 0 for i = 1, ... ,n, j = 0, ... , Vi - 1. Prove that, for each t,

Itp(t)1 = sup {1/(t)l: I EW~, lIJ<r)lIoo = 1, II .. = O}.

Hint. Follow the method used in the proof of Corollary 6.5.

Exercise 8.3.2. Let N be an even number. Suppose that the spline tp from PrN equi-oscillates at N + 1 points, Le.,

k=O, ... ,N,

for some points 0 ~ to < tl < ... < tN < 211". Let IItp(r)lIoo = 1. Show that

Hint. Assume the contrary. Then there exists a perfect spline P E PrN with IIP(r)lIoo = 1, such that IlPllc < IItpllc. Then the function g(t) := tp(t) - P(t), and

Ch. 8, § 8.3] Favard's Problem 131

consequently g(r)(t), will have at least N sign changes in [0,2'11-). On the other hand, every translation tp( t + 8) of tp( t) equi-oscillates at N + 1 points. We can then assume without loss of generality that the smallest interval (say [ek,ek+l ]) between two consecutive knots of tp is entirely contained between two consecutive knots of the spline P. Then the function g(r)(t) has no sign changes in (ek- l ,ek+2)' Besides, g(r)(t) van-

ishes or has the sign of tp(r)(t) in each ofthe intervals (el, 6), ... , (eN- l , eN)' (eN' 6 + 211'), which implies that g(r)(t) has N - 2 sign changes at most in [0,211'), which is a contradiction.


There is not much written about periodic perfect splines. The presentation by Bernoulli's polynomials was proved in particular cases and used by Zensykbaev [1974], and Korneichuk [1984]. For further properties of the Bernoulli polynomials we refer to the book of Krylov [1967]. The B-spline basis for periodic functions is mentioned in Schumaker [1976], [1981]. The characterization of the Hermite interpolation problem is done by Melkman [1974] on the basis of a Budan-Fourier type theorem for splines. Schumaker [1976] derived the interpolation theorem from an estimation for the number of zeros of a periodic spline. The present proof of Theorem 8.6 is taken from Bojanov [1992], where the more general case of interpolation by splines with Birkhoff knots is considered. Section 8.3 is based on Bojanov [1977], [1990].

Chapter 9

MULTIVARIATE B-SPLINES AND TRUNCATED POWERS

§ 9.1. A Geometric Interpretation of Univariate B-Splines and Truncated Powers

In the following, it is convenient to accept slightly a different normalization for Bsplines. Namely, a B-spline with a knot set fJ = {to, ... , tr }, a ~ to ~ ... ~ tr ~ b is defined by the rule

M(t) = M(t I fJ) = r[to, ... ,tr](·_t)+-l (9.1.1)

and therefore (see 3.2.1) b J M(t)dt = 1.

a

The relation (9.1.1), at the same time, is the B-spline representation via truncated powers. In the case of different knots we have

M(t I fJ) = ~ r(t. - t)+-l = ~ r(t - t.)+-l . ~ II (t. - t .) ~ II (t· - t .) ,-0 J ,-0 'J - O~j~r, - O~j~r,

j~' j~i

(9.1.2)

For the first equality see Theorem 1.2 and (1.1.2). In order to check the second equality it is enough to note that

(ti-t)+-l + (_l)r-l(t - ti)+-l = (t.-tt- 1

and [to, ... ,tr](.-tt-1 = o. In this section, we present a geometric interpretation for this formula, which in

cludes interpretations both for B-splines and truncated powers. First, we shall deal with the result concerning B-splines due to Curry and Schoenberg [1966].

To this end, we recall the two familiar integral representations for divided differences. The first representation with the new notation of B-spline is (see Definition 3.2)

b

[to, ... , trlj = ~ J M(t I fJ)j<r)dt, r.

j E Wl[a, b]. (9.1.3) a

132

Ch. 9. § 9.1) Geometric Interpretation of Univariate B-Splines 133

Until the next representation, let us introduce a basic multivariate notation we shall often use:

J 1 := J 1:= J 1 (1I0XO + ... + Vrxr) dill ... dllr, (9.1.4)

[X] [:r;o •...• :r;r] Sr

where X = {xo, ... , xr} C ~k, I: ~k -+ ~ and

sr = {(110,' .. ,lIr): 110 + ... + IIr = 1, IIi ~ 0, i = 0, ... ,r}

is the standard r-simplex. Or, carrying out a surface integral of the second kind to the multiple integral:

J 1= J I[Xo+lIl(xl_xO)+ ... +lIr(Xr-xo)]dlll ... dllr, (9.1.5)

[:r;0 •...• :r;r] Qr

where Qr = {(Ill, ... , IIr): III + ... + IIr ~ 1, IIi ~ 0, i = 1, ... , r}. Now the second integral representation of divided differences, i.e., Hermite-Genocci

formula (see (1.3.9)) can be written in the form

[to, ... ,tr]f= J j<r), 1 E C"[a,b]. (9.1.6)

[to •...• t r ]

The representations (9.1.3) and (9.1.6) yield the following basic relation:

J M(t)/(t) dt = r! J 1 (9.1.7)

]I [to •...• trl

for any 1 E Co(~)· The idea of Curry and Schoenberg is based on the following nice change of variables

in the right-hand side of (9.1.7):

(9.1.8)

where yO, . .. ,yr E ~r are the vertices of a proper simplex whose first coordinates agree with to, ... ,tr respectively, i.e.,

i = 0, ... ,r, (9.1.9)

with u = [yO, •.. , yr] the convex span of yO, ... , 11. It is clear, that y 11.:= Yl = 1I0to + ... + IIrtr and the absolute value of the Jacobian is r! volru. Therefore

J 1 = I 11 J x~(Y)/(y II.) dy = ~ J I(t)cp(t) dt, r.vo rU r. [to •...• t r ] I.r I.

134 Multivariate B-Splines and Truncated Powers [Ch. 9, § 9.1

where

(t) __ I_ J (I)d - voI,.-l{VEO': vl.=t} If' - Xu , U U - •

voI,.O' voIr 0' :ar - 1

Hence, on account of (9.1.7), we obtain

1 VOlr_1 {y E 0': vi. = t} M(t 0) = I,. , vo 0'

(9.1.10)

where the simplex 0' satisfies (9.1.9), i.e., M(t 1 0) is the ratio of the (r - I)-volume ofthe cross section of 0', orthogonal to the Y1 axis at the point t over the r-volume of CT. Let us mention a significant fact we have obtained - the ratio in the right-hand side of (9.1.10) is independent of the choice of 0' satisfying (9.1.9). Because of the geometric interpretation (9.1.10), B-splines are called also simplex splines. Let us illustrate (9.1.10) in the case r = 2

t

Fig. 9.1.

Let to, t1, t2 be three points in the real line IR with to ~ t1 ~ t2, to < t2. We choose the points yO, y1 , y2 as the vertices of the triangle T in the plane lying arbitrarily on lines orthogonal to IR with bases to, t1, t2, respectively. Let also the line orthogonal to IR at the point t intersect the triangle T with segment I. Then according to (9.1.10) we have

length {I} M(t I to,t1,t2) = area{T}.

To find a similar geometric interpretation for truncated powers, we notice that the

Ch. 9, § 9.1) Geometric Interpretation of Univariate B-Splines 135

B-spline M(t I 0) and the first truncated power in the second sum in (9.1.2)

r(t - to)±-l Tto(t) = Tto(t Itt, ... ,tr) = ( ) ( ) tl - to ... tr - to

coincide in a neighbourhood of to. This yields that this truncated power satisfies the relation (9.1.10) with simplex (1' in the right-hand side ratio's numerator replaced by the cone C, which coincides with the simplex (1' at the neighbourhood of the vertex yo:

C = C(yO I yl _ yO, .. . , yr _ yO)

:= {Yo + Vl(yl - yO) + ... + vr(yr - yO): Vi ~ OJ, (9.1.11)

i.e.,

(9.1.12)

A similar relation can be obtained for the truncated power with respect to the last (greatest) knot t r , using the first sum in the representation (9.1.2), respectively. Hence, one can state (ignoring the condition of the monotonicity of the knots) that (9.1.12) holds iff

(9.1.13)

It is worthwhile to note in this context that the last condition is necessary and sufficient for the ratio on the right-hand side of (9.1.12) to be a finite-valued function of t. Because of the following obvious relation

Tto (t I tt, ... , tr) = To(t - to I tl - to, ... , tr - to),

we will later often deal with O-centered truncated powers - missing the subscript 0 in its notation: T(t) = T(t I t l , . .. , tr). In this case (9.1.13) looks like 0 ~[tlJ ... , tr].

The relation (9.1.12) implies the following equality similar to (9.1.7)

J T(t)f(t) dt = r! J f(Vltl + ... + vrtr) dVl ... dVr (9.1.14)

11+ II'

for any locally integrable function f. Let us note that the condition 0 ~[tl"'" tr] is necessary and sufficient for the right-hand-side integral in (9.1.14) to be finite for locally integrable functions f and for the existence of a function T satisfying (9.1.14).

Let Y = {yO, ... , yr} C Rr be defined as in (9.1.9) and

r-l

E:= U L i ,

i=O

where Li is the (r - I)-dimensional hyperplane containing the convex span of Y\lI. We turn next to the following striking representation of a characteristic function of the simplex (1' by the same functions of the appropriately chosen cones:

r

X,,[Y](y) = ~) -1)iXc(yilYi)(Y)' (9.1.15) i=O

136 Multivariate 8-Splines and Truncated Powers [Ch. 9, § 9.1

where yi := {yi _ yO, ... , yi _ yi-l, yi+l _ yi, ... , yr _ yi}. The case r = 2 of this relation is illustrated below.

Fig. 9.2.

To prove (9.1.15) let us denote

1'1 = l' (yO , ... , yl I yl+l, . .. , yr) = { Po Aiyi: Po Ai = 1, Ai ~ 0

for i = 0, ... , I; Ai ~ 0 for i = 1 + 1, ... , r}. We have for i = 1, ... , r - 1

C(yilYi) = {yi + vO(yi _ yO) + ... + Vi-l (yi _ yi-l)

+ Vi+l (yi+l - yi) + ... + Vr (yr _ yi): Vj ~ O} = { - VWo - ... - vi_lyi-l + yi(1 + Vo + ... + Vi-l - Vi+l - ... - vr)

+ Vi+lyi+l + .... + vryr: Vj ~ O} =: 'Yi-l U 'Yi.

For i = 0 and i = r, the analogous relations are

Taking into account that 'Yi-ln'Yi eLi, i = 1, ... , r-l, and un'Yo C Lo we conclude

i = 1, ... ,r - 1,

XC(!l0IYO)(y) = XO'(Y) + X-Yo(y) for y E IRr\Lo,

XC(!lrlyr)(y) = X-Yr-l (y) for y E IRr.

Ch. 9, § 9.2) Multivariate B-Splines and Truncated Powers 137

Summing up these relations we easily come to the result (9.1.15). The relation (9.1.15) on account of (9.1.10) and (9.1.12) is the desired geometric

interpretation of relation (9.1.2). We present a more detailed discussion of this subject in the multivariate case.

§ 9.2. Multivariate B-Splines and Truncated Powers

Now we will consider a natural generalization of B-splines to the multivariate case due to de Boor.

Let X = {zo, ... , z"} C lit, r ~ k + 1, and volt [X] -:f O. It is not difficult to show that there exists a proper simplex (1' C lit with vertices yi having the same first k coordinates as zi, i = 0, ... , r, respectively, i.e.

i = 0, .. . r; (9.2.1)

vol,.(1' -:f O.

DEFINITION 9.1. The B-spJine M(· I X) with the knot set X is defined as follows (r ~ k + 1):

I) ( I ° ) vol,._t {y E (1': yill" = z} M(z X = M z z, ... , z,. := I .

vo ,.(1' (9.2.2)

Similar to the univariate case, the change of variables (9.1.8) implies that the following relation which is the multivariate analogue of (9.1.7) is equivalent to the previous definition:

f l(z)M(z I X) dz = r! f I, (9.2.3)

11" . [Xl

for all I E Co(lit ). This relation establishes the independence of M(. I X) for the choice of (1' satisfying (9.2.1).

For (9.2.3) to be still valid for the case r = k put

M( l Ot) X[~o, ...• ~lol(z) z z, ... , z := 1 [ ° t] . vo t z , ... ,z

(9.2.4)

The following properties of the B-spline M (. I X) for every X C li, VOlk [X] -:f 0 directly follows from (9.2.2) and (9.2.3):

i)

ii)

iii)

supp M('I X) = [X], f M(:c I X)dz = 1,

11"

M(·-yIX)=M(·IX+y), YElik, 1

M(z I AX) = detA M(A-1z I X), z E lit,


for the arbitrary nonsingular matrix A of order k. In Fig. 9.3, we illustrate the connection between univariate and bivariate B-splines. Let xl E [XO, x 2, x3 ] C ~2 (see Fig. 9.3). We put yi = xi, i = 0,2,3, and choose

y1 in the space on xl (i.e., on the perpendicular), such that vola[yO,y1,y2,y3] = l. Then, by (9.2.2) the side surface of the pyramid coincides with the space graph of M(x I xO,x1,x2 ,x3), i.e., it equals 1- the length of the segment [x,y]' where y is in the side face situated on x, while the value M(t I x~, xL x~, x~) is S - the area of the axes perpendicular to Xl at t cross-cutting the pyramid.

Fig. 9.3.

This relation in higher dimensions indicates the formula (9.2.6). To prove this let

i = 0, .. . ,r. (9.2.5)

Ch. 9, § 9.2] Multivariate B-Splines and Truncated Powers 139

Applying (9.2.3) to the function !(u) = rp(u IlIk) == rp(x) we get

On the other hand

r! f ! = r! f !(vouO + ... + VrUr)dlll ... dllr [uo •...• u r] s ..

= f rp(voxO + ... + VrXr)dlll ... dVr = f rp(x)M(x I X) dx. S.. J.k

Since the above relation holds for every locally integrable function rp, we obtain

M(xlxO, ... ,xr)= f M(uluo, ... ,ur)duk+1 ... dum J.m-Ic

provided that (9.2.5) holds. Note that the geometric interpretation (9.2.2) is a special case of (9.2.6). Indeed,

to obtain (9.2.2) it is sufficient to put m = r in (9.2.6) and note that according to (9.2.4)

M( l or).- X[uo ..... ur](u) u u, ... , u .- I [ ° ] . vo r U , ••. , ur

(9.2.7)

The following lemma is an immediate corollary of (9.2.2) and shows that the Bspline is continuous with respect to knots.

LEMMA 9.2. Let X = {xO, xr}, X[v] = {xO[v], ... , Xr[lI]} and

lim xi[v] = xi, 11_00

i = 0, . .. ,r.

Then lim M('I X[v]) = M('I X).

11-00

In analogy to the relations (9.2.2) and (9.2.3) for multivariate B-splines, one readily obtains the following equivalent definitions for k-variate truncated powers.

Let X, = {xl, ... , xr}, xO = 0 rt. [X'] and yi, i = 0, ... , r, are chosen as in (9.2.1).

DEFINITION 9.3. The Truncated power T(· I X') with knot set X' is defined as follows (r ~ k + 1):

T(. I X') .- T( I 1 r) _ volr-A:{y E C: YIJ.k = x} .- x x, ... , x - l,. , VO iT

x E ~k, (9.2.8)

where C = C(yl, ... ,yr) := {Vlyl + ... + IIryr: Vi ~ o}.


The equivalent definition is:

+00 +00 J T(x I X')/(x) dx = r! J ... J /(Vlxl + ... + VrX")dVI'" dVr (9.2.9)

mk ° ° for any / E Co(IRAl The xO-centered truncated power is obtained by means of the parallel shift:

Suppose that all knots in X are pairwise distinct and ordered, such that

i = 0, .. . ,r, (9.2.10)

Exercise 9.3.1. Check that one can achieve this by ordering X with the following rule:

i) (xOh ~ ... ~ (x"h, ii) (X')i = (x'+! )i, i = 1, ... , m -1, 0 ~ I ~ r - 1, implies (x')m ~ (x'+1 )m' By the representation (9.1.15) using (9.2.2) and (9.2.8), we obtain the following

generalization of (9.1.2):

,. M(x I X) = L(-I)iT(x - xi I Xi) for almost all x E IRk, (9.2.11)

i=O

provided that (9.2.10) holds. Indeed, according to (9.1.15), this equality can be violated at x only if

for some i = 0, ... , r - 1, (9.2.12)

which means that the (r-k)-dimensional hyperplane {y E IRr: y Im"= x} is contained in the (r - I)-dimensional hyperplane Li. This, in turn, implies that the equation of Li must resemble '\Iyl + .. . '\Wk = c and the set of all x satisfying (9.2.12) is the (k - I)-dimensional hyperplane in IRk given by '\IXI + ... '\kxk = c. Thus, the set of all points x at which the equality (9.2.11) does not hold, consists of a finite number (at most k) of (k - I)-dimensional hyperplanes and, hence, has zero measure. Of course, this equality must hold at each point x where the functions of (9.2.11) are continuous. At the end of this section, we present some results about the constructions of B-splines and truncated powers. Here, we will deal also with B-splines M(x I X) with

X C IRk, vo1m+! [X] = 0, volm[X] i= 0, m<k.

Ch. 9, § 9.2) Multivariate 8-Splines and Truncated Powers 141

In this case, we define

I ) vol,._m {y E u: Y\Jlk = x} M(x X = ,

vol,.+m_k U (9.2.13)

where Yi E tI1\,.+m-k yi \ - ",i ; 0 r' ~ , Jlk - '" , • = , ... , ,

U = [yO, ... , y,.], vol,.+m_k U f; O. (9.2.14)

Of course, we have that supp M(. I X) = [X] and the analogue of the relation (9.2.3) in this case is: J f(x)M(x I X)ds = r! J f (9.2.15)

L [:t"o ........ r1

for all functions f locally integrable on L with ds being the volume element in L.

THEOREM 9.4. Let X = {XO, ... , x,.} C IRk and

Then

where U is the distance of xO from LX\:t"o - the (k - I)-dimensional hyperplane containing X\xo.

Proof. Let Y = {yO, ... , y,.} satisfies (9.2.1). Using (9.2.2) we readily obtain for x E LX\:t"o (see Fig. 9.4)

(9.2.17)

It is easy to check that vol,.-l [Y\yO] r

= -; vol,.[y] f}

(9.2.18)

indeed, by the hypothesis of the theorem we have

Je (t)"-l vol,.[Y] = e VOI,._l [Y\yO] dt.

° On the other hand, using a geometric argument again on account of (9.2.2) we

have that on the segment [xO, x]:

o ~ t ~ 1, (9.2.19)

where C:t" does not depend on t. Now (9.2.17)-(9.2.19) yield (9.2.16).

142 Multivariate B-Splines and Truncated Powers

M(xIX)

Fig. 9.4.

Using the same arguments we readily obtain the following

THEOREM 9.5. Let X = {xO, ... , xr} C ]Rk,

volk[X] :f; 0, xO :f; [X\xO] ,

[Ch. 9, § 9.2

and let L be a (k-l)-dimensional hyperplane which intersects [XO, xi] at xt:f; XO for i = 1, ... , r. Then

i) M(x I X) = cM(x I xl. ... ,xL) for x E L,

ii) M(xIX)=c'M(xlxo,xl, ... ,xL) for xE]Rk, [xO,x]nL=0,

iii) Tzo (tx + (1- t)xO I X) = :. tr- k M(x I xl, ... , xL), for x E L, 0 ~ t ~ 00. (}

Let us point out that the relation iii) combined with (9.2.11) provides a recurrence relation of B-splines with respect to the dimension k.

§ 9.3. Recurrence Relations for B-Splines

In this section, we present recurrence relations for a B-spline with respect to the number of knots. We start with the following integral relation.

THEOREM 9.6. Let X = {xo, ... ,xr} C]Rk and volk[X\xO] :f; O. Then 00

M(x I X) = r J rrH-l M(tx + (1- t)xO I X\xO)dt.

1

(9.3.1)

Ch. 9, § 9.3] Recurrence Relations for B-Splines 143

Proof. For every f E Co(lRk) we have

J f(vozO + ... + VrZr)dVl ... dVr = J f(vozO + ... + vrzr)dvo ... dVr_l Sr Sr

= j h'-'{ J !(zO+h t.v;("J -.O))dV, .. dV,_}h ° vl+···+vr=l -

= j hr-k-1{J f(z)M(h-1(z-zO) Iz1-zO, ... ,zr-zO)dz}dh

° J.k

= J f(Z){j hr-k-lM((1_h-l)zr+h-lzlzl ... ,zr)dh}dz.

]lk ° It remains now to carry out the change of variable t = h-1 in the last integral.

N ow we present a basic relation for B-splines which is the generalization of Tchaka-101£'s formula (3.4.6) to the multivariate case.

THEOREM 9.7. Let X = {zo, ... ,zr} C Im.k and volk[X] =f. O. Then

D:t;i_:t;M(z I X) + (r - k)M(z I X) = rM(z I X\zi), (9.3.2)

if the function M(zi + t(z - zi) I X\zi) is continuous at t = 0, and

D±(:t;i_:t;)M(z I X) + (r - k)M(z I X) = r lim M(zO +t(z - zO) I X\zi), (9.3.3) f_±O

in the opposite case.

Proof. The case r = k is obvious. If r > k, we can assume that volk[X\zi] =f. 0 (using the continuity of a B-spline with respect to its knots, see Lemma 9.2).

Let i = O. By virtue of (9.3.1) we have

M:=M(z+s(xO-x) IX) 00

= r J Cr+l:- 1 M(1- t)xO + tx + ts{xO - x) I X\xO)dt 1 00

= r J C,.+l:-l M{(1- (1- s)t)XO + t(1- s)x I X\xO)dt. 1


Carrying out the change of variable T = (1 - 8)t and using (9.3.1) we get

00

M = r(1 - 8t-1: J t-rH- 1 M(I- T)ZO + TZ I X\ZO)dT

1-.

~ r(1 - sr' [ j .-_+'-1 M «1- T).' + TZ I X\.')dT + ~ M(. I X) l 1-.

Hence (1- 8)-rH M(z + 8(ZO - z) I X)

1

= M(z I X) + r J rrH-1 M(I- T)ZO + TZ I X\ZO)dT.

1-.

This equality can be written in the form:

~ [(1- 8)-rH - I]M(z + 8(ZO - z) I X) + ~ [M(z + 8(ZO - z) I X) - M(z I X)] 8 8

1

= ; J rrH- 1M(I_ T)ZO + TZ I X\ZO)dT.

1-.

It remains to pass to the limit when 8 -+ 0 or 8 -+ ±O. From Theorem 9.7, we readily obtain the following recurrence relations for multivariate B-splines.

THEOREM 9.B. LetX={zo, ... ,zr}clm./:. Then

for

for

r

M(z I X) = r ~ k LAiM(Z I X\zi) i=O

r

Z = LAiZi, i=O

r

DIIM('I X) = r LAiM(·1 X\zi) i=O

r

Y = LAiZi, i=O

if the functions M(· I X\zi) with nonzero coefflcients are continuous at z.

(9.3.4)

(9.3.5)

To prove the theorem it is enough to multiply both sides of (9.3.2) by Ai and sum up for i = 0, ... ,r.

Ch. 9, § 9.3) Recurrence Relations for B-Splines 145

In particular, we get from (9.3.4)

(9.3.6)

DEFINITION 9.9. We say that the set of knots X has the degeneration d = degen (X), o ~ d ~ IX 1- k - 1, if d is the smallest integer such that voh: [Y] i= 0 for every subset Y ofk + 1 + d elements of X.

If d = 0, we say that X is in a general position.

DEFINITION 9.10. Let X E]Rk and volk[X] i= O. Regions which are bounded but not intersected by convex spans [Y] with Y ex, volk[Y] = 0, VOlk-l[Y] i= 0 we call b-regions.

It follows from Theorems 9.8 and 9.4 that M(· I X) is a piecewise polynomial function. Namely, it is a polynomial of total degree ~ r - k in each b-region. Moreover,

In particular, maximal smoothness is attained, if the knots of X are in a general position, then

(9.3.7)

On the other hand, the discontinuous B-spline M(. I X) we have in the case of X with maximal degeneration which has been described in Theorem 9.l.

Note that if we replace M by DOt M, lal = m, then the relations (9.3.5) and (9.3.6) remain valid while (9.3.1)-(9.3.4) hold with (r - k - m) instead of (r - k).

Now let us discuss the barycentric coordinates which enables us to check the coefficients Ai in the recurrence relation (9.3.4) (as well as in (9.3.5».

Let V = {vo, . .. , vk } C ]Rk, volk [V] i= O. Assume that Li is the (k -1 )-dimensional hyperplane passing through the points of V\ {vi} and is given by the following equation:

Alzl + ... + A~Zk + >'~+1 = 0,

Then one has for any x = {Xl, ... , Zk} E ]Rk

with

k

Z = ~biVi, i=O

i = 0, ... ,k.

bi = bi(Z) := A~Z~ + ... + >'~Z~ + >'~+1 >'ivi + ... + >'1: v1: + >'1:+1

being the barycentric coordinates and with slightly changed coefficients:

where

k

Z = ~Civi, i=O

(9.3.8)

(9.3.9)


Since both sides of the equalities (9.3.8) are linear functions, it is sufficient to check them for x = vi, j = 0, ... , k. In this case, we have

bi (vi) = Di,i'

which obviously implies (9.3.8). The equalities (9.3.9) follow directly from the following relations:

k k k k

~)bi - Ci) = ~bi(O) = 1 and ~(bi - Ci)Vi = ~ bi(O)vi = O. i=O i=O

It is worthwhile to noting that

where

bi(X) = e(x.' Li) e( v·, Li)

i=O

and () e(x, L?) Ci x = (. )' e v·, Li

i=O

i = 0, .. . ,k,

(9.3.10)

is the signed distance of x from Land c = ±1 is chosen such that that e(x, L;} ~ 0 for x E [V] and LO is the parallel shift of L containing the origin. The sign c is the same for Li and L?

Exercise 9.10.1. Check that for i = 0, ... , k

i) bi(x)=det v, ... ,v ,x,v , ... ,v det v, ... ,v , ( 0 i-1 i+1 k) / (0 k) 1, ... ,1, 1, 1, ... ,1 1, ... ,1

Ci( x) = det v , ... , v , x, v , ... , v det v , ... , v ( 0 i-1 i+1 k) / (0 k) 1, ... ,1, 0, 1, ... ,1 1, ... ,1

and

ii)

where ni is the normal of Li (directed toward vi) and r ELi.

Exercise 9.10.2. Let X consist of(k+1) distinct knots Vi, repeated with multiplicities J.&i + 1, i = 0, ... , kj

k

~(J.&i + 1) = n + 1, ;=0

Then

i)

k

ii) if vO = 0, X' = {v E X: v =F vol, ~(J.&; + 1) = n' + 1, then ;=1

Ch. 9, § 9.4) Ridge Functions 147

§ 9.4. Ridge Functions

In this section, we observe the so-called ridge functions: k-variate functions of the form

!(X) = !(X1, ... ,XI:) = g{A1X1 + ... + AI:XI:) = g(A' x),

where 9 is a univariate function and A E ]il:. Later, we present the method of generalization of univariate features to the multivariate case based on ridge functions.

From the forthcoming lemma, the denseness of the linear span of ridge functions in C{O) for any compact 0 C ]il: follows in particular.

LEMMA 9.11. The linear span of polynomials (1 + A . x)n, A E ]iI:, coincides with 1rn {]iI:), n = ~+.

Proof. First, we show that

"/ E]il:,

belongs to K, the linear span of mentioned polynomials, where the directional derivative is taken with respect to the variable A. This is true since K is a closed linear subspace of 1rn{]iI:) while

D-y{l + A . xt lim (1 + (A + t-y) . x)n - {1 + A· x)n t_O t

and the above fraction belongs to K. Therefore

for arbitrary ,,/1, ... , ,,/n E ]il:. This implies that all the monomials of degree n belong to K. On the other hand, (1 + A . x)n-1 E K, A E ]iI:, since

It can be easily checked that ridge functions satisfy the following relations:

D,d = (A . y)g'{x), (9.4.1)

as well as

(9.4.2)

Let us now demonstrate the ridge function method on some examples. On account of the relations (9.1.6) and (1.3.8) we have the following univariate formula:

J [(t - tj )!(t)](r) = J lr-1),

[to, ... ,t r] [{to, ... ,tr}\t;]

148 Multivariate 8-Splines and Truncated Pawers [Ch. 9, § 9.4

which reduces to (with f(r-l) replaced by !p)

j (t - tj)!p'(t) + r· j !p = j !p. (9.4.3)

[to, ... ,t r] [to, ... ,t r] [{to, ... ,tr}\t;]

This relation is easily generalized to multivariate functions f: ]Rk -+ ]R and xi E ]Rk, i = 0, ... ,r. Explicitly, we have:

(9.4.4)

The equality (9.4.4) is the integral analogue of the recurrence relation (9.3.2). Each can be easily obtained each from other, using (9.2.3). On the other hand, (9.4.4) can be easily obtained from (9.4.3). Indeed, by Lemma 9.11 it is enough to check (9.4.4) for a ridge function f. In this case, it follows from (9.4.3) because of the relations (9.4.1) and (9.4.2)

Let us prove now the following equality, which we will use in Chapter 11.

(9.4.5)

with X = {xC, ... ,xr} C ]Rk, volk[X]:F O. Using (9.2.3) we obtain from (9.4.2) that

j(1+>..x)-r-1 M(X I X)dx=r! j (1+t)-r-l

)\1< [>.ozO,ooo,>.ozr]

= (-It j ( _1 )(r). l+t

[>,ozo,ooo,>,ozr]

Because of the Hermite-Genocci formula (9.1.6), the last integral is equal to

[ or] 1 (-lr >. . x , ... , >. . x -1- = ~r::---'---'---+ t n (1 + >. . xi)

i=O


Multivariate B-splines were defined by de Boor [1976b] generalizing the geometric interpretation (9.1.10) of univariate B-splines due to Curry and Schoenberg [1966]. Micchelli [1979] discovered the recurrence relations (9.3.1) and (9.3.4). The relation (9.3.5) for a directional derivative of a B-spline is proved in Dahmen [1980] and Micchelli [1980]. The proof based on the equality (9.3.2) presented here is due to Hakopian [1982]. The truncated power was introduced in Dahmen [1980] where the equality (9.2.11) was established. The relation (9.4.5) is due to Dahmen and Micchelli [1983].

Chapter 10

MULTIVARIATE SPLINE FUNCTIONS AND DIVIDED DIFFERENCES

§ 10.1. Multivariate Spline Functions

In this section, we consider linear combinations of B-splines. Let X = {Zl, ... , zn} be an arbitrary finite set of knots in IRk (not necessary distinct) with volk[X] :f. O. Denote

XCv) := {Y C X: IYI = v}, v= O, ... ,n. (10.1.1)

DEFINITION 10.1. The space S:;' x of spline functions of order m, 1 ~ m ~ n - k, with a knot set X is defined as a'linear span ofthe system

n~,x := {M(·I Y): Y E X(m+ k), volk[Y]:f. O}. (10.1.2)

It is not difficult to see that B-splines in n~,x are linearly dependent. In the following two theorems we construct bases for the space S!. x consisting of B-splines from n~ x. We start with the case when the knot set X is in a general position, i.e., every subset Y E X(k + 1) forms a proper simplex: volk[Y] :f. O.

THEOREM 10.2. Let X be in a general position, 2 ~ m ~ n - k and Z E X(m). Then the system

1IE~,x := {M(·I ZUA): A C X\Z, IAI = k}

forms a basis for S!.,x.

Proof. Let us prove first that

M(·I Y) E (IIE~ x) ,

for all Y E X(m + k). We will use induction on v(Y) := IY n AI.

(10.1.3)

(10.1.4)

In the case v(Y) = m, we have M(·I Y) E IIE~ x. Suppose now that v = v(Y) < m and (10.1.4) holds if v(Y) > v. Then IY\ZI ~ k + 1 and any knot w E Z\Y can be written in the following form:

w = L:: AtJ v, tJEY\Z

149

L:: AtJ = 1, vEY\Z

150 Multivariate Spline Functions and Divided Differences

since X is in a general position. Now, using the relation (9.3.4) with

y=W- L AvV=O vEY\Z

we obtain:

(Ch. 10, § 10.1

O=DyM(·IYUw)=m ~_I(M(.IY)- L AvM(.IYUW\V)). + vEY\Z

Since v(Y U w\v) = v(Y) + 1 > v, the induction hypothesis implies that (10.1.4) holds.

To prove the linear independence of the system I1B~,x we need the following

LEMMA 10.3. Let C and A j , j = 1, .. . ,1, be arbitrary finite subsets of~l: satisfying

j = 1, ... ,I.

Then the systems

{M(·ICUAj): j=I, ... ,/} and {M(·I Aj): j = 1, ... ,I}

are independent simultaneously.

Proof. We can assume without loss of generality that ICI = 1. Let C = {w}, W E ~l:. We need to prove that the equalities

I I

LAjM(·lwUAj) =0 and LAjM(-1 Aj) = 0 (10.1.5) j=l j=l

are equivalent for arbitrary constants Aj. According to (9.3.2) we have

Dw-z: (i;:AjM(X I Aj U w)) = (q + 1) (i;:AjM(.1 Aj))

I

-(Q+l-k)([;AjM(XIAj UW))

for x E ~l:, hence the first equality in (10.1.5) implies the second one. The converse statement easily follows from the relation (9.3.1).

By Lemma 10.3 it is sufficient to prove the independence of the system I1B~ x in the case m = 1. Suppose Z = {w}, wE X, X' = X\w, and '

L AAM(·lwUA)=O, (10.1.6) AEX'(l:)

for some constants AA. Because of (9.2.4), each B-spline in (10.1.6) is discontinuous on the boundary of [A]. Since X is in a general position, this implies that the lefthand-side sum in (10.1.6) is a discontinous function if AA i' 0 for some A E X'(k).

Ch. 10, § 10.1) Multivariate Spline Functions 151

COROLLARY 10.4. Let X be in a general position and 1 ~ m ~ n - k. Then

. Ie (n-m) i) dlmSm,x = k with n = IXI,

ii) ifY C X, v = IYI < m and the system

M(·I Ai), Aj C X\Y, j = 1, ... ,jo,

forms a basis for S!._II,X\Y' then the system

M(·I YUAj), j = 1, ... ,jo, (10.1.7)

forms a basis for S!. x. , Indeed, i) directly follows from (10.1.3); to obtain ii) it is sufficient to note that

by Lemma 10.3 the system (10.1.7) is linearly independent and according to i)

. . Ie (n-v-(m-v)) (n-m) . Ie )0 = dlmSm_ lI,x\y = k = k = dlmSm,x,

The relations (9.2.2) and (9.3.7) imply that if X is in a general position then every spline function s(x) from the space S~ x has the following properties:

i) supp s C [X], , ii) s is a polynomial of total degree ~ (m - 1) in each b-region of X (see Defini

tion 9.10), iii) s E cm-2(~Ie).

In the following theorem, we show that under some restriction these properties are also sufficient for s to be in S~,x'

THEOREM 10.5. Let the knot set X be in a general position and such that for every (k - 2)-dimensional hyperplane h spanned by some knots of X the number of (k - I)-dimensional hyperplanes spanned by knots of X which contain h is not greater than m, 2 ~ m ~ n - k. Then the function s( x) is in S~ X if and only if it has properties i)-iii). '

To prove the theorem we need the following simple

LEMMA 10.6. Let L be an arbitrary (k -I)-dimensional hyperplane in ~k. If the function f E cm-2(~Ie) vanishes on one half-space of f and is a polynomial P of degree ~ (m - 1) on the other one, then

P(x) = c[U(x,L)]m-l,

with some constant c and U(x,L) the signed distance o{x from L.

Proof. Let I be an arbitrary line orthogonal to L. Considering the polynomial P on I yields

P(x) = c'[U(x, L)]m-l, x E I,

with some constant c,. Since deg P ~ m - 1, the constant c, does not depend on I.

152 Multivariate Spline Functions and Divided Differences [Ch. 10, § 10.1

Proof of the Theorem. We use induction on k. The case k = 1 is proved in Theorem 3.9. Suppose that k > 1 and that the theorem is true in ~k-l.

Let the function s satisfy i)-iii). Choose wE X such that w ~[X\w] and the (k - I)-dimensional hyperplane L such that wand X\w are in the different halfspaces of L (see Fig. 10.1).

v w

Fig. 10.1.

Denote for v E X\ w

vL := L n [v, w] and XL:= {vL : v E X\w}.

Then the induction hypothesis implies that s\L E S~~iL' i.e., s can be presented on L as follows:

s\L = LAyLM(-\yL), (10.1.8) yL

where the sum is taken over yL with yL E XL(m+k-l), volk_tfyL] f. o. Consider now the spline function

ql = LAyLCyLM(x I y U w) E S~,x yL

with Y := {v E X\w: vL E yL} and the constants CyL chosen such that (see Theorem 9.5, i»

cyLM(x I y U w) = M(x I yL) for x E L.

Ch. 10, § 10.1) Multivariate Spline Functions 153

Then, for 81 = 8 - ql, we have 811L == 0 which combined with Lemma 10.6 implies

supp 81 C [X\w). (10.1.9)

Using Lemma 10.6 and the properties of the knot set X one can show that 81 is a polynomial of total degree:::;; (m - 1) in each b-region of X\w. Of course, we have also 81 E Cm - 2 (JItk). Thus, 81 has the properties i)-iii) with the knot set X\w. Therefore, using similar arguments, we can construct sequence of spline functions qi

and knots Wi E X, i = 1, ... , v, with v = n - m - k + 1 such that

i

where Wi = U Wi,

j=l

and the functions 8i := 8 - ql - ... - qi satisfy i)-iii) with the knot set X\Wi, i = 1, ... , v. It remains to note that 8" == 0, since IX\W"I = k + m - 1 (this can be easily obtained from the univariate Lemma 3.4).

Now we are going to construct a basis for S~ X with an arbitrary set of knots X C JItk, volk[X) :f O. Let .

X = {y l , ... , yr } , 1'1 I'r

r

El'i = n = lXI, i=l

(i.e., X consists of distinct points yi repeated with multiplicity I'i). Suppose, without loss of generality, that

1= 2, ... ,r with yl:= { yl, ... , yl } 1'1 1'1

and let p be determined by the following conditions:

(10.1.10)

Consider for each 1= p, ... , r the (k - I)-dimensional hyperplane L{/} such that yl and y'-1 are in different half-spaces of L{/} and for A C y'-1 denote

A{/}:= {y{/}: YEA} with y{/}:= L{/} n [y, yl].

Now we are in position to present the theorem which gives an induction method (with respect to k and IXI) to construct a basis for S~ X (for the univariate case,

see Theorem 3.8). By 1IB~~1 we denote a basis for the s~ace s~~l. THEOREM 10.7. For an arbitrary set of knots X with volk[X) :f 0 and m, 1 :::;; m :::;; IXI- k, the system

r 1',

lIB~.x:= U U {Mel {yl, ... ,yl}UA): M(·IA{/}) ElIB~-~P+l.YI-l{I}} I=p p= 1 "'-..--"

p

forms a basis for s~.x' (10.1.11)

The proof of theorem is based on the following


LEMMA 10.8. Suppose that the knot sets Z = {zi}~=l and V = {J}}=l are in the different half-spaces of the (k - I)-dimensional hyperplane L. If ZU V is in a general Position and lIBA: v is a basis for SA: v then the following system forms a basis for m, m,

S~,zuv :

with A{i} := {[zi, v] n L: v E A} for A C V.

Proof. Using Corollary 10.4, we obtain that

A: t-m t+z-m- t+a-m . A: ( ) 1 (. 1) ( . )

IlIBm, ZuV 1= k + tt k - 1 = k = dlmSm, ZuV,

hence we need only to prove that the system lIB~ ZuV is linearly independent. Sup-pose that '

(10.1.12)

MeJ~,zuv

with some constants cM' Let us denote lIB i = lIB~~li+l, V{i} and consider the equality (10.1.12) in the half-space Q of L which contains Z. Since each B-spline in (10.1.12) with the knot set from V vanishes in Q, we have

/

E xEQ. i=1 MCIA{i} )eJi

Taking differential operators DZi - Z , i = 1, ... ,1- 1, and using (9.3.2) yields

M('IA{/})eJI

By Theorem 9.5, i) this implies

E cMc~M(x I A{/}) = 0, M(-IA{l} )ell

xE Q.

x E L,

with constants c:W- =f 0 depending on M. Because ofthe independence of lIB/ we obtain cM = 0 for ME lIB, . Similarly, one can prove that CM = 0 for all ME lIBi , i = 1, ... , I.

Proof of the Theorem. We prove first that S!, x is a linear span of lIB~ x' Choose the sequence of knot sets ' ,

X[v] = {{x~[V]}~~1}r , J J- ;=1

v = 1,2, ... ,

Ch. 10, § 10.1] Multivariate Spline Functions 155

such that Xlv] are in a general position and

lim Xlv] = X, II-CO

i.e., lim x~[v] = yi,

11_00 1 j = 1, ... , JJi, i = 1, ... , r. (10.1.13)

Using Lemma 10.8 and the construction of the system J1B~ x' given by (10.1.11), we

can construct {M('IAn[v])}~=l - the basis for the space S!,X[II] satisfying r

i)An[v]={{x~[V]}~!l}r , O~mr~JJi' ""mr=m+k, n=l, ... ,N, 1 1- i=l L....i

i=l ii) for every neither M(·I An) E J1B~,x or volk[An] = 0, where

An = n n = hm An[v], { yl, ... ,yr} . m1 mr II-CO

n= 1, ... ,N.

Assume now that M('I C) is an arbitrary B-spline from the space S!,x with

C = { yl, ... , if } , VI vr

r

o ~ Vi ~ JJi, L Vi = m + k, ;=1

and let C(v] = {{ x; [v]}l~l H=l, V = 1,2, .... Then

• M(·I C(vD = L cn[v]M('1 An[vD (10.1.14)

n=l with some constants cn[v]. We choose a subsequence vp , p = 1,2, ... , such that the sequence Cn [vp ] converges for each fixed n. Using the continuity of a B-spline with respect to knots (see Lemma 9.2) and ii) we obtain from (10.1.14) when V = V tends

. fi . p to In mty:

n

Cn = lim Cn [vp], p-co

where the sum is taken over all n with M (·1 An) E J1B~ x. The linear independence of the system J1B~ X can be ~roved as in the Lemma 10.8. ,

COROLLARY 10.9. If degen (X) ~ m - 1 (see Definition 9.9), then

dim S!,x = (n - ; + 1) . The corollary can be easily obtained from (10.1.11) by induction on k. One need only note that the condition degen (X) ~ m - 1 implies that in (10.1.11)

degen (yl-l{l}) ~ m - 1, 1= p, ... , r.

Exercise 10.9.1. Prove that if degen (X) ~ m, then

• k (n-m+l) dlmSm,x < k .


§ 10.2. Multivariate Divided Differences

In this section, we observe multivariate divided differences and its connection with B-splines. We start with the theorem which describes the construction of B-splines M(·I X) when the knot set

X={XO, ... ,xr }

is in a general position (see Definition 9.9). Since M(·I X) is a polynomial of total degree::;; r - k in every b-region E of X

(see Definition 9.10), DaM('1 X) is constant on E for every 0: E ~~, lad = r - k. We denote this by DaM IE' If E and G are neighbouring b-regions with a common side contained in [Y] for some Y E X(k) then we set

~YI~M:= ~YI~M(.I X) = DaMIE - DaMIG'

Let us denote also

d(x, "y, Y) = det

Xl yt ... yf

xk

"y

y~ 1

k , ... Yk

1

for Y = {y1, ... , yk} C X, yi = (yi, ... , YO. It is easy to see that the (k - 1)dimensional hyperplane containing Y is determined by the equality d(x, 1, Y) = O. By the right(left)-hand side of [Y] we mean the half-space {x E ~k: d(x, 1, Y) > O}, « 0). We assume below that E is in the right-hand side of [Y].

THEOREM 10.10. Let X = {xo, ... ,xr} be in a general position, 0: = (0:1 ... O:k) E ~t, lal = r - k and E, G be neighbouring b-regions with a common side contained in [Y], Y E X(k). Then

Proof. Assume that v E X\Y is on the left-hand side of [Y] and r> k. Let X be in the interior of G and let it be such that:

i) the half-line {v + t (x - v): t > 1} first intersects the side [Y], ii) it intersects different sides [V], V E X(k) at different points. Since r> k we have a j > 0 for some j. Then using the equality (9.3.1) yields

00

Da-ejM(xIX) = jt-r+k-1Da-ejM(V+t(x_v) IX\v)dt

1

1 ( . = rM(x I X) + L t Da- e' M(v + (tv + O)(x - v) I X\v) VeX(k) v

- Da- ej M(v + (tv - O)(x - v) I X\v)) ,

Ch. 10, § 10.2] Multivariate Divided Differences 157

where tv is the value of t at which the half-line {v + t(x - v): t> I} intersects the side [V]. If s is sufficiently small then by ii) the half-lines {v + t(x - v): t> I} and {v + t( x + sej - v): t > I} intersect the same sides. Therefore

!(Da-eiM(x+sei IX)-Da-eiM(xIX)) =rL:!(-1 -~) s v s t,v tv

x (va- ei M(v + (tv + o)(x - v) I X\v) - va- ei M(v + (tv - o)(x - v) I X\v)) .

Since d(v + tv(x - v), V, 1) = d(v + t,v (x + sei - v), 1, V) = 0,

it follows that

Hence

1 (1 1 ) d(ej , 0, V) -; t,v-tv =-d(v,I,V)·

DaM(x I X) = -r L: d(ej,O, V) v d(v, 1, V)

x (va- ei M(v + (tv + O)(x - v) I X\v)

- Da- ei M(v + (tv - O)(x - v) I X\v)) .

Now by i) we get

(10.2.1)

We now pass to the case r = k, a = 0. Since v is on the left-hand side of [Y], Y = X\v, then

LlYI~M(·IX) = d(V,~,y)" (10.2.2)

It remains to combine relations (10.2.1) and (10.2.2). Notice, since

the relations (10.2.1) and (10.2.2) hold unchanged if v is on the right-hand side of [Y].

COROLLARY 10.11. Let X be in a general position and Y E X(k). Then:

i) LlyM := LlYI~ M(·I X) := LlYI~ M(-I X) = Lly I~: M(·I X), lal = r - k,

ii) Llys := Lly I~ s(·1 X) := Lly I~ s(·1 X) = ~y I~: 5(·1 X), lal = m - 1,

for s E S~ x' where E, G and E', G' are pairs of neighbouring b-regions with a common side contained in [Y], E, E' being on the right-hand side of [Y].


DEFINITION 10.12. Let X = {ZO, ... , Zr} C IRk, volk[X] i' 0, a E Zi, lal = r - k + 1, and let f be sufficiently smooth. Then the k-variate a-divided difference of f at X is (see (9.1.4) and (9.2.3»

(10.2.3)

For IXI = k, i.e., r = k - 1, a = 0, we use the notation [Xl! too:

[X]f:= [X]Of = (k - I)! J f = VOlk~l[X] J fdmk-l. [Xl [Xl

The following theorem is a generalization of the univariate equality (1.3.5), when X is in a general position.

THEOREM 10.13. Let X be in a general position volk[X] i' 0, a = (at. ... ,ak ) E Zi, lal=r-k+1. Then

[X]Q f = L Cv [V]f, (10.2.4) VeX(k)

where k n [d(ei , 0, y)]Qi

r-k+l r! i=l cv =(-I) a!(k-l)! n d( lY)· v, ,

tlex\y

Proof. Suppose ai > ° for some j, 1 ~ j ~ k. The relations (10.2.3) and (9.2.3) imply

Denote by n the set of all b-regions of X. Then we have

Using Corollary 10.11 and Stoke's theorem yields

1= L (Lly-ejM)d(ei,O,Y)[Y]J, YeX(k)

which combined with Theorem 10.10 implies (10.2.4).

Ch. 10, § 10.3) Polyhedral Splines 159

COROLLARY 10.14. Let X be in a general position, voh,[X]:I 0, a = (al, ... ,ak ) E ~i. Then

i) J DO f(x)M{x I X) dx = (_1)'"-k+1 L (~y-ei M)[Y]f, lal = r - k + 1, lI\k YeX(k)

ii) J DO f(x)s{x I X) dx = (_I)m L (~y-ej s)[Y)f, lal = m, Ilk YeX(k)

where s E S~,x.

§ 10.3. Polyhedral Splines

For a given convex polyhedron Q C IR, n = k + s we define a polyhedral spline as follows

(10.3.1 )

provided that the right-hand-side expression has a finite value for every x E IRk. In particular, taking n-simplex u = [yO, ... ,yn] in (10.3.1) we obtain a simplex spline with differs from the corresponding B-spline by a constant multiplier (see (9.2.2»:

(10.3.2)

where Xu = {xO, ... ,xn} consists of projections of vertices of u on IRk :

i = 0, ... ,no

Choosing the barycentric coordinates (see (9.3.8» of x E IRk with respect to {yO, ... ,yn} C IRn :

x = bo(x)yO + ... + bn(x)y"

and projecting onto IRk we have

x = bo(x)xo + ... + bn(x)xn.

Therefore, using (Exercise 9.10.1) and (10.3.2) we obtain from (9.3.4) the following recurrence relation for simplex splines:

Mu = n~k t{·_(i).niMu;, i=O

(10.3.3)

where Ui = [{yO, ... ,yn} \if] are the (n - I)-faces of U with the ni-the inner normal of Ui and (i E Ui, i = 0, ... , n.

Let us consider now a triangulation T for a given polyhedron Q C IRn , such that for any (/, (/' E T the intersection u' n u" is empty or a common face. Applying (10.3.3) to each of the simplexes U E T and summing up yields

MQ = n ~ k L {._(}niMq;,

i

(10.3.4)


where the sum is taken over all (n - I)-faces of the polyhedron Q with the ni-the inner normal of qi and (i E qi, since the two coefficients corresponding to a common face differ only with sign.

In a similar way, we obtain from (9.3.5):

(10.3.5)

Expanding on an idea of de Boor [1976b], one may easily construct a partition of unity consisting of polyhedral splines.

Let D be any measurable subset of]W.$, S = n - k, and assume that the collection of convex polyhedra {Q : Q E .6.} forms a partition of]W.k x D provided that for any bounded n c ]W.k only a finite number of Q's intersect n x D. It is clear from (10.3.1) that the following sum

L aQMQ(x) QE~

is well-defined for any x E ]W.k and any sequence of real numbers aQ . Moreover, the polyhedral splines MQ, Q E .6., form a partition of unity:

L MQ(x) = vol$ {y E IRk x D: Y\llIk = x} = vol$D, QE~

By a standard argument, this implies the denseness of the family of spaces ({MQ: Q E .6.j }) in C(n), where n c IRk is any compact, provided that

hj := max diam( QI:m") --+ 0, QE~;

as j -+ 00.

In fact, we have for any continuous function f and xQ E QllIk

where Woo (J, n, h) :=sup{lf(x)-f(y)l: x,yEn, Ix-yl~h}.

Let us consider now the case when D is the standard s-simplex:

and let T be a collection of n-dimensional simplexes which forms a triangulation of IRk X S$:

UU=]W.kxs·;u~nug=flJ forul,u2ET; \{uET:YEu}\<oo, allyElRn .

<TET

Ch. 10, § 10.3) Polyhedral Splines 161

We denote by Yq := {y~, . .. , y~} the set of vertices of the simplex u E T and assume that

We are interested in the polynomials contained in S(T) - the linear span of B-splines M(·I Xq), U E T, where X q := Yq IlIk . For (E IRk define the mapping

G«z,u) = (z,(1 +(.z)u): IRk X S· -+ IRk X IR'.

The following theorem is the multivariate analogue of Marsden's identity (see Exercise 3.41) and combined with Lemma 9.11 shows that S(T) does contain all polynomials of total degree ~ s.

THEOREM 10.15. If all B-splines M(·I Xq), U E T, are continuous at z E IRk, then

(1 + (·zY = L cq«()M(z I Xq), (10.3.6) qeT

where

with IIG«Yq)1I the (n + 1) x (n + 1) matrix with columns

[G(?)] , and sign (Yq) E {-I, I} chosen so that Cq(O) is positive.

Proof. For a fixed z both sides of (10.3.6) are polynomials in ( and we may therefore assume that ( is small. Small perturbations of the vertices do not change the combinatorial structure of a triangulation. Moreover, G( maps the hyperplanes which form the boundary of IRk x S· onto hyperplanes. Therefore, for fixed z and small (, using the relations

(z,S') = (z,IR') n ( U 0"), ~eqlWt

we obtain that

(z,(I+(.z)S') = Gdz,s') = (z,IR')n ( U [G«(yq)]), ~eqIR"

(10.3.7)

where G«Yq) := {G«y): y E Y q}. Computing the volume of both sides of (10.3.7), it follows from (10.3.1) that

1 ,(I+(.z)' =vol.(z,(I+(.z)S') = L vol.{YE [G«(Yq)]: Y\ ,,::;z} s. II

~eqlWt

= L (vain [G«(Yq)l)M(z I Xq), qeT


which implies (10.3.6).


The basis (10.1.3) for the space S!a,$ with X in a general position was constructed independently by Hakopian [1984] and Dahmen and Micchelli [1983]. The Theorems 10.5, 10.7 and 10.10 are due to Hakopian [1984], [1982b]. The multivariate divided differences (10.2.3) were introduced by Hakopian [1982b] and differ from the one presented by Cavaretta, Micchelli, and Sharma [1980] by the value of the modulus of lY. Theorem 10.13 was proved by Hakopian [1982b]. The recursion relations (10.3.4) and (10.3.5) for polyhedral splines were derived in de Boor and Hollig [1982] with the aid of Stoke's theorem. The identity (10.3.4) was proved by Hollig [1982] and Dahmen and Micchelli [1982]. It is the basis for the construction of dual linear functionals and local approximation schemes. In two variables, the identity is due to Goodman and Lee [1981], who also obtained a more explicit formula for the coefficients c,,().

Chapter 11

BOX SPLINES

§ 11.1. Definition and Basic Properties

Let X be an arbitrary set of knots (not necessarily distinct) containing a basis for ~k :

We will at times think of X, equivalently, as a real matrix of order k x n. Denote

n

X(t) = L:>ixi for t = {t1, ... ,tn} E ~n, i=l

X(V) := {X(t): t E V} for V C ~k.

DEFINITION 11.1. The Box spline B(x I X) is a function defined by the rule

J J(x)B(x I X) dx = J J(X(t)) dt, (11.1.1)

ll\k [0,1)"

where [0, l]n is the unite cub in ~n.

The following properties are immediate corollaries of the definition:

suppB(x IX) =X([O,I]n), J B(x I X) dx = 1, (11.1.2)

ll\k

and for the arbitrary nonsingular matrix A of order k

B(x I AX) = Ide!AI B (A- 1x I X), AX := {Av: v EX}.

In the case n = k, the change of variable y = X(t) in the right-hand-side integral of (11.1.1) provides:

J J(X(t)) dt = IJI· J J(y) dy, (11.1.3)

[0,1]k X([O,1]k)

163

164 Box Splines

where J is the Jacobian. Since (see (11.1.3) with f == 1)

we get from (11.1.1) and (11.1.3)

B(:c I X) = XX([O.lt)(:C) VOlkX([O,l]k)

(IXI = k).

(Ch. 11, § 11.1

(11.1.4)

(11.1.5)

To obtain a geometric interpretation similar to (9.1.10) and (9.1.12) for the Box spline let us consider the parallelepiped Q in IRn :

Q := {tti yi : 0 ~ ti ~ 1, i = 1, ... , n}, .=1

voInQ i= 0,

with the vertices yi satisfying yi Ililk = :ci . Then by a change of variable in the righthand-side integral of (11.1.1):

we get

J f(X(t))dt= J f(u(t)llIIk)dt= [voInQ]-l J f(U1, ... ,uk)du1 ... dun

[0.11" [0.11" Q

= [volnQ]-l J f(U1, ... ,U,J J dU"+1···dundu1 ... duk

JIIk {uEQ: ulllt,,=(u1 •...• U k )}

= [volnQ] -1 J f(:c)voln-d u E Q: ul lll" = :c} d:c. ]\k

Hence, according to (11.1.1)

voln_k{u E Q: ul k =:c} B(:cIX) = I Q 111 , :CEIRk.

VOn (11.1.6)

Note that in view of (11.1.1), the fraction in the above geometric interpretation does not depend on the choice of parallelepiped Q.

The relation (11.1.6) implies that the Box spline is a spline function in the sense of Definition 10.1, i.e., it is a linear combination of B-splines since every parallelepiped Q can be presented as a sum of finite number simplicities with disjoint interiors.

Below, if A is an arbitrary set of not necessarily distinct elements a with 1'(0') being the multiplicity of a, then by A U a, (A\a) we denote the same set with a having multiplicity J.I(a) + 1, (J.I(a) - 1).

Ch. 11. § 11.1)

THEOREM 11.2. If

then

Definition and Basic Properties

n

n

X = LAiXi,

i=l

D~B('I X) = LA;[B(-I X\Xi) - B(.-xi I X\Xi)], i=l

165

(11.1.7)

Proof. It is enough to prove (11.1.7) when x coincides with one ofthe knots xi. Let x = xn. Using the equality

a o/(tx) = D~f(tx) (11.1.9)

we obtain from (11.1.1) that for arbitrary f E CWm,k)

J f(x)D~"B(x I X) dx = - J D~"f(x)B(x I X) dx = - J O~n f(X(t)) dt llk llk [0,1]"

= J f(t1 x1 + ... + tn_1 Xn - 1) dt1 ... dtn- 1

[0,1]"-'

- J f(t 1x1 + ... + tn_1xn- 1 + xn) dt1 ... dtn [0,1],,-1

= J f(x)B(x I X\xn) dx - J f(x + xn)B(x I X\xn) dx

ll" Ilk

This finishes the proof of (11.1.7). The equality (11.1.8) readily follows from (11.1.7) and the relation:

n

D~B(x I X) = (n - k)B(x I X) - LB(x - xi I X\xi). (11.1.10) i=l

In order to do this, we use integration by parts:

J f(x)D~B(x I X) dx llk

k J a k J a = ?= f(X)Xi OX; B(x I X) dx = - ?= OXi (x;f(x))B(x I X) dx ,=1 llk ,=1 llk

(11.1.11)

= -k J f(x)B(x I X) dx - J D~f(x)B(x I X) dx. llk Ilk

166 Box Splines [Ch. 11, § 11.1

On the other hand, by (11.1.1) and (11.1.9) we have

J DIJI(x)B(x I X) dx = J DX(t)/(X(t» dt = t J ti ~/(X(t» dt Ilk [0,1]" 1=1 [0,1]"

= t J ~i (td(X(t»)) dt - t J I(X(t» dt 1=1 [0,1]" 1=1[0,1]"

= t J I( (X\Xi)(T) + xi) dT - n J l(x)B(x I X) dx 1=1 [0,1],,-1 Ilk

= t J I(x + xi)B(x I X\X i ) dx - n J l(x)B(x I X) dx 1=1 11k lIIk

= t J l(x)[B(x - xi I X\Xi) - nB(x I X)] dx. 1=1 Ilk

This combined with (11.1.11) yields (11.1.10). Denote

\1d(·) := 1(,) - 1(' -t) and \1tuY 1(·):= \1.{\1y l(· )}, t E~,

where Y is a finite subset of~. We get from (11.1.7) in particular

i = 1, .. . n.

Hence, we have for an arbitrary Y eX

DyB(·1 X) = \1yB(·1 X\Y) .

Exercise 11.2.1. Let Y C ]Rk, IYI = m. Prove, that

a) for an arbitrary P E 1rm(]Rk)

DyP(x) = \1yP(x),

b) for an arbitrary 1 E Cm(]Rk)

x E ~k,

if Dy I(x) == 0, then \1y l(x) == 0.

(11.1.12)

(11.1.13)

(11.1.14)

(11.1.15)

It follows from Theorem 11.2 and (11.1.5) that the Box spline B(x I X) is a piecewise polynomial function with support X([O, l]n). It coincides with a polynomial of total degree (n - k) in each region which is bounded but not intersected by hyperplanes of the form

x + (Y), where Y C X, dim(Y) = IYI = k - 1,

x= LCiXi, CiE{0,1}. lJ'tI!y

(11.1.16)

Ch. 11, § 11.1] Definition and Basic Properties 167

For a further discussion of the properties of Box splines we need the following notation:

Y = Y(X) := {Y eX: (X\Y) =F J.k},

Z = Z(X) := {z C X: (X\Z) = J.k },

V=V(X):= {/: Dy!=OforallYEY},

d = d(X) := min {WI: Y E Y}-1.

It is clear that X E Y(X), 0 E Z(X), ° ~ d ~ n - Ie and

By Theorem 11.2 and (11.1.5)

B{z I X) E Cd-1{J.k)\Cd{~k)

(11.1.17)

(11.1.18)

and B(z I X) is a polynomial from V(X) in each region, which is not intersected by hyperplanes of the form (11.1.16).

CO -linear c1 - quadratic Fig. 11.1.

In Fig. 11.1 we illustrate the supports of the Box splines B(z I zl, z2, z3) and B(z I zl,z2,z3z4), Z E ~2, with knots zl = (0,1), z2 = (1,0), z3 = (1,1), z4 = (-1,1). The first Box spline is continuous in ~2 and linear on each triangle, the second one is in C1(~2) and quadratic.

168 Box Splines [Ch. 11, § 11.2

§ 11.2. Integer Translates of a Box Spline

In this section, we consider the space S(X) spanned by the integer translates of Box spline B(x I X) :

S(X):= ({B(.-O' I X): 0' E ~k}}. (11.2.1)

This space plays an important role in multivariate spline approximation theory. We start with the theorem which shows that 7ro(~k) e S(X). Let us denote for X e ~k

JE(X) = {vex: IVI=dim(V}=k}. (11.2.2)

THEOREM 11.3. a) Let V = {xl, ... , xk} E lJB(X). Then

( k .) 1 L B x - ?= O'iX * I X == Idet VI'

aezk *=1

b) If X e ~k\{O} then

L B(x - 0' I X) == 1. (11.2.3) aezk

Proof. By the definition of a Box spline and (11.1.4) we have

J !(x) [ L B(X- to'ixi I X)] dx = L J !(x + to'ixi)B(X I X) dx 1I\k aez" *=1 aez" 1I\k *=1

L J!( (t1 + 0'1)x1 + ... + (tk + O'k)xk + tk+1Xk+1 + ... + tnxn) dt1 ... dtn aezk [0,1]"

= J (J !(t1x1 + ... + tnxn) dt1 ... dtk) dtk+1 ... dtn

[0,1]" 11\"

= J !(t1x1 + ... + tkxk)dt1 ... dtk = Ide! VI J !(x) dx,

~ ~

which implies a). To prove b) we need the following

LEMMA 11.4. Let Y = {y1, . .. , yk} e ~k, (Y) = ~k and

b(Y) = {t,tiyi : (tb ... ' tk) E [0, l)k } n ~k, a(Y) = {t,O'iyi: 0' E ~k } .

Then

i) U [a(Y) +,8] = U [b(Y) + 0'] = ~k, ,Be b(Y) aeo(Y)

whereA+,8:={O'+,8: O'EA}, ii) for an arbitrary,8, ,8' E b(Y), 0', a' E a(Y) the equality a +,8 = a' +,8' holds

if and only jf,8 = ,8', 0' = 0",

iii) Ib(Y)1 = volkY([O, l)k) = Idet YI.

Ch. 11, § 11.2) Integer Translates of a Box Spline 169

Proof. i). Since (Y) = IRk we can present an arbitrary vector 1 E IRk in the form of

k

1 = L aiyi, ai E IR. i=l

Then denoting by ai = [ail - the greatest integer not extending ai and ti = ai - ai, we will have

k k

1 = Laiyi + Ltiyi =: 11 + 12, i=l i=l

11 E a(Y), ti E [0,1), i = 0, ... ,k.

From this follows i), since 1 E Zk implies 12 = 1 - 11 E 7i}. ii). Assume that

k k k k Laiyi + LtiYi = La~yi + Lt~yi, i=l i=l i=l i=l

(11.2.4)

(11.2.5)

where ai, a~ E Z, ti, t~ E [0,1), i = 0, ... , k. Then, because of the independence of Y, we get ai + ti = a~ + t~ which is possible only if

i = 1, ... ,k. (11.2.6)

iii). Let us denote

k

Q = {trtiyi: (tl, ... ,tk) E [0, l)k},

VN={a: Q+aC[O,N)"}, N=1, ....

Since (11.2.5) implies (11.2.6), the sets Q + a, a E a(Y), are disjoint and according to (11.2.4)

[m,N - m)k C U (Q +a) C [O,N)k, aeVN

where m = [diamQ] + 1. From this, using the equalities

volk ([m, n)k) = Hm, n)k n tlk 1= (m - n)k,

volk ( U (Q + a)) = IVNlvolkQ = IVNlvolkY([O, 1)k) , aeVN

I U (Q + a) n 7lk l = IVNIIQ n 7lk l = IVNllb(Y)I, aeVN

we get

170 Box Splines [Ch. 11, § 11.2

Therefore

which implies volkQ = Ib(Y)I. Let us prove now the equality (11.2.3) for Xc Zk\{O}. Choose Y E J1B(X). Then,

according to a), we have

1 L: B(x-O'IX)= L: B(x-.B-O'IX)=ldetYI' aEa(Y)+.8 aEa(Y)

for arbitrary .B E b(Y), x E Rk. Hence by Lemma 11.4

1 L: B(x - 0' I X) = L: L: B(x - 0' I X) = Ib(Y)lldet YI = 1. aEZk .8Eh(Y) aEa(Y)+.8

THEOREM 11.5. If Xc Ilk\{O}, (X) = Rk, then

Proof. Let JHI(X) be the set of all hyperplanes of dimension (k -1) which are spanned by some knots from X. Denote

qX):= U U(h+O'). (11.2.7) hEM(X) aEZk

It follows from (11.2.3) and Theorem 11.2 that if the polynomial P belongs to S(X), then

DyP(x) = 0, Y E Y(X) (11.2.8)

for x E Rk\qX), i.e., almost everywhere. Since P is a polynomial then (11.2.8) holds everywhere, which means that 7r(Rk) n S(X) C V(X). The inverse inclusion follows from the following

THEOREM 11.6. For the arbitrary set of knots X C Rk, (X) = Rk,

a) V(X) C 7rn _k(Rk ), n = lXI,

b) if X c Ilk, then the mapping

TI:= L: I(O')B(. -0' I X) aEZk

is one to one and onto the space D(X).

Ch. 11, § 11.2) Integer Translates of a Box Spline 171

Proof. a). Let us prove that, for arbitrary IE V(X), e E IRk

if r> n - k, (11.2.9)

from which it follows that I is a polynomial (see Lemma 9.11). Note that for every Z E Z(X) the vector e can be presented in the form

and therefore

e = E a(y)y lIEX\Z

(Der Dz = (De)II-1 E a(y)DzulI , lIEX\Z

v = 1,2, ....

Applying this equality for v = r, r - 1, ... , and Z = 0, {y}, ... , respectively, we conclude

(Der = E a(Y) (Dyr-1Y1Dy + E a(Z)Dz. (11.2.10) YEY(X) ZEZ(X) IYI~r IZI=r

This implies (11.2.9) since Dy 1= 0 for Y E Y(X) and the second sum is taken over the empty set if r > n - k.

b). Note first that the function TI is well defined for an arbitrary I, since the Box spline has compact support (see (11.1.2». By (11.1.13) we have that for every f3 E X C Ilk

aEZ"

= E [/(a)-/(a-f3)]B(.-a I X \,8.) aEZ"

(11.2.11)

= E (\1 p/)(a)B(. -a I X\f3). aEZ"

Therefore, if IE V(X), then according to (11.1.15)

Dy(Tf)(x) = E (\1y l)(a)B(x - a I X\f3) = 0, aEZ"

which means that TI E V(X). It remains to prove that the mapping T I is one to one, i.e. T I = 0 implies I = 0

(since T is linear). In order to do this it is enough to show that

Q:= P - TP E 1I'r_l(IRk) for every P E 1I'r(IRk) n'D(X), r = 1,2, ... , (11.2.12)

172 Box Splines [Ch. 11, § 11.2

(see (11.2.3) for r = 0). On view of Q E V(X) we have from the relations (11.2.3), (11.2.10), and (11.2.11):

(D(rQ(x)= L a(Z)[(DzP)(X)- L (V'zp)(a)B(x-aIX\Z)] ZEZ(X) ewEZ" IZI=r

L a(Z) L [(DzP)(x) - (V' zP)(a)] B(x - a I X\Z) = 0, ZEZ(X) ewEZ" IZI=r

since deg P ~ rand IZI = r (see (11.1.14». Hence Q E 1I'"r_l (IRk). Theorem 11.6 and (11.1.18) imply

COROLLARY 11.7. If Xc 7l}, then

(11.2.13)

§ 11.3. A System of Partial Differential Equations Connected with 1>(X)

Let X be an arbitrary finite subset of IRk \ {OJ. Denote

yO(X) := {X\h: h E JHI(X)} C Y(X).

Since each Y E Y(X) contains some subset Y' which is in yO(X), then the space V(X) can be considered as a set of solutions of the following system of homogeneous differential equations:

Dyf = 0, (11.3.1)

This leads to the investigation of the following system of partial differential equations

Dyf = IPy, (11.3.2)

where IPy are sufficiently smooth real functions. The following relations which obviously are necessary for the solvability of this system we call conditions of accordance:

Y, y' E yO(X). (11.3.3)

The starting conditions for the system (11.3.2) we give first by means of the sufficiently smooth function t/J:

Dzf(O) = Dzt/J(O), Z E Z(X) , (11.3.4)

(we mean Dzf = f if Z = 0).

THEOREM 11.8. If(X) = IRk, then the system (11.3.2) with conditions of accordance (11.3.4) has a unique solution satisfying the starting conditions (11.3.3).

Ch. 11, § 11.3] System of Partial Differential Equations 173

Proof. Let v E X and Y = X\v\h E yO(X\v), where h E IHI(X\v). Then hE IHI(X) and we have either Y = X\h E yO(X) (if v E h), or Y u v = X\h E yO(X) (if v fJ. h). Denote

_ {<PYUV' <Py,v - D If'}

vry,

if YUv E yO(X),

if Y E yO(X). (11.3.5)

The following theorem plays a crucial role in the proof of Theorem 11.8 and gives a recurrence method for solving the system (11.3.2).

THEOREM 11.9. Let V E J1B(X). Then the system (11.3.2) with starting conditions (11.3.4) is equivalent to the system

v E V, (11.3.6)

with the single starting condition

1(0) = 1/1(0), (11.3.7)

where Fv, v E V, is the solution of the system

DyFv = <pYv' , (11.3.8)

with starting conditions

Z E Z(X\v).

Proof. If 1 is a solution of the system (11.3.2) with (11.3.4), then denoting Dvl = Fv, v E V, we easily obtain (11.3.6)-(11.3.9).

Suppose now 1 is a solution of (11.3.6) with (11.3.7) and Fv, v E V, satisfying (11.3.8), (11.3.9). Then for any Y E yo(X) there exists v E Y n V for which we have Y\v E yO(X\v) and hence according to (11.3.6) and (11.3.8)

(11.3.9)

i.e., 1 is a solution of the system (11.3.2). It remains to prove that f satisfies the starting conditions (11.3.4). For Z = 0 it follows from (11.3.7). If for Z E Z(X) there exists v E Z n V, then Z\v E Z(X\v) since (X\v)\(Z\v)) = (X\v) = ~k. Hence (11.3.9) implies

Assume now that Z n V = 0 and let Z = w U Zl. Then in the representation

W = LAvV vEV

we will have Av = 0 if Zl fJ. Z(X\v). Therefore

Dzf(O) = L AVDZ1FV(0) = L AVDZ1UV¢(0) = Dz1UW 1/I(0) = Dz1/l(O). vEV: ZlEZ(X\v) vEV

174 Box Splines [Ch. 11, § 11.3

Proof of Theorem 11.8. We use induction with respect to n = IXI. The case n = k is well known. Let us suppose that the theorem is true for the systems with IXI < n and prove it for IXI = n. By Theorem 11.9 it is enough to show that for an arbitrary V E IlB(X) the systems (11.3.8) and (11.3.6) have unique solutions with starting conditions (11.3.9) and (11.3.7), respectively. For this, according to the induction hypothesis, we need only to prove that the conditions of accordance of those systems are satisfied, i.e.

Y, Y' E yO(X\v), (X\v) = ~k, (11.3.10)

and DwFv = DvFw, v, wE V. (11.3.11)

(Note that if dim(X\v) = k - 1 for some v E V, then {v} E yO(X), yO(X\v) = {0}, Z(X\v) = 0 and (11.3.8) contains only one (solved) equation Fv = <P{v}). Let us prove (11.3.10). By (11.3.5) and (11.3.4) we have:

1) if Y, Y' E yO(X), then

DYI\Y<Py,v = DvDy,\y<Py = DvDY\YI<Pyl = DY\Y1<Pyl,v'

2) if Y U v, Y' U v E yO(X), then

Dyl\y<py,v = D(y/uv)\(Yuv)<Pyuv = D(yuv)\(y/uv)<Py/uv = Dy\yl<pyl,v'

3) if Y U v, Y' E yO(X), then (v rt Y U y')

Dyl\y<py,v = Dyl\yuv<pyuv = D(yuv)\yl<Pyl = Dy\yIDv<pyl = Dy\yl<pyl,v·

Equation (11.3.11) follows from (11.3.6), (11.3.7), and (11.3.4). Indeed, 1) ifdim(X\v) = dim(X\w) = k-1, then {v}, {w} E yO(X) and

DwFv = Dw<P{v} = Dv<P{w} = DvFw,

2) if X\v = ~k, then by Theorem 11.9 both the functions DwFv and DvFw are solutions of the system

Dy F = <pY,v,w'

with starting conditions

y E yO(X\v\w),

DzF(O) = DzuvuwtP(O), Z E Z(X\v\w),

and hence according to the induction hypotheses they coincide. Expressing the solution of the system (11.3.6) with (11.3.7) by means of integration

over the path we obtain the following recurrence relation for the solution 1 of (11.3.2): 'I:

I(x) = tP(O) + J L Fv av, ° vEV

(11.3.12)

where {v: v E V} is the dual basis of V, i.e., V·W = 0 if v =P wand V·W = 1 if v = w (v, wE V).

The next result immediately follows from Theorem 11.9 and (11.3.12) by induction on IXI. It shows, in particular, that the solution of the system (11.3.2) is a polynomial if the right-hand-side functions of (11.3.2) are such.

Ch. 11. § 11.4] Further Properties of the Spaces 175

COROLLARY 11.10. The class of functions satisfying the relations

coincides with 1I"n-k+m(]RI:) (n = lXI, m = 0,1, ... ). From this, taking m = 0, we clearly obtain'D(X) C 1I"n_l:(lRl:), since V(X) is the

class of solutions (11.3.1) (see the statement a) of the Theorem 11.6). Let us consider now the polynomial space

with

P(X) := ({ Pz(x): Z E Z(X)})

Pz(x) = II z· x if Z f:: 0 and Pz(x) == 1 for Z = 0. zez

Theorem 11.8 implies

COROLLARY 11.11. If Xc ]RI:\{O}, (X) =]RI:, then 1) dim 'D(X) = dim P(X),

(11.3.13)

2) for an arbitrary function 1/J E Cn-l:(lRl:) there exists a unique polynomial q E "D(X) such that

P(D)q(O) = P(D)",(O) for all P E P(X).

The spaces P(X) and 'D(X) are dual to each other in the following sense. Each q E "D(X) gives rise to a linear functional q* on P(X):

q*(P) = P(D)q(O), P E P(X).

The second statement of Corollary 11.11 shows that the mapping

"D(X) --+ (P(X»)*: q -+ q*

is one to one while the first statement implies that it is onto, hence we obtain the following:

COROLLARY 11.12. The polynomial spaces P(X) and'D(X) are dual to each other.

§ 11.4. Further Properties of the Spaces 1>(X) and 'P(X)

Our next purpose is to replace the function", in (11.3.4) by arbitrary values. To this end, we need to choose linearly independent starting conditions from (11.3.4), i.e, to construct a basis in the space P(X). We have the following decomposition of P(X) by the direct sum of spaces of homogeneous polynomials:

n-I: P(X) = Ep,(X), n=IXI,

,=0

176 Box Splines [Ch. 11, § 11.4

where

'P,(X) := ( {PZ(Z) = II z· z: Z E Z(X), IZI = s} ). . :rez

Let us consider now the family of (k - I)-dimensional hyperplanes

£x={I,,: VEX}, (11.4.1)

where I" is given by the equation V • z = O. In what follows, we assume that an arbitrary (k - 1 )-dimensional hyperplane I is

determined by the equation

[ij(z) := n,' z = 0,

with n, being the unit normal of I. Of course, we have [ij = e(·, I) with e(z, I) the signed distance of z from I.

Taking into account that for any V C :IRk the condition (V) = :IRk is equivalent to

n I" = {OJ, "ev

we can present the polynomial space 'P.(X) in the form

with PL := II[ij for L '" 0 and Pm = 1.

'eL

(11.4.2)

Let now £ be an arbitrary family of (k - I)-dimensional hyperplanes (not necessarily distinct) satisfying

m := 1£1 > k, n I = 0,

'e.c

Denote

for Lee, ILI=k.

for s = 0, ... , m - k - 1 and 1',(£) = 1',(£, k) := {OJ for s ~ m - k.

(11.4.3)

(11.4.4)

It is not difficult to show that £x can be completed by the hyperplane h such that £x U h satisfies (11.4.3). Since

n 1= {OJ 'e.cx

Ch. 11, § 11.4] Further Properties of the Spaces 177

we have from (11.4.2) and (11.4.4) that

s = 0, ... , IXI - k - 1. (11.4.5)

Hence, our problem is reduced to a construction of a basis in the space P.(.c) (s = 0, ... , m - k - 1). This is just the space of polynomials isomorphic to the following space of spline functions (see Definition 10.1):

where Xc = {VI },ec,

provided that the hyperplane I is defined by the equation v,. x + 1 = 0 (here we assume without loss of generality that the hyperplanes I E .c do not pass through the origin). The isomorphism is given by the following formula (see (9.4.5)):

II(v,.x+l) j(1 + x.y),-mM(y I {v,},eL)dy= II (v"x+l), lec:mk leC\L

(11.4.6)

where

L C.c, ILl = m - s, n I i= 0. (11.4.7) leL

(Note that (11.4.7) is equivalent to VOlk[{VI},eL] i= 0 with

[A] := { L: AflV: L: Afl = 1, Afl ~ 0 } fleA fleA

the convex span of the set A (see (9.4.5)). Let us now introduce some notation. Let p(x) be the multiplicity of the point

x E IRk in the family .c:

p(x) = p(x,.c) := 1{1 E.c: x E I}I, (11.4.8)

and p(/) - the multiplicity of the hyperplane IE .c in .c:

p(1) = p(1,.c):= I{h E.c: h = I}I·

The set of all distinct elements of an arbitrary set A we denote by Ad, then

P II means the trace of the polynomial P on the hyperplane I, and for the class of polynomials n we denote

ni, := {pl , : PEn}, p·n:= {PQ: Q En}.

178 Box Splines [Ch. 11, § 11.4

Let us note, that if (11.4.4) holds then by (11.4.3) the set £\L contains at least k + 1 distinct hyperplanes, i.e., 1'(1, £\L) :::; I£\LI- k = m - s - k for any 1 E £ and hence 1'(1, L) = 1'(1, £) - 1'(1, £\L) ~ 1'(1) - m + s + k. Therefore if

11(1) > 0 with 11(1):= 11(1, s, £) = 1'(1) - m + s + k,

then every polynomial PL in (11.4.4) has the factor [W(l)and hence

11(1) > O. (11.4.9)

(From now on by

we denote the set which consists of elements 1 of the set L with multiplicity 11(1)). We start the investigation of the space P 6 (£, k) for the case k = 1. Let T be an

arbitrary finite subset of ~, which contains at least two distinct points. Then the space P6 (T) can be defined as follows (see (11.4.4)):

( { Il (. -t): LeT, ILl = s, T\L contains at least} ) P6(T) = P 6(T, 1) = tEL.. •

two dIstmct pomts

with s = 0, ... , ITI- 2. The following lemma gives the complete characterization of the space P6 (T, 1).

LEMMA 11.13. For an arbitrary T C ~ and s, 0:::; s :::; ITI- 2,

P6(T) = ( II (. _ty(t)+) 1rN(~) , tETd

(11.4.10)

where

lI(t) = lI(t, s, T) = /J(t) - ITI + s + 1, N = N(s, T) := s - L lI(t)+. (11.4.11) tETd

Proof. According to (11.4.9) we have

{ t } Tl=T\ , lI(t)+ tETd

and lI(t, s, TI) :::; 0, t E T1. Therefore, we need only to prove that if lI(t) :::; 0 for all t E T then

(11.4.12)


To prove the inverse inclusion, we use induction on ITI. If ITI = 8+2, then JJ(t) = 1 for all t E T and for fixed rET the polynomials

II (.-t), Z E T\r, tE(T\T)\1:

belong to P,(T). Since they are the fundamental polynomials for the Lagrange interpolation with the node set T\r (see (1.1.2» we obtain (11.4.12).

Suppose now the lemma is true for ITI = m - 1 and prove it for ITI = m. Let T' = {t E Td: v(t) = O}. For every rET' we have

v(t, T\r) = {~'O ~ ,

if t E T'\r, if t E (T\T') U r.

Then by the induction hypothesis

( II (. -t)}r'-IT'I+1 (IR) = p,(T\r) C P,(T). tET'\T

Again using the Lagrange interpolation we conclude that P,(T) contains any polynomial P of the form P = PlP2 with Pl E 1I"IT'I-l(IR), P2 E 1I".-IT'I+l(IR), hence 1I".(lR) C P.(T).

COROLLARY 11.14. For an arbitrary T C IR the following equality

holds.

Proof. Since

we get from (11.4.10)

ITI-2 L dimp,(T) = L JJ(t)JJ(r)

t,TET", t<T

L JJ(r) = ITI, TET"

ITI-l ITI-2 ( ) (ITI- l)ITI L dimP.(T) = L 8+ 1- L v(t,8,T)+ = 2 .=0 .=0 tET"

_ L 1I:2 (p(r) + 8 -ITI + 1)+ = 1~12 _ I~I _ L p(r)(JJ;r) - 1)

TET" .=0 tET"

= ~ [( L p(r») 2 - L JJ(r)2] = L JJ(t)JJ(r). tET" tET" t,TET",t<T

Let us return to the polynomial space P.(.e) with .e satisfying (11.4.3) in the multivariate case. We denote by .ella the trace of.e on the hyperplane h E.e:

(11.4.13)

180 Box Splines [Ch. 11, § 11.4

It follows from (11.4.3) that .elh is a set of (k - 2)-dimensional hyperplanes in the (k - I)-dimensional space h with properties similar to (11.4.3):

n 6 = n I = 0, n 6 :f 0 for H C .e I h' IH I = k - 1. 6E.Cj" lEe 6EH

Hence .elh gives rise to the polynomial space 'P$(.elh) on h. Moreover, we have

s = 0,1, ... , (11.4.14)

which follows from the equality of the polynomials:

[fj Ih = [I n h] for I E.e, l:f h, (11.4.15)

and the equivalence of the conditions

n 1= 0 and n (I n h) = 0 IEL IEL. lth

for h E L C .e (here we mean that the hyperplane In h is given in h by the equality [I n h](u) = 0, [I n h] E 7rl(h). The equality (11.4.15) holds, of course, with some constant).

The equality (11.4.14) enables us to use Lemma 11.13 in order to characterize the space 'P(X) in the case k = 2.

COROLLARY 11.15. Let X C 1R2 , (X) = 1R2. Then

i) 'P$(X) = ( II (x.v)"(v)+ )7rN (1R2),

vEX 4

where v(v) = Jl(v) - IXI + s + 1,

N=s- L v(v), s=0,···,IXI-2,

ii) dim V(X) = dim P(X) = IIlB(X)I.

Proof. Let .ex be defined as in (11.4.1). Then by (11.4.5) we have

p$(X) = p$(.ex U h), s = 0, ... , IXI- 2, (11.4.16)

where h is an arbitrary line which does not pass through the origin and intersects all the lines Iv, v EX. Hence (11.4.9) implies that

'P$(X) C ( II (x .v),,(v)+ )7rN (1R2). (11.4.17) vEX 4

Assume now P is a polynomial from the right-hand-side space in (11.4.17) and T = (.ex U h)lh = {Iv n h}vEX. Then v(v) = v(t,s,T) for t = Iv n h, v E X, and Lemma 11.13 implies that Plh E P$(T). Hence, according to (11.4.14) there exists a polynomial Q E P $(.e) such that Qlh = Plh. Since P and Q are both homogeneous polynomials of degree s, this implies that P = Q, i.e., P E P$(X), which combined with (11.4.17) finishes the proof of i).

The equalities ii) follow from Corollaries 11.11, 11.14 and i), since Jl(lv n h, T) = Jl(lv,.ex ).

Below we will often use the following simple lemma:


LEMMA 11.16. Let L be (k - I)-dimensional hyperplane in IRI: with the equation L(x) = 0 and P E 1rn (IRI:). H

(D>.)m P(x) = 0 for x E L, m = 1, ... , s - 1,

where D>. is the normal derivative with respect to L, then

Proof. Since this decomposition is independent of thp. choice of coordinate system we can assume, without loss of generality, that L coincides with the hyperplane Xl = 0 and hence

for Xl = 0, m = 0, ... , s - 1.

Let us present P(x) in the form

6-1

P(x) = p(Xl, ... 'XI:) = L: xl" Pm (X2, ... 'XI:) + x~p.(Xl, ... , xl:)' m=O

where Pm E 1rn _ m (IRI:-l), m = 0, ... , s. Then Pm == 0 for m = 0, ... , s - 1 and we get the desired decomposition.

Let us note now that if P.(£} i~ defined in (11.4.4) with (11.4.3) and Jl(x) = Jl(x,£) ~ m - s for some X E IR then every polynomial PL in (11.4.4) contains at least Jl( x) - (m - s - 1) factors vanishing at x. Therefore for any P E 1'. (£) we have

lal~Jl(x)-m+s, for XEIRI:, Jl(x)~m-s. (11.4.18)

It appears that this condition characterizes the space 1'.(£).

THEOREM 11.17. For an arbitrary family of hyperplanes £ with (11.4.3) and s = 0, ... ,m-k-1 the polynomial P E 1r.(IRI:) is in 1'.(£) ifand only if(1l.4.18) holds.

Proof. Because of the relation (11.4.9) we can assume, without loss of generality, that

Jl(/) ~ m - s - k for every 1 E £. (11.4.19)

The proof of the theorem is based on the following

LEMMA 11.18. a) H £ satisfies (11.4.19) then the polynomial P E 1r.(IRI:) is in 1'.(£) if and only if for every 1 E £

j = O, ... ,min{I'{I) -l,s}, (11.4.20)

where n(/) is the normal of I.

b) H P = [h]Q for some h E £, then P E 1'.(£) ifand only ifQ E 'P._l(£\h).

182 Box Splines [Ch. 11, § 11.4

Proof. We will use induction on k. In ]Rl, the statements of the lemma follow from the Lemma 11.13. Suppose that they hold in ]Rk-l and prove in ]Rk.

a). For every polynomial Pz in (11.3.13) and y E ]Rk we have

DyPz = L: CzPZ\z E P(X) , zEZ

since an arbitrary subset of Z E Z(X) is in Z(X). Therefore

for P E P,(£)

and using (11.4.14) we obtain (11.4.20).

(11.4.21)

Assume now the polynomial P E 1I",(]Rk) satisfies (11.4.20). We will prove by induction on ILl, that for every L C £ there exist polynomials pL E P,(£), qL E 1I"('_ILI)+(]Rk) such that

P _ pL = qL II[/] . (11.4.22) lEL

Then taking L = £ in (11.4.22) we will have qC = 0 since s < 1£1 and hence P = pC E P,(£).

For L = 0, we put pL = 0, qL = P, then (11.4.22) holds with

II[/] = 1. lEL

Suppose the polynomial pL satisfies (11.4.22), h E £\L and let us construct the polynomial pLUh (remember that the multiplicity of h in £ can be greater than one and in our notation h can be in both of the sets Land £\L). Since both of the polynomials P and pL satisfy (11.4.20), then

c (qL II [~) I = D~~~),L) (p - pL) Ih E P'-/J(h,L) (£Ih) . lEL, l~h h

Hence, using the induction hypothesis for £Ih (dim h = k - 1), we obtain from the statement b) and (11.4.15) that

qL Ih E P,-ILI (£\L) Ih) . Together with (11.4.14) this implies that there exists a polynomial G E P .-ILI(£\L) such that Glh = qLlh. Now, using Lemma 11.16, we can present G - qL in the form G - qL = [h]qLUh, qLUh E 1I"(,-ILuhl)+ (]Rk) and denoting

we obtain (11.4.22) for L U h:

pLuh = pL + G II[~ lEL

p - pLuh = (G - qL) II[~ = qLUh II [I]. lEL lELuh

Ch. 11, § 11.4) Further Properties of the Spaces 183

Let us prove now the statement b) of the Lemma. Note first that according to the induction hypothesis and the proved part a) we are allowed now to use the statements a) for ~I: and b) for ~1:-1. Because of a) it is sufficient to show that the conditions

j=O, ... ,min{Jl(/,.e)-l,s}, IE.e, (11.4.23)

and (Dn(I»)i QI, E P.-1-i (.e\hl ,),

j = 0, ... ,min {Jl(/,.e\h) -1, 8}, IE .e\h, (11.4.24)

are equivalent provided that P = [h]Q, h E .e. For this, we will use the following equalities:

(D )i pi = { C(Dn(I)~i-1QI, + [h](Dn(I»)iQII' if I f h, (11.4.25)

n(l) I (D ))-1 QI if 1= h, n(l) I'

and (1.e) {Jl(/, .e\h), if I f h, (11.4.26) Jl, = Jl(1,.e\h) + 1, if 1= h.

(Here D-1 f == 0). Suppose (11.4.24) holds. Then (11.4.25) implies

(Dn(h»)i Ih E P'- l -U-l}(.e\h)lh) C P'-i (.el h) .

If If h then according to the statement b) (induction hypothesis on I, diml = k -1) and (11.4.14) we have ([h](Dn(I»)iQ)11 E P'-i(.eh) and (11.4.23) follows from (11.4.24) and (11.4.25).

Suppose now (11.4.23) holds. The equality (11.4.24) with I = h directly follows from (11.4.25) and (11.4.26). We prove (11.4.24) with If h by induction on j. If j = ° then using b) on I, we get QII = PI, E P.(.el,). Under the assumption that (11.4.23) with j - 1 holds, (11.4.23) and (11.4.25) imply

[h](Dn(I»)i QI, = (Dn(I»)i pl , - C(Dn(I»)iQI, E p.-i(.el ,),

hence the statement b) on I implies

(Dn(I»)i QI, E P.-1-i (.e\hl ,) .

Proof of the Theorem. To prove the sufficiency of the conditions (11.4.18) for the polynomial P E 1I-.(~1:) to be in P.(L) we use induction on k. For k = 1, it follows from Lemma 11.13. Suppose the sufficiency is proved in ~1:-1 and prove it in ~I:. By Lemma 11.18 we need only to prove that for any hE.e and j = 0, ... , min{Jl(h)-1,s}

(11.4.27)

Since Jl'(x) := Jl(X,.elh) = Jl(x)-Jl(h) for x E hand l.elhl = m-Jl(h) then according to (11.4.18) we have

DQQ(x) = 0, 10:1 ~ P'(x) -I.elhl + (8 - j) for x E h, P'(x) ~ l.elhl- (8 - j),

and the induction hypothesis implies (11.4.27).

184 Box. Splines

COROLLARY 11.19. Let X C ]Rk\{O}, (X) =]Rk. Then

i) if I'(z) < m - s for any z E]Rk, then

ii) if d = d(X) is defined as in (11.1.17), then

Proof. Choosing the hyperplane h as in (11.4.5) we obtain

IXI-k P(X) = L P. (Cx U h) .

• =0

[Ch. 11, § 11.4

(11.4.28)

Here P.(CxUh) consists of homogeneous polynomials of degree s, hence the number d satisfying (11.4.28) can be determined as follows:

d=max{jE[O,IXI-k]: P.(CxUh) =1r~(]Rk) for s=O, ... ,j}.

On the other hand, by Theorem 11.17, the necessary and sufficient condition for the polynomial P E 1r~(]Rk) to be in P.(Cx U h) is

with III: the subspace of]Rk given by t· z = O. Therefore, the equality P.(Cx U h) = 1r~(]Rk) holds iff s < IX\/I for every (k - I)-dimensional subspace I of]Rk, i.e.,

d = min {IX\/I: dim 1= k -I} - 1 = min {IYI: Y E yO(X)} -1 = d(X).

The next theorem gives a recurrence method to construct a basis for the space P.(C).

THEOREM 11.20. Let hE C, 0 < s < m- k, I'(h) ~ m-s-k. If the systems {Pi} and {qi} are bases for the spaces P,_l(C\h) and P,(Clh)' respectively, then the system

(11.4.29)

is a basis for P.(C), where Qi are chosen by (11.4.14) such that

(11.4.30)

Ch. 11, § 11.4) Further Properties of the Spaces 185

Proof. Let P E P,(.e). Then by (11.4.14) there exist numbers ai such that Plh = Laiqi and (11.4.29) implies that P(x) - LaiQi(X) = 0 for x E h. Hence, according to Lemma 11.16

(11.4.31)

Now using the statement b) of Lemma 11.18 we get that Q E P,_l(.e\h) and can find constants bi such that Q(x) = LbiPi which, combined with (11.4.31), proves that P is in the span of the system (11.4.29).

To show the linear independence of (11.4.29) suppose that

(11.4.32)

Putting x E h in this equality we get Li aiqi(x) = 0 and ai = 0 for all i, since qi are independent on h. Then (11.4.32) implies

hence LbiPi = 0 and b, = 0 for all i.

COROLLARY 11.21. For an arbitrary h E .e and s, 0 < s ~ m,

dim P,(.e) = dim P._1(.e\h) + dim P, (.el h ) •

Indeed, for I'(h) ~ m - s - k the corollary follows from Theorem 11.20. If I'(h) > m - s - k then by (11.4.9)

P.(.e) = [h]P._1(.e\h), p.(.el h ) = {O}.

COROLLARY 11.22. For an arbitrary .e with (11.4.3)

m-k-l L: dimP.(.e) = I lIB (.e) I , (11.4.33) .=0

where

1IB(.e) := { L C.e: ILl = k + 1, n 1= 0} . IEL

Proof. In the case k = 1 the equality (11.4.33) follows from Corollary 11.14. Suppose that it holds in ~k-l and prove in IRk. We will use induction on m = l.el. For m = k + 1 we have Po(.e) = ?ro(~k), 1IB(.e) = {.e}. Assuming that (11.4.33) is true for l.el = m - 1 we obtain from Corollary 11.21 that

m-k-l m-k-2 m-k-l L: dimP,(.e) = L: dimp,(.e\h) + L: dimp,(.elh )

= 11IB(.e\h)1 + 11IB(.elh ) 1= 11IB(.e)1·

COROLLARY 11.23. For an arbitrary Xc IRk\{O}, (X) = IRk,

dim V(X) = dim P(X) = 11IB(X)I. (11.4.34)

186 Box Splines [Ch. 11. § 11.4

Proof. Using (11.4.5) we get from Corollaries 11.11 and 11.21 that

IXI-k IXI-k dim V(X) = dim 1'(X) = I: dim1'.(X) = I: dim1'.(£x U h) = 11IB(£x U h) I

= I{ L C ex: ILl = k, n 1 = {O}}I = {V C X: IVI = k, (V) = m.t} = 11IB(X)I· leL

Now when we know the dimension of the space 1'(X), we can easily construct a simpler basis for it.

Let Av , v E X, be arbitrary constants. Denote by T y , for V E lIB(X) the unique common point of the hyperplanes V· x- Av = 0, v E V. It follows from the definition of 1'(X) that the polynomials

V E lIB(X) , (11.4.35)

are in 1'(X), if the constants Av are chosen such that the following conditions hold:

for V I V', V, V' E lIB(X) . (11.4.36)

Moreover, they are independent, since

and qy(Ty,)=O for VIV', V,V'ElIB(X). (11.4.37)

Hence, using (11.4.34) we obtain the following

COROLLARY 11.24. The system (11.4.35) with (11.4.36) forms a basis for the space 1'(X). Moreover,

for every P E 1'(X).

Let us denote now

P= I: P(Ty)qy Ye:BI(x)

1)1.(£):= {x E JW.k: e(x) ~ O} with e(x) = e(x, s, £) := J.t(x, £) - m + s.

The following result shows that in the case when 1'11.(£)1 < 00, the conditions in (11.4.18) are independent and one can easily determine the dimension of 1'.(£).

COROLLARY 11.25. If 11)1,(£)1 < 00 for some s = 0, ... ,m - k - 1, then

(11.4.38)

Ch. 11. § 11.4] Further Properties of the Spaces 187

Proof. For k = I, the equality (11.4.38) directly follows from Lemma 11.13. Suppose it is proved in IRk- 1 and consider the case of IRk. Now we use the induction on the quantity

e(s,.c):= E e(z,s,.c). ~e91.(.c)

The case e(s,.c) = 0 is obvious because of Corollary 11.19, i). Let e(s,.c) > 0, z E 'Jl.(.c) and choose h E .c with z E h. Then we have

( _ 1 .c\h) _ {e(z,s,.c) -I, e z,s , - ( .c) e z,s, , if z E h, if z rt. h,

if z E h, if z rt. h.

Since e(z,s - 1,.c\h) ~ e(z,s,.c) -1 and 1'Jl.-l(.c\h)1 + 1'Jl.(.chl)1 < 00 then by induction we obtain from Corollary 11.21

Using the isomorphism of the space of spline functions and P.(.c) given by formula (11.4.6) one can obtain the analogues of Corollaries 11.22 and 11.25 for the spaces of spline functions. Let X C IRk, volk [X] i= 0, with [X] the convex span of X. Then, under an appropriate choice of the origin, the family .c of the hyperplanes defined by the equations v . z + 1 = 0 for v E X satisfies the conditions (11.4.3). Denote by 'Jl.(X), s = 0, ... , IXI- k - 1 the set of all (k - I)-dimensional hyperplanes 1 with s - IX\ll ~O. Then the condition 1'Jl.(.c) I < 00 is equivalent to 1'Jl.(X)I < 00 and the isomorphism (11.4.6) implies

COROLLARY 11.26. For an arbitrary X C IRk with volk[X] i= 0 we have m-k-l

a) E dimStxl_k_.,x = I{Y C X: WI = k + I, volk[Y] i= O}I, .=0

b) if 1'Jl.(X)I < 00 for some s, s = 0, .. "IXI- k - I, then

d' Sk _ (S+k) '" (s-IX\ll +k) 1m IXI-k-.,x - k - L..J k .

le91.(X)

188 Box Splines (Ch. 11, § 11.5

§ 11.5. Linear Independence of Translates of a Box Spline

Throughout this section, we assume that X is a finite subset of /lk\ {O} with (X) = ~k. As we already know from Theorem 11.5 in this case

V(X) = 7I"(~k) n S(X) , (11.5.1)

where S(X) is a space of functions f of the form

f = E aa B (. -a 1 X), aEZ~

We are interested in whether this representation is unique or, equivalently, whether the condition

E aaB (x - a 1 X) = 0 (11.5.2) aEZ~

for all x E ~k implies

for all a E /lk. (11.5.3)

DEFINITION 11.27. We say that the integer translates

B(.-a 1 X), a E /lk, (11.5.4)

of a Box spline B(·I X) are (globally) linearly independent, if the condition (11.5.2) with all x E ~k implies (11.5.3).

The translates (11.5.4) are said to be locally linearly independent iffor any domain n C ~k the condition (11.5.2) with all x E n implies that aa = 0 for all a satisfying

nnsuppB(.-a 1 X) i= 0. (11.5.5)

We have the following:

THEOREM 11.28. Let X be a finite subset of /lk\{O}, (X) = ~k. The following statements are equivalent:

i) X is unimodular, i.e.,

Idet Vi = 1 for all V E lIB(X) ,

ii) the translates (11.5.4) are locally linearly independent,

iii) the translates (11.5.4) are globally linearly independent.

(11.5.6)

Ch. 11. § 11.5) Linear Independence of Translates of a Box Spline 189

Proof. It is obvious that ii) implies iii). Because of Theorem 11.3, we have

~ Idet VIBC- to:ivi IX) - I: B(.-O: I X) == 0 aEZk i=l aEZk

for every V = {vI, ... vk } E W(X). Hence iii) implies i). It remains to prove that i) implies ii), i.e., if (11.5.6) holds then for any domain n c ~k the following system T is linearly independent on n:

T:= {B(·-o:IX): O:Eb(nlx)},

where b(n I X) is the set of all 0: E /Zk for which (11.5.5) holds. We can assume, without loss of generality, that n is a connected domain with

nnqx) = 0,

where qX) is defined as in (11.2.7). Then we have b(n I X) = b(t I X) for any tEn with

b(t I X) := {o: E /Zk: t E suppB(.-o: I X)}, (11.5.7)

since in this case (11.5.5) implies n c supp B(· -0: I X). Thus, we must prove that the condition

~ aaB(X - 0: I X) = 0, x E n, (11.5.8) aEZk

implies that for all 0: E b(t I X) . (11.5.9)

We need several lemmas.

LEMMA 11.29. Let X C ~k\{O}, (X) = ~k, and vEX. Ify and y+ v are both in the support of B(·I X), then y E suppB(-1 X\v).

(Note that the Box spline B(·I X) is defined in (11.1.1) under the assumption (X) = ~k. Here and below if (X) I ~k we assume that the set supp B(-I X) is defined as in (11.1.2)).

Proof. By (11.1.2), there exist y', y" E suppB(·1 X\v) and A', A" E [0,1] such that

y = y' + A'V, y+ v = y" +A"V.

Hence, y' = y - A'V, y" = y + (1 - A")V and from the convexity of supp B(·I X\v) we get

1 - A" 1 - A' y = 1 _ A" + .AI y' + 1 _ A" + .AI y" E supp B (. I X\ v) .

LEMMA 11.30. Let X be unimodular, V = {xl, ... ,xk} E IlB(X) and

k

W = L!iXi E X\V. i=l

Then for any i we have either !i = 0, or I!il = 1, i = 1, ... ,k.

190 Box Splines (Ch. 11. § 11.5

Proof. It is sufficient to prove for i = 1. Suppose that

We can assume without loss of generality that (V\X1) coincides with the coordinate hyperplane Xl = 0, i.e.,

If €"1 :f:. 0 then (V\x1) U wE I1B(X) and

o o

with 1811 = 182 1 = 1. Therefore

hence hi = 1. Before beginning the next lemma, we need some notation. For 11' 12 E ben I X)

we write 11 '" 12 iff

We say that 0' and j3 are equivalent and write 0' ~ j3 iff there exist

such that

0' = 11' j3 = Ii and Ii '" Ii+! for i = 1, ... ,j - 1.

Clearly, ~ is an equivalence relation on ben I X). LEMMA 11.31. Let X = {xi}~=l C ~k\{O}, (X) = ~k, and X is unimodular. If 0', j3 E bet I X) for some t E ~ , then 0' ~ j3.

Proof. We use induction on n. The case n = k is obvious. Suppose n > k and the lemma is true for IXI < n. Choose V E I1B(X) such that

Ch. 11, § 11.5) Linear Independence of Translates of a Box Spline 191

We can suppose, without loss of generality, that V = {Xl, ... , xk} and for xk+1 we have

k

",k+1 = ""' ""s'''';, "" ..... 0 '" L.J<- '" <-; ~ , for i=I, ... ,k. ;=1

Then by Lemma 11.30, for any i = 1, ... , k we have either C; = 0 or C; = 1.

for some m, 0 ~ m ~ k. (11.5.10)

According to the induction hypothesis, any two vectors in b(t I X\x k+1) are equivalent. Hence it is enough to prove that for an arbitrary a E b(t I X) there exist -y E b(t I X\xk+1) such that a ~ -y. According to (11.5.7) and (11.1.2) we have

where

n m

o ""' i ""'( ) i Y := t - a = L.J Ai x = L.J Ai + Ak+1 X + Z,

i=l i=l

Z = L Ai x;, m+1~i~n

i#+1

AiE[O,I], i=I, ... ,n.

Without loss of generality we may assume that 1 ~ Al ~ ... Am ~ O. If Al +Am +1 ~ 1 then yO E supp B('I X\xk+1), i.e., a E b(t I X\xk+1). Let Al + Am+1 > 1 and q is the largest integer such that q ~ k and \ + Am+1 > 1. Denote

j m

yi = L (Ai + Ak+1 - l)xi + L (Ai + Ak+1)xi + Z, j = 1, ... , q. (11.5.11) i=l i=j+1

It is easily seen from (11.5.11) that

. '-1 j . Y' = Y' - X, J = 1, ... , q. (11.5.12)

On the other hand, by (11.5.10)

j-1 m

yi = L (Ai - Aj)x; + L [1 - (Aj - Ai)]xi + (Aj + Ak+1 - l)xk+1 + Z,

i=l ;=j+ 1

which shows that yi E suppB(·1 X\xk+ 1). This combined with (11.5.12) implies that the vectors -yj := t - 11 satisfy the following relations:

which means that a ~ -ym. Let us prove now (11.5.9) with tEn provided that (11.5.8) holds. The proof

proceeds by induction on IXI. The case IXI = k is trivial since by Lemma 11.4 and (11.5.6) we have Ib(t I X)I = IdetXI = 1. Suppose that IXI > k and (11.5.8) implies

192 Box Splines [Ch. 11, § 11.5

(11.5.9) for any X' with IX'I < IXI. It follows from (11.5.8) that for any v E X with (X\v) = jRk

DV(LaaB(z-aIX))=o, zEn. aEZk

Hence, according to (11.2.11)

L ('VVa) (o:)B (z - a I X\v) = 0, zEn,

where 'Vva(a) = a(o:) - a(a - v). By the induction hypothesis this implies

a(a) = a(a - v) for all a E b(t I X\v) . (11.5.13)

We want to prove now that a(a) = a(p),

if 0: '" p. Let a = P+ v with a,p E b(t I X), vEX. Then

t-o:, t-a+vEsuppB(·IX)

(11.5.14)

(11.5.15)

and by Lemma 11.29 we have t - 0: E supp B('I X\v), i.e., a E b(t I X\v). On the other hand, the condition (11.5.15) implies that (X\v) = jRk, since if (X\v) :f jRk then suppB('1 X\v) C qX) and hence t - a E qX) which contradicts the condition t ¢ qX). Thus we obtain (11.5.14) for 0: '" p. Then (11.5.14) obviously holds also for a ~ {3 and, because of Lemma 11.30, for an arbitrary a, p E b(t I X). Using (11.2.3) we get from (11.5.8)

ap = ap L B(t - 0: I X) = L aaB(t - a I X) = ° aEZk aEh(tIX)

for any p E b(t I X).

§ 11.6. Interpolation by Translates of a Box Spline

In this section, we investigate the interpolation properties of S(X) - the space spanned by integer translates of a Box spline B(. I X), and V(X) - the space of polynomials in S(X).

DEFINITION 11.32. Let X C ~k, (X) = jRk, and t E jRk\qX). The Lagrange interpolation problem from the space V(X) with the node set b(t I X) is to find a (unique) polynomial P E V(X) satisfying

P(o:) E Aa for all a E b(t I X) ,

for given values An, a E b(t I X). The necessary condition for the regularity of such an interpolation is

dim V(X) = Ib(t I X)I.

The dimension of the space V(X) we already know by Corollary 11.23:

dim V(X) = PIB(X) I ' and the cardinality of b(tIX) we determine in the following

(11.6.1)

(11.6.2)

Ch. 11. § 11.6] Interpolation by Translates of a Box Spline 193

THEOREM 11.33. Let X C ;;ZA:\{O}, (X) = ]W.A:, and t E ]W.A:\qX). Then

Ib(t I X)I = E Idet Vi· (11.6.3) VeJ(x)

Proof. Let us denote

Q(X) :=suppB(·IX) = {EA"V: 0 ~A" ~ 1}. "ex

The equality (11.6.3) follows from Lemma 11.4 and following

LEMMA 11.34. Let Xc ]W.A:\{O}, (X) = ]W.A:. Then for each Y E B(X) there exists f3y E ]W.A: such that

i) Q(Y) + f3y and Q(Y') + f3y' have disjoint interiors for each Y, Y' E B(X),

ii) U (Q(Y) + f3y) = Q(X). YeJ(X)

Moreover, each f3y has the form

f3y = E c"V "ex\Y

with c"E{O,1}.

Proof. We proceed by induction on IXI. When IXI = k there is nothing to prove. Suppose the assertion holds for any X with k ~ IXI ~ n. We will prove now it holds for the set Xw = X U w. For this purpose, we define

r := {x E Q(X): x + tw ft Q(X) for all t > O} .

Then r is a closed subset of the boundary of Q(X). Furthermore, any closed line segment in Q(X) containing a point of r in its interior lies in r. Therefore, r is partitioned by some collection of (k -I)-faces of Q(X). By the induction hypothesis there is a collection of parallelepipeds

n(x) = {Q(Y) + f3y : Y E(X)}

satisfying i) and ii). In particular, the (k - 1)-faces of some of the parallelepipeds in n(X) being (k - 1)-parallelepipeds must induce a partition of r. Let G denote this partition. We now construct a partition of Q(Xw) by appending to n(X) the following set of parallelepipeds n1. Each element of G has the form

Q(V) + f3v , V E X(k -1), (V Uw) = ]W.A:,

where f3v is some extreme point of Q(X). In addition, from the decomposition ii) it follows that

f3v = E c"V, "ex\V

c" E to, I}.

194 Box Splines [Ch. 11. § 11.6

The corresponding parallelepipeds in 0 1 are then obtained by forming the sets

Q (V U W) + ~V = {Z = v + tw: v E Q(V) + ~V' 0:::; t :::; I} .

Clearly, the set H := U{Q: Q EO U Od is contained in Q(Xw). In order to show that Q(Xw) ~ H it is, in view of the induction hypothesis, sufficient to show that any z E Q(Xw)\Q(X) is also in H. Such an z has the form

z = y + tw, y E Q(X), 0 < t :::; 1.

Let L denote the line segment connecting z and y. L must intersect the boundary of Q(X) since otherwise z E Q(X). Let z = v + tow, where 0 < to :::; 1. By definition, z lies therefore in some element of 0 1. This proves

Theorem 11.33 and (11.6.2) imply that for the regularity of the Lagrange interpolation from the space V(X) it is necessary that X be unimodular. The following theorem shows that this condition is also sufficient.

THEOREM 11.35. Let Xc Ilk\{O}, (X) = ~k, and t E ~k\qX). Then the Lagrange interpolation from the space V(X) with the node set b(t I X) is regular if and only if X is unimodular.

Proof. We need to prove the sufficient part of the theorem only. Let X be unimodular. Then by Theorem 11.33 and (11.6.2), the condition (11.6.1) holds, hence it is enough to prove that if

P(Q) = 0, all Q E bet I X) , (11.6.4)

for some P E V(X), then P == O. Let 0 be a neighbourhood oft such that onqX) = 0. Then the equality

b(z I X) = bet I X)

holds for all z E 0 and (11.6.4) implies

(TP)(z):= L P(Q)B(z I X) = L P(Q)B(z I X) = 0 O'Eb(xIX) O'Eb(tIX)

for all z E O. According to Theorem 11.6, the mapping P ---. TP is one to one and onto V(X) C 7r(~k). Therefore (TP)(z) = 0 for all z E ~k and hence P == O.

Let us consider now the cardinal interpolation from the space S(X) which spanned by integer translates B(· -Q I X), Q E Ilk, of a Box spline B(·I X).

From now on it is convenient to consider a O-centered Box spline instead of B (. I X). Let

with w = ~ LV. vEX

Ch. 11, § 11.6) Interpolation by Translates of a Box Spline 195

It is obvious that B(x) can be defined by the rule

f f(x)B{x I X) dx = f f{X(t») dt, (11.6.5)

II" [-1/2,1/2]"

DEFINITION 11.36. Let X C Zl:, (X) = lRl:. The Cardinal Interpolation Problem (CIP) from the space SeX) is to find a (unique) sequence a = {aa} E loo(ZI:) satisfying

B * a(fJ):= E aaB(fJ - a) = Ap, fJ E Zl:, (11.6.6) aez"

for given values >'p, fJ E Zl: with A = {>.p} E loo (Z").

Clearly a necessary condition for the regularity of CIP is that the translates of a Box spline B(· ) are linearly independent, i.e., X must be unimodular (see Theorem 11.28).

Associated with the cardinal interpolation problem is its characteristic polynomial defined by

P(x):= Px(x):= E B(a)exp(iax), x ElR. aez"

The solution of the multivariate difference equation (11.6.6) with

if fJ = 0, otherwise,

(11.6.7)

we call (if it exists) a fundamental solution of (11.6.6) and denote by al = {a!}:

E a~B(fJ - a) = 6p , fJ E Zl:. (11.6.8) aez"

Multiplying both sides of (11.6.6) by exp(ifJ·) and summing over fJ E Z" leads to the equation

PA=L, (11.6.9)

where A and L are formal Fourier series with coefficient sequences a and A, respectively. It follows that aa are the Fourier coefficients of L/ P.For a fundamental solution A = {6 }, we have L = 1 in (11.6.9) and PAl = 1. Therefore, if CIP is regular then, at feast formally, the characteristic polynomial P has no zeros and the fundamental solution al is the sequence of Fourier coefficients of 1/ P:

I 1 f exp(-iau) d aa = (2'11")1: P(u) u, (11.6.10)

[-"'."']"


196 Box Splines [Ch. 11. § 11.6

THEOREM 11.37. The Cardinal interpolation from S(X) is regular if and only if the characteristic polynomial P does not vanish on ~k. Moreover, the fundamental solution a! is given by (11.6.10) and satisfies

la~ I = o( exp (-clal)) (11.6.11)

for some positive constant c, and for any sequence a E 100 (Zk) the unique bounded solution of (11.6.6) is given by

a E Zk. (11.6.12)

Proof. Suppose P(O) = 0 for some 0 E ~k. Then

E exp ( -iaO)B(f3 - a) = exp (-iaf3) E exp ( -iaO)B( a) = exp ( -ia/3)P( 0) = 0

for all /3 E Zk, which means that the translates B(· -a) are linearly dependent on Zk.

Suppose now that Px does not vanish. Then 1/ Px is a continuous function on [-"/I","/I"]k and the sequence a! defined by formula (11.6.12) is a fundamental solution satisfying (11.6.9). Therefore, the sequence a in (11.6.10) is well defined for any bounded sequence {A,6} and is a solution of (11.6.6):

Remark. Note that Theorem 11.37 remains valid with B(X) an arbitrary bounded compactly supported function defined on ~k.


Box splines were introduced by de Boor and De Vore [1983]. The basic properties (Theorems 11.2, 11.3, 11.5, 11.6) were established by de Boor and Hollig [1982b]. The system of partial differential equations were studied in Hakopian and Sahakian [1988], [1989], where Theorems 11.8, 11.9, 11.17, 11.20 were obtained. Earlier, the equality dim V(X) = 11IB(X) I (see (11.4.34» was proved by Dahmen and Micchelli [1985]. The spaces P(X) and V(X) were studied in Dyn and Ron [1990], [1990b], where Corollary 11.24 was obtained, as well as the local approximation order of some exponential spaces and an interpolation problem induced by V(X) were considered. In the paper of de Boor, Dyn, and Ron [1991] the space P(X) is characterized as a joint kernel of differential equations

( ) IX\hl Dh.J. f = 0, hE lHI(X),

Ch. 11. § 11.6) Interpolation by Translates of a Box Spline 197

(see also Dahmen and Micchelli [1989], de Boor and Ron [1991]). Theorem 11.28 was proved by Dahmen and Micchelli [1983b], [1985] and Jia [1984], [1985]. Here, we have presented the proof of Jia. Theorems 11.33 and 11.35 are due to Dahmen and Micchelli [1985]. The Cardinal Interpolation Problem (see Definition 11.36) was considered in de Boor, Hollig, and Riemenschneider [1985], where the regularity of CIP for any unimodular X was proved in R2 and in Rk, k ~ 3, with the additional condition that each knot appears in X with even multiplicity. It was proved in de Boor, Hollig, and Riemenschneider [1985c] that the linear independence of translates of B(. I X) is, in general, not sufficient for the regularity of CIP (see, in this context, also Jetter and Riemenschneider [1987], Chui, Jetter, and Ward [1987], Riemenschneider, Scherer [1987]).

Chapter 12

MULTIVARIATE MEAN VALUE INTERPOLATION

§ 12.1. Mean Value Interpolation of Lagrange Type

Let T = {to, ... , tr } C ~ be an arbitrary set of knots (not necessarily distinct) and JLj := {i : ti = tj} - the multiplicity of the point tj in T. The Hermite interpolant of a function f is defined as a unique polynomial Pj E 7l'r(l~) satisfying (see Chapter 1)

(12.1.1)

where til' ti 2 , ••• ,ti, are all the distinct points in T. In the case when all the points ofT are distinct, i.e., JLj = 1 for all j, the polynomial

Pj is a Lagrange interpolant of the function f and the conditions (12.1.1) look like

j = 0, . .. ,r.

Moreover, the Lagrange interpolant can be presented in the form

r

Pj(t) = LJ(tj)lj(t), (12.1.2) j=O

where lj(t) are fundamental polynomials of Lagrange interpolation:

II (t-ti) lj(t):= (t.-t.)'

i=O,i¢j} • j, i = 0, ... , r. (12.1.3)

If T is an arbitrary set of knots, Pj is given by the following formula of Newton:

r

Pj(t) = l)t - to)· ... · (t - tj-d[to, ... , tjl!, (12.1.4) j=O

with [to, ... , tj l! the divided difference of f with knots to, ... ,tj. In the following theorem we, present a multivariate mean value interpolation of

Lagrange type. Let us denote

XU):={vcX:IVI=j}, O~j~IXI,

for an arbitrary finite set X C ~k.

198

Ch. 12, § 12.1] Mean Value Interpolation of Lagrange Type 199

THEOREM 12.1. Suppose that the node set X = {ZO, ... ,Zr} C ]RI: with r ~ k + 1 is in a general position. Then for arbitrary values

~V, VEX(k),

there exists a unique polynomial P E 1I"r+l-I:(]RI:) such that (see (10.2.3»

[V]P = ~v, V E X(k).

Proof. Let us consider the mapping

C : 1I"r+l-1: (]RI:) _]RN with N = IX(k)1 = (r; 1) , defined by the rule

CP := {[V]P}VEX(I:)'

Since C is linear and dim 1I"r+1-I:(]RI:) = dim ]RN,

it is sufficient to show that C is one to one. Suppose that [V]P = 0, all V E X(k), for some polynomial P with m := deg P ~ r + 1 - k. Then, by Theorem 10.13, we have

[A] a P = 0 for all A E X(k + m) and a E 7i} with lal = m.

This means that either deg P < m or P == o. Since deg P = m, then P == o. We call the interpolation described in Theorem 12.1 the mean value interpolation.

The interpolation parameters here are integral means taken over a convex span of k points of the set X:

(V]P = I 1 [V] J Pdml:_l, VOI:_l

[V]

In the univariate case, the divided difference with one point coincides with the value of the function in that point. Thus, mean value interpolation is a generalization of the univariate Lagrange interpolation. Theorem 12.1, in particular, implies that for each locally integrable function f there exists a unique polynomial

PJ := PJ,x E 1I"r+1_I:(]RI:)

satisfying

[V]PJ = [V]f for all V E X(k). (12.1.5)

It follows from (12.1.5) and (10.2.4) that

J DaPJ = J Daf, lal=IAI-k, (12.1.6)

[A] [A]

for all A C X, IAI ~ k. In the following theorem, the analogue of the representation (12.1.4) for mean value interpolation is given.

200 Multivariate Mean Value Interpolation [Ch. 12, § 12.1

THEOREM 12.2. Let the node set X = {ZO ... , Zr} C ~k with r ~ k + 1 be in a general position. Then

r

P"x(z) = L i=k-l lal=i+l-k

where Xi = {zo, ... , zi}.

Proof. Consider the following representation:

P"Xi(Z) = L za[Xi]a P"Xi + P'(z) lal=Hl-k

(12.1.7)

(12.1.8)

with P' E 1ri_k(~k). Since Pp. Xi-1 = P', taking the mean value interpolant in (12.1.8) with node set Xi-I, w~ obtain

P"Xi-1(Z) = L POc>,Xi-1(Z)' [xi]a P"Xi + P'(z), lal=i+I-k

hence, according to (12.1.8) and (12.1.6)

P"Xi(Z) - P"XH(Z) = L (za - p(.)c>,Xi_l(Z»)[xi]a f. lal=Hl-k

Summing up (12.1.9) over j = k - 1, ... , r yields (12.1.7).

Exercise 12.2.1. Prove that

i) P,(z) = (k - 1)! t L J II D~_!l/' i=k-l YeX(j-k) [Xi] !leY

(12.1.9)

k-l (k -1) J ii) /(z) - P,(z) = ~ j L. II D~_yf.

J=O Yex(J) xu{~, ... ,~} yex\y ---k-i

Let us denote by Pv, V E X(k), the fundamental polynomials of the mean value interpolation, i.e., Pv E 1rr+l_k(~k) and

[YjPv = {~: if Y = V, if Y # V,

Y, V E X(k). (12.1.10)

In the following theorem, we obtain a formula for fundamental polynomials; a generalization of the formula (12.1.3).

Ch. 12, § 12.1] Mean Value Interpolation of Lagrange Type 201

THEOREM 12.3. Let X be in a general position. Then, for each V E X(k) and v E V

Pv{x) = ((k-1)! II d{v-y,O,V))-ID!(v\( II d{X-Y,O,v)) (12.1.11) !lEX\V !lEX\v

with n{V) the unique normal of L{V) - the (k - 1)-dimensional hyperplane containing V and directed towards the half-space d{x, 1, V) > 0.

(The function d{x,-y, V) is defined as in Section 10.2). For the proof, we need the following:

LEMMA 12.4. Let X be in a general position and for sufficiently smooth functions Ii, i = 1, ... , m,

m

Lqi{D)1 = 0, (12.1.12) i=1

where qi(D) are constant coefficient homogeneous differential operators of order 1. Then

Proof. Denote

m

Lqi(D)P1i = 0. i=1

m

P := L qi(D)Pli E 7rr+1_k_l(~k) i=1

and note that according to (12.1.12) and (12.1.6)

J P = J t qi(D)PJ; = J t qi{D)/i = ° [A] [A] ,=1 [A] ,=1

for all A E X (k + 1). Therefore

J DOIP = 0,

[A]

101 = j -1,

(12.1.13)

(12.1.14)

for all A E X(k+ j) and j, 1 ~ j ~ r+ 1- k, since the left-hand side of (12.1.14) is a linear combination of integrals in the form of (12.1.13). Since DOl P E 7rr+1_k_l(~k), it follows from (12.1.13) that P = 0.

Proof of Theorem. Let us denote by Lv(y), Y E ~k, the (k - 1)-dimensional hyperplane passing through Y and parallel to Lv. Of course, Lv ( v) = Lv for v E V. Let f1(x,L) be the signed distance ofx from L (see (9.3.13)). Then (12.1.11) can be written as

Pv = C1 D!(v) ( II U(·, Lv (y)) . !lEX\v

(12.1.15)

Let us present Pv in the following form:

202 Multivariate Mean Value Interpolation [Ch. 12, § 12.1

PV(X) = L: cOtX Ot + PI(x) with COt = [X]Ot Pv, pI E 1rr_k(~k). IOtI=r+l-k

Since [Y]Pv = 0 for all Y E (X\v)(k), taking the mean value interpolant with the node set X\ v yields

hence

0= L: COt Po ''', X \v (x) + PI(x), IOtI=r+l-k

IOtI=r+l-k

(12.1.16)

Since (12.1.15) does not depend on the coordinate system, we can assume, without loss of generality, that the set V belongs to the coordinate hyperplane Xl = O. Then by Theorem 10.13 the relation (12.1.16) reduces to

Pv(x) = C2<P(X), X E ~k,

where ( ) . r+l-k P ( ) <P X .= xl - zr+l-k X\v X ,

1 •

Using Lemma 12.4, we obtain that Pzr+l-k X\v(x) depends only on Xl, i.e., 1 •

<p(x) = <p(xt}, X = (Xl, ... , Xk) E ~k.

Hence, the conditions (12.1.7) imply

t/J(t):= J <p = J <p = 0 [tUV\v] [IIUV\v]

for t = ylll' Y E X\ V. Since t/J(t) is a polynomial in t of degree r + 1 - k the latter means that

t/J(t) = C3 II (t - ylll'). IIEX\V

Now, using (1.3.11) and (1.3.12) we obtain

PV(x) = C2<P(X) = C4C~ f- l II (Xl - ylll) , 1 IIEX\V

which proves (12.1.15). To determine Cl in (12.1.15), we notice that

1 = Pv(v) = clD~(.,) ( II e(x, Lv(y») I = cl(k -1)! II e(v, Lv(y» . IIEX\v z=v IIEX\V

As a corollary, we obtain the following generalization of the univariate representation (12.1.2)

PJ = L: ([V]f)Pv . VEX(k)

Ch. 12, § 12.2) Kergin Interpolation and the Scale of Mean Value Interpolations 203

§ 12.2. Kergin Interpolation and the Scale of Mean Value Interpolations

In this section, we study the generalizations of the univariate polynomial interpolations based on the ridge function method considered in Section 9.4. Recall that the k-variate function / is called a ridge function if it can be presented in the form

fez) = g(,x·z), (12.2.1)

with 9 : ~ _ ~ and ,x E ~k • We construct the multivariate interpolation by "lifting" the univariate one. Let

be an arbitrary node set. Then for ridge functions such as (12.2.1), which are dense in C(~k) (see Lemma 9.11), we define

1I./(z) = Hg(,x·z), (12.2.2)

where H is the Lagrange-Hermite interpolation operator with node set ,x. X := p. zo, ... ,,x. zr} C ~. If, for a pair of linear operators, the relation (12.2.2) holds, we say that 11. is the lift of the univariate operator H to ~k. It was proved by Kergin [1980] that the Lagrange-Hermite interpolation operator can be lifted to ~k for all k. Thus, the Kergin interpolation is a lifted univariate Lagrange-Hermite interpolation.

It appears that the multivariate mean value interpolation can also be obtained as a lift of some univariate interpolation operator. Moreover, the lifted univariate operators depend on k. These operators form the univariate scale of interpolations.

For the node set T := {to, ... , tr } C ~ they are defined satisfying the following conditions (g is an arbitrary sufficiently smooth function):

HmgE1I"r_m(~)' m=O, ... ,r,

HOg(ti) = 9(ti), i = 0, ... , r, (12.2.3)

i = O, ... ,r- mj m= 1, ... ,r.

(Ho = H is the Lagrange-Hermite interpolation operator). It is not difficult to see that

1 = 0, ... , r -I, 1 = 0, ... , r. (12.2.4)

The following theorem shows that this operator can be lifted to ~k for k ~2 and form the multivariate scale of interpolations.

204 Multivariate Mean Value Interpolation [Ch. 12. § 12.2

THEOREM 12.5. Let X = {ZO, ... , Zr} C IRk and 0 ~ m ~ r. Then, for each sufliciently smooth function f, there exists a unique polynomial1f.m f E 7rr _ m (IRk) satisfying

J V a1f.mf = J Va f, lal = IAI- m - 1, (12.2.5)

[AI [AI

for each A C X with IAI ~ m + 1. Moreover, 1f.m is the lift of Hm and

1 J ~ J -m! 1f.m f( z) = f + L.J DfI:-fl:; f

[fl:0 •...• fl:ml ;=0 [fl:0 •...• fl: m+ 11

+ L J DfI:_fl:;l DfI:_fl:;' f + ... O~h<;,~m+l [fl:0 •...• fl:m+'1

(12.2.6)

+ L J DfI:_fl:;l' •. DfI:_fl:;r-mf.

o~h<···<;r_m~r+l [fl:0 •...• fl:rl

Proof. Let us prove first that the linear operator 1f.m given by (12.2.6) is the lift of Hm. According to (12.2.4) and (12.1.4), we have for the interpolation operator Hm with node set p.zo, ... ,'\.zm}:

1 -,Hmg(t) = m.

J g + [(t - ,\. zO) ... (t - ,\. zm)] J [>'.fl:0 •...• >,.fl:m] [>..fl:o •...• >..fl:m+l]

+ [(t - ,\. zo) ... (t - ,\. zr-l )](m) J g.

[>..fl:o •...• >..fl:r]

Hence, using the properties (9.4.1) and (9.4.2), we obtain that

i.e.,1f.m is the lift of Hm.

g

It remains to prove (12.2.5). For this, we again use the above-mentioned properties of ridge functions and (12.2.3):

J V a1f.mf = J va Hmg('\· z) =,\a J (Hmg)(lol) =,\a J gOal) = J f· [A] [AI [>.·A] [>.·A] [AI

If X is in a general position, taking m = k-1 in the Theorem 12.5 we obtain a mean value interpolation of Lagrange type considered in the previous section (cf. (12.1.6». It is important to note that the interpolation parameters in (12.2.5) are linearly dependent, in contrast with Theorem 12.1, where they are linearly independent.

Ch. 12. § 12.2) Kergin Interpolation and the Scale of Mean Value Interpolations 205

When m ~ k and degen (X) ~ m - k (see Definition 9.9) the linearly independent parameters in (12.2.5) can be selected as follows. According to Theorem 10.13, the conditions (12.2.5) hold iff

J 1im!= J Va! for all A E X(m + 1), (12.2.7)

[A] [A]

which, because of (9.2.3) is equivalent to

J 1im !(X)M(xIA) dx = J !(X)M(xIA) dx for all A E X(m + 1).

lI\k Jlk

Therefore the parameters

{l AEn} are linearly independent parameters of (12.2.7) (and (12.2.5)) if and only if the system {M(·IA) : A E O} forms a basis in the space of spline functions S!-k+l.X' thus, we can use Theorems 10.2 and 10.7 to select linearly independent parameters in (12.2.5).

Exercise 12.5.1. i) Prove the multivariate analogue of (12.2.4):

ii) Prove the analogue of Lemma 12.4 for interpolation operators 1lm .


The mean value interpolation was first considered by Kergin [1978], [1980] (the case m = 0 in Theorem 12.5), while the corresponding representation (12.2.6) is due to Micchelli and Milman [1980]. The case m = 1 of Theorem 12.5 was proved by Cavaretta, Micchelli, and Sharma [1980]. The case m = k - 1 was studied in Hakopian [1981], [1982b], [1983], where the results of Section 12.1, as well as similar results for the mean value interpolation of Hermite type (when X is not in a general position), were established. In the form presented, Theorem 12.5 is due to Goodman [1983].

Let

Chapter 13

MULTIVARIATE POLYNOMIAL INTERPOLATIONS ARISING

BY HYPERPLANES

Lj C IRk, dimLj = k -1,

be an arbitrary collection of (k - I)-dimensional hyperplanes. In this chapter, we consider the problem of the existence of a polynomial P with given traces on the sdimensional hyperplanes (0 ~ s ~ k -1) which are intersections of some hyperplanes from C. We start with the case s = 0, i.e., pointwise interpolation with a node set arising by the hyperplanes L E C.

§ 13.1. Pointwise Interpolation

Assume that the hyperplanes L E C are determined by the equalities:

U(x, L) = 0, (13.1.1)

where

U(x, L) = U(Xl, ... , Xk, L) = AIXI + ... + AkXk + Ak+l, A~ + ... + A~ = 1,

is the signed distance of x E IRk from the hyperplane L. Let Ck be the set of all (distinct) points being intersections of k hyperplanes

from C:

Ck := {(1{ E IRk: (1{ = n L, 1i E C(k)} LE1{

with C( s) the collection of all subsets of C of cardinality s:

C(s):={1iCC: 11iI=s}, s=O, ... ,p.

We denote by p«() the multiplicity of ( E IRk in C:

p«() = J.l«(,C):= I{L E C: (E L}I, p«() ~ k.

(13.1.2)

(13.1.3)

(13.1.4)

DEFINITION 13.1. The set of hyperplanes C is said to be admissible, if every k hyperplanes from C have a unique common point:

1 n LI = 1 for all 1i E C(k). LE1{

We say that C is in a general position if C is admissible and

p«() = k for all (E Ck •

206

Ch. 13, § 13.1) Pointwise Interpolation 207

DEFINITION 13.2. The Hermite interpolation problem (C) is to find a unique polynomial P = PA E 1I"/A_k(~k) satisfying

Dap«) = At, (13.1.5)

for a given set of values

A = {At: lal ~ J1«) - k, (E Ck }.

If C is in a general position, then the interpolation parameters in (13.1.5) are the only values and we obtain the Lagrange case (Chung-Yao interpolation).

THEOREM 13.3. i) The Hermite interpolation problem (C) is regular for any admissible set C = {Ll, ... ,L/A} of (k - I)-dimensional hyperplanes in ~k.

ii) If C is in a general position then the unique polynomial P = PA E 1I"/A_k(~k) satisfying

can be presented in the form:

P = L: A,P" (ee.·

where p( are the fundamental polynomials:

p. ( ) ._ II U(x, L) _ { 1, ( x .- - 0

Lee., (ILL U«, L) ,

if x = (, if XEC k \{(}.

(13.1.6)

Proof. We will use induction on k. In the case k = 1, we obtain a univariate Hermite-Lagrange interpolation (see Theorem 1.1). Suppose that k> 1 and the theorem is true in ~k-l. It is not difficult to check that

IAI = dim(1I"/A_k(~k)) = (n· Therefore, to prove i) it is sufficient to show that the equalities

Da P«() = 0, lal ~ J1«() - k, (E Ck ,

imply P == O. For each (fixed L) the set of (k - 2)-dimensional hyperplanes

L n C := {L n h: hE C}

(13.1.7)

is admissible on L. Hence, the induction hypothesis implies that the Hermite interpolation problem (L n C) is regular (on L) and then (13.1.7) implies that P IL= O. Now, using Lemma 11.16 yields

P(x) = c II U(x, L), Lee.

Since deg P < J1 = ICI, this implies c = 0, i.e., P == O. Let us denote by PI = PI.e. the unique polynomial satisfying

PJ E 1I"/A_k(~k), Da PJ«) = Daf«), lal ~ J.l«) - k, (E Ck ,

for given sufficiently smooth f. In the following theorem, we present the analogue of the univariate recurrence relation (12.1.6).

208 Multivariate Polynomial Interpolations Arising by Hyperplanes [Ch. 13, § 13.1

THEOREM 13.4. Let C be admissible and 'H. E C(k + 1) be in a general position. Then

with

Pj,.c(X) = L bL(x)Pj,.c\d x ), LE'H.

e(x, L) bL(X) = ('" L)' e .. 'H.\L,

L E 'H.,

the barycentric coordinates of x with respect to'H. (see (9.3.8)).

Proof. Suppose first that C is in a general position. Denote

Q(x) := L bL(X)Pj,C\L(X), LE'H.

Since Pj,C\L E 1I"JJ_t_k(JRk) and bL E 1I"1(JRk) then Q E 1I"JJ_k(JRk) and we need only to check that

Q«() = f«()

Let (E Ck. For every L E C we have either (E Land bL «() = 0 or (E (C\L)k and Pj,.c\d() = f«(). Therefore

Q«() = L h«()f«() = J«(), LE'H.

smce

L bL«() = 1 LE'H.

(see (9.3.8)). The general case can be obtained by applying a continuity argument. Let us consider now the case of the Lagrange interpolation in more detail. For an

arbitrary node set

. h (n + k) Wlt 8 = k ' (13.1.8)

the Lagrange interpolation problem is to find a unique polynomial P E 1I"n(JRk) satisfying

i = 1, ... ,8,

for given values Ai. If this problem is regular, we say that U is a regular node set (for Lagrange interpolation by 1I"n(JRk)).

The problem of description (for an arbitrary positive integer n) of regular node sets for the Lagrange interpolation remains still open and is solved only for k = 2, n = 2 (the case n = 1 is obvious). Theorem 13.3 gives a way to construct regular node sets. Another construction is given in the following:

Ch. 13, § 13.2) Polynomial Interpolation by Traces on Manifolds 209

THEOREM 13.5. Let for the node set U given by (13.1.8) there exist (k - I)-dimensional hyperplanes Lo, ... , Ln such that for j = 0, ... , n the node set

{ j-l }

Uj:= u E U: u E Lj\ W Li 1=0

is regular for the Lagrange interpolation from 7rn _j(Lj). Then U is a regular node set for the Lagrange interpolation by 7rn {lRk ).

Proof. Since dim 7rn(lRk) = lUI, it is sufficient to show that the unique polynomial p E 7r n (lR k) satisfying

i = 1, ... ,8, (13.1.9)

is P == O. Because ofthe regularity ofthe node set Uo (on Lo) it follows from (13.1.9) that P ILo= 0, hence by Lemma 11.16

P(x) = Co£l(x,Lo)P1(x),

Now, (13.1.9) implies that P1«() = ° for ( E U1 , hence we obtain from the regularity of the node set U1 (on Ld, that

Continuing this process yields

n

P(x) = c II e(x,Lj), j=O

which means that c = 0, since deg P ~ n while deg £1(', Lj) = 1.

§ 13.2. Polynomial Interpolation by Traces on Manifolds

In this section, we are interested in the interpolating of polynomials given on (k - 8)dimensional linear manifolds obtainable as an intersection of «k - I)-dimensional) hyperplanes from a given (multi)set 1f. of hyperplanes in]Rk. We treat this problem in full generality, taking into account the multiplicities by the corresponding matching of derivative information and also taking into account the information at infinity in the case of empty intersection of the corresponding hyperplanes.

We give necessary and sufficient conditions in terms of consistency of the given data, for the existence and uniqueness of an interpolant from 7rn = 7rn (]Rk), or from 7r~ = 7r~(]Rk) := homogeneous polynomials of degree = n, under the assumption that data intended to prescribe some derivative of order r on some linear manifold is indeed a polynomial of degree ~ n - r. Let us start with some notation.

For P E 7rn (]Rk) and i = 1, ... , n we denote by p[il the homogeneous component of P of degree i, hence, we have

n

P= LP[i1, i=O

210 Multivariate Polynomial Interpolation Arising by Hyperplanes [Ch. 13. § 13.2

Let Dy be the directional derivative along y E ~k. Define

D~P := p[n-ij for PE7rn and i=1, ... ,n.

Note that the definition of D:x, depends on n. Since DyP E 7rn -l (~k) for P E 7rn(~k), we have that

D~DyP = DyD~P, i = 1, ... , n.

Now let 1i be a given multiset of hyperplanes in ~k. This means that each H E1i is of the form

H = {x E ~k: nH·x = aH }

for some nonzero k-vector n H and some number aH . Note that we do not exclude here the possibility of repetitions. For H E 1i we denote by HO the (k - 1)dimensional subspace parallel to H, i.e.,

It is helpful to consider HO as a set of improper points of H or points of H "at infinity". Besides, the points of H itself we call proper. Denote by

C =C1t

the collection of all linear manifolds obtainable as intersections of hyperplanes from 1i and include here even linear manifolds corresponding to the empty intersections, i.e., having improper points only in a manner to be clear in a moment. Thus'\ E C if and only if

,\ ='\M:= n H HEM

for some M ~ 1i. For ,\ E C we denote by ,\0 its improper part, l.e., the linear subspace

parallel to it. We call ,\ proper in case ,\ '1= " and call the others improper. We identify the

improper manifolds by their improper part, i.e., set ,\ = A in case ,\0 = AO. In what follows, the expression (linear) manifold means proper or improper manifold.

Note that improper manifold ,\ does not contain any proper manifold (including proper points) and is contained in the manifold A iff ,\0 C A ° .

We classify manifolds by their dimension, defined as follows:

d. \._ {dim,\O, lm".- dim'\o _ 1,

if ,\ is proper, if ,\ is improper.

Now we obtain the following nice relation which also is a motivation for the above classification:

d· (\ H) {dim,\, 1m "n = dim'\ _ 1,

if ,\ C A, otherwise,

Ch. 13, § 13.2] Polynomial Interpolation by Traces on Manifolds 211

for any manifold A and hyperplane H. For a proper manifold A we denote by Aoo the improper manifold for which {Aoo)O =

A 0 ; Aoo is the unique improper manifold of A of dimension dim A-I. We write

C! = 4 := fA E .e : dim A = k - s}

for the collection of all linear manifolds in .e of codimension s, s = 1, ... , k. In particular,

.e1 = {H E1t}.

The multiplicity of A E .e plays a central role. It is determined by the multiset

.e>. := {H E1t : A C H}.

Let A E .e •. Then I.e>. I ~ s with I.e>. I = s, the simple case. In any case we intend to prescribe on A E .e- all derivatives in lRi , normal to A and of order ~ 1£>.1- s.

Moreover, in the inductive proof we will deal with the situation when lRi is replaced by another manifold A with A C A. For this purpose we choose for

A C A, m:= dim A - dimA, mo:= dimAo - dimAo

some orthonormal coordinate system

for the orthogonal complement

if m = mo,

if m = mo + 1,

for a polynomial P and multiindex a = (a1, ... , am). Note that m = mo if both A and A are proper or improper, and m = mo + 1 if A is improper and A is proper.

Further we will omit the manifold A in the notations iff A = lRi. For proper A E .e we denote by

1I"n{A)

the space of all polynomials of degree ~ n on A in the coordinates with respect to any particular coordinate system on A. Correspondingly,

is the subspace of 1I"n{A) of all homogeneous polynomials of degree = n on A. In order to fix the latter class of polynomials, we choose the projection 0>. of the origin o E lRi into A as a coordinate origin on A. For improper A we take


We put also 1I"n(A) := JR if A is a point. pi.>. means the trace of the polynomial P on A, if A is proper, and we define

for improper A. For proper 1 EC we denote by 1I"n(A)' and 11"~ (A)' the corresponding classes of

polynomials with coefficients being functions on 1. It is clear, that for proper A C A a polynomial P E 1I"n(A) can be considered in a natural way as a polynomial from 1I"n(A).>..1.(A), which we will denote by

(13.2.1)

Suppose, C· =F 0 for some fixed s, 1 ~ s ~ k. Consider the following sequence of polynomials

This sequence will serve as a data on manifolds of C·, which means that the interpolation problem is to find a polynomial P E 1I"n such that

for all P.>..a E V· (13.2.2)

with A.1. := A.1.(JRk). This restoration of the polynomial P will be carried out gradually, beginning on restoration of traces of P and its some derivatives on larger manifolds. Namely, assuming that the polynomial P with (13.2.2) exists, we will try to find polynomials of the sequence

i.e., express them in terms of V·, where we assume

(13.2.3)

We can easily obtain some information about the polynomials of V·- 1 at this point. In particular, we can easily find the polynomial

j = 0, ... , IC.>.I - s - 1.81, (13.2.4)

for arbitrary proper manifolds A C A, A E C6, A E C6-1, and multiindex .8 E ~+-1, 1.81 ~ ICAI - s + 1. In fact,

pf,A,j = L c","(P,>,,","(, "'"(EZi-, hl=I.8I+j

(13.2.5)

Ch. 13, § 13.2] Polynomial Interpolation by Traces on Manifolds

where the coefficients Cy are to be found from the relation

D~.LD{.L(A) = L c-yDI.L·

-yez+, hl=IPI+;

213

If the above A is improper, which in this case means A = Aoo, then we can determine the polynomial

P p '- D'.' P Aoo,A,;'- 00 A,p, j = 0,. ·.,1£>.1- s -1,81, (13.2.6)

using the equality

(13.2.7)

where a A is the vector 00 A' which lies on A.l:

Now, in order to combine the obtained information about the fixed PA,p E 1)6-1 with proper A, we associate with it the following collection of manifolds:

£"(A) = £"(A, 1t) := P E £" : A C A}, v = s,s + 1.

The collection £6(A) can contain at most one improper element, viz. Aoo. In any case we set

and /l .- {/lA' 00'- 0 00

if Aoo E £6(A), otherwise.

We call /l~1 the 1,8I-multiplicity of A and it corresponds to the upper limit of j in (13.2.4) or in (13.2.6). Next, denote

Ii>. = 1i~1 := I{H E 1t : A C H, Act. H}I,

and, for 1 E£6+l ,

Ii(A):= L Ii>., >.e.c·(A)

/l(A):= L /l>. >.e.c·(A)

£i(A) := P E £6 : 1 CAe A},

1i,(A) = liIP1(A):= L Ii>., >.eq(A)

Note that

/l,(A) = /lIPI(A):= L /l>.. >.eq(A)

214 Multivariate Polynomial Interpolation Arising by Hyperplanes [Ch. 13, § 13.2

Let us consider now the data on manifolds of .c'(A), i.e., the polynomial sequence

V' (A) = V',P(A) := {P>.,; : A E .c'(A), j = 0, ... ,1'>. - 1},

where P>.,; := pf,A,j and let

be the unit normal of A in A. The interpolation problem here is to find a polynomial q E 7rn (A) (which is sup

posed to be PA,p, see (13.2.4» such that

for all P>.,; E V' (A). (13.2.8)

This problem as compared with the problem (13.2.2) is much more simple. Indeed, the multiindex a and the system of vectors of Ai in (13.2.2) are replaced here by the nonnegative integer j and the single vector n>.. Actually we have here the case of co dimension one, since the co dimension of manifolds A E .c'(A) in A equals to one.

Essentially the method, we use here, is to reduce the problem (13.2.2) to a number of problems of the type (13.2.8), i.e., to the problems of co dimension one.

In order the problem (13.2.8) (or (13.2.2» to be regular, we need some conditions on data, which we will call consistency conditions, to be satisfied. For instance, if we have manifolds A1, A2 E .c'(A) with 1 = A1 n A2 E .c,+1 to be proper then, of course, we need the condition P>'l,oh = P>'2,ol, for the problem (13.2.8) to be regular.

This is the essence ofthe consistency condition (a), which we will introduce bellow. The second consistency condition (b) concerns the case of improper 1 (i.e., the case

when A1 and A2 are parallel) and guarantees, for instance, the relation

The third consistency condition (c) arises only if Aoo E .c'(A) and is similar to the previous one. In order to introduce the consistency conditions in full generality, denote

and, for 1 E.c,+l, v,(A) = vlPI (A) := min {w, J.t,(A) - 2},

where w := J.t,(A) + rnA + uA - 2 = l.cd - (s + 1) -1,81 + UA ·

DEFINITION 13.6. We say that the sequence of polynomials

with 1 ~ s ~ k - 1, ,8 E fZ~-l, 1,81 ~ I.cAI- s + 1,

is consistent in proper A E .c,-1 iff the following conditions (a), (b), (c) hold: (a) For every proper 1 E.c,+l(A) there exist polynomials

v = 0, ... , v,(A),


satisfying the relation

. fJ II! "-i I u"z. Q",,(t)= (11 _ i)! D, P>',i, (13.2.9)

for ~ E .cHA), t E ~o n 11. (A), 0 ~ i ~ min (11,1'>. - 1). (b) For every improper I E .c'+1(A) there exist polynomials

fJ 1.,0 Q"" E 11",,(1 (A» , 11 = 0, ... , v,(A),


(13.2.10)

for each proper ~ E .c;(A), t E ~o n 11. (A), i = 0, ... , min {II, 1'>. - I}.

(c) If ~ = Aoo E .c' then the polynomials Qf" defined in (b) satisfy the conditions ,

(13.2.11)

forO~i~lI, O~II~J.'>.-l.

Remark 13.6.1. It is not difficult to check that the conditions (a), (b) and (c) are necessary for the interpolation problem (13.2.2) to be regular. Indeed, if there exists a polynomial P satisfying (13.2.2) then the polynomials Qf" can be chosen M~~ ,

in CMe (a):

(to check (13.2.9) it is sufficient to use the relation DaF(z) = mDaD,;,-l f with F(z) = Dr;' f)j

in CMes (b) and (c) (see (13.2.1»:

Qr,,,(t) = D~Q'(t),

Remark 13.6.2. It is evident that all the interpolating parameters in the righthand sides of (13.2.8) and (13.2.10) (with fixed 11) are uniquely determined by 11+1 hermitian parameters of them. This follows by the univariate Hermite interpolation (on the line in CMe (b) and on the circle by homogeneous polynomials in CMe (a), see Theorem 13.15 below for k = I).

This clarifies also that conditions (13.2.9) and (13.2.10) do not impose any restriction in the CMe 11 > J.',(A) - 2.

DEFINITION 13.7. We will say that V',fJ(A) is fully consistent if the condition (c) holds and conditions (a) and (b) hold for 11 = 0, ... ,J.',(A)-2 (i.e., v,(A) ~ J.',(A)-2).

Let us now consider the case s = k (pointwise interpolation) and let A E .ck- 1 be a finite line with directional unit vector dA •


DEFINITION 13.8. We will say that the sequence of numbers

with P E 7l~-1, IPI ~ I£AI- k + 1,

is consistent in finite line A E £k-1, if I' (A) ~ n + 1 - IPI or (d) there exists a polynomial PA,P E 1I"n-I'oo-IPI(A) satisfying the equality

(13.2.12)

for j = 0, . " ,I'). - 1 and proper point ,\ E £k(A), where

(13.2.13)

Let us notice that in this case P). E lR and the number of conditions in (13.2.12) equals to J.l(A) - Poo. Hence for p(A) ~ n + 1 - IPI, according to the Hermite interpolation, there exists a polynomial PA,p (unique in the case J.l(A) = n + 1-IPI) satisfying the conditions (13.2.12). It is clear that the polynomial

has the following properties:

for P).,i E Vk (A), ,\ is proper, (13.2.14)

and ( p. )[n-ij = pP .

A,p Aoo ,l for j = 0, ... ,1'00 - 1. (13.2.15)

Remark 13.8.1. The necessity of condition (d) for the interpolation problem (13.2.2) to be regular is evident. The choice in this case is:

In order to reduce the definition of consistency in improper A E £.-1, 2 ~ s ~ k, to the finite case, denote

11.0 := {HO : H E 11.}.

Note that there is no improper element in £1t 0 ' Consider the following polynomial sequence on £1tO

DEFINITION 13.9. We will say that V·,P(A) is consistent in improper A, if the sequence V~~ (A 0) is consistent in A 0, which in this case is equivalent to the condition (a) only.

Ch. 13. § 13.2) Polynomial Interpolation by Traces on Manifolds 217

DEFINITION 13.10. The sequence V' is said to be consistent ifthe sequence V',f3(A) is consistent in A for every

A E C,-1,

THEOREM 13.11. Let V' = 1>1-£ be the polynomial sequence introduced earlier for the fixed s = 1, ... , k. The necessary and sufficient condition for the existence of a polynomial P E 1Tn(lRI:) such that

(13.2.16)

is the consistency of the class V' . The polynomial P is unique iff 11t1 ~ n + s. Our next purpose, is to complete the proof of Theorem 13.11 in the case s = 1, k ~

2, (the case k = 1 reduces to the Lagrange-Hermite univariate interpolation). Let

'1.1 {H H} { L1 , •• • , Lr } n= b"" I' = , 1'1 I'r

where we assume H1 = L1 , HI' = Lr (the hyperplanes Li are different with multiplicities I'i and with the unit normal ni) and

V(1t):= V~ = {Pili : j = 0, ... ,I'i -1, i = 1, ... , r}.

(Here we have 1'00 = 0). Actually, we will discuss in the case s = 1 a more general setting; namely, we put

I'i =: lii + m, i = 1, ... ,r,

where m is an arbitrary integer with

and, respectively,

O~m< min I'i 1:E;;i:E;;r

u := (n + 2 - li - m)+· sign m r

with li = Llii' i=1

THEOREM 13.12. Let V(1t) be consistent {i.e., conditions (a), (b) with A = ]RI:, s = 1 and (J = 0 hold). Then there exists a polynomial P E 1Tn such that

j=0, ... ,l'i-1, i=l, ... ,r. (13.2.17)

The polynomial P is unique iff 11t1 ~ n + 1.


Proof. The uniqueness of the polynomial P in the case I-' ~ n + 1 follows from Lemma 11.16. On the other hand, if I-' ~ n the polynomial

II e(·, H) E 1I"n(I~k) He'H

satisfies the condition (13.2.17) with Pi,; = 0 for all i, j, which means that the polynomial P satisfying (13.2.17) is not unique.

Let us construct now the interpolating polynomial P in the case s = 1. First we consider the case I-' ~ n + 1. We have

u = (n + 2 - Ji - m) . sign m,

since I-' = Ji + rm ~ Ji + m. In the case m ~ 1 this implies

Ji( I) + m + u - 2 = Ji( I) + n - I-' ~ Ji( I) + I-' - Ji - 1 ~ Ji( I) + rm - 1 ~ 1-'(1) - 1

and therefore V(C) is fully consistent in jRk.

For proper L in jRk of dimension k - 1 and q E 11" n (L), denote by ij the polynomial from 1I"n satisfying the conditions

Let us prove, by induction on 1-', that there exist polynomials qi E 1I"n-i+b i = 1, ... ,1-', such that the polynomial

satisfies the conditions (13.2.17). For I-' = 1 we put P l = ql = Pl,o. Let ql, ... , qp-l

be known and suppose Pp - l satisfies the conditions:

IY.a;~-ll = Pi,;, Lj

j = 0, ... , Jli - 1, i = 1, ... , r;

(i,j) i= (r,l-'r -1).

It is sufficient to find a polynomial qp E 1I"n_P+1(jRk) such that

D~~-l [pp- l + e(· , Hd·· ... e(· , H p-d· qpllLr = Pr,Pr-l·

It is clear that

and therefore (13.2.19) reduces to

- Po DPr-lp I - r,Pr- l - nr p-l Lr

•

Lr

(13.2.18)

(13.2.19)

(13.2.20)

Ch. 13, § 13.2)

Let

Polynomial Interpolation by Traces on Manifolds

10 := {i : 1 ~ i ~ 1', H? = H~, H; =/; HI'}'

It := {i : 1 ~ i ~ 1', H?=/;H~},

0;:= 11;1, i=O;1.

219

Now the condition (b) of full consistency of V(Ji) in ~k, Remark 13.6.1 and the induction hypothesis yield:

Pr,l'r-1 - D~;-IPI'-IILr E 1rn-l'r-6o+1(Lr).

Indeed, according to (13.2.20) for v = 0, ... ,00 + I'r - 2, the 00 upper homogeneous components of Pr,l'r- 1 are uniquely determined by means of the quantities of the right-hand side of (13.2.18). On the other hand, the polynomial

D l'r- l p I nr 1'-1 Lr

has the same 00 upper components, because of the necessity of conditions (13.2.10) (see Remark 13.6.1) and (13.2.18). Denote

{L · n L .' I} -' { It, ... , lr' } • I' . z E 1 -. , VI Vr'

r'

LV; = 01. ;=1

According to condition (a) of consistency and (13.2.18), as in the previous case, we have

i=0, ... ,v;-1, i=1, ... ,r'.

Hence, by Lemma 11.16 we get the factorization:

where q E 1rn _I'+I(Lr ) since 00 + 01 + J.tr = 1'. It is evident that we will have (13.2.19), taking

where

Consider now the case I' ~ n + 2. We start with choosing a sub collection

r

LO';=n+1, ;=1


such that

i=l, ... ,r, if ii~n+1 (case 1),

and

i= 1, ... ,r, if ii~ n+ 1 (case 2).

Let us check that the polynomial sequence V(il) is fully consistent in ]RA:. This is obvious in case 1. For case 2 it is sufficient to notice that

smce

r

~)Ui - iii) = n + 1- ii < m + u, i=1

i) if u = 0, i.e., n + 2 - ii - ffi ~ 0, we have n + 1 - ii ~ m - 1, ii) if u 10, i.e., n + 2 - ii - m = u, we have n + 1 - ii = m + u - 1.

Hence V(C) is fully consistent in ]RA: and using the previous construction we can find a polynomial P E 11" n, satisfying the conditions:

~iplLi = Pi,;, i = 0, ... , Ui - 1, i = 1, ... , r.

Let us prove that P is the desired polynomial, i.e., conditions (13.2.17) with P = P are satisfied. Let

Pr,;o E V(1i),

Assume, that (induction on io)

~rPILr =Pr ,;, i=O,···,io-1.

Let us define for 1i similar to the case of 1i :

fo := 10 n {i : Hi E il},

Then by the condition (b) of consistency of V(1i), we have that

Pr,;o - ~orP E '1rn-;o-'1(Lr),

where

Precisely, we have

or 11 = L iii + (iir + m + u - io - 1). ie10

(13.2.21)

in case 1,

in case 2.

Ch. 13. § 13.2) Polynomial Interpolation by Traces on Manifolds 221

This follows from the fact that in case 1) the sub collection

{ L1, ... ,Lr - 1 , . Lr }

0'1 O'r-1)0 + 1

is fully consistent, since jo + l-lir ~ I'r -lir = m and in case 2) it is fully consistent iff

2:( O'i - Iii) + jo + 1 - lir ~ m + u. iEi

Let us denote now

{H.nH .' [- }_{i1 , ... ,lro} • p.IE 1 - I' I' '

.. 1 .. ro

ro

2:(i = 61 ,

11>=1

and ni = it(H), i = 1, ... , ro . Then (13.2.21) and the consistency of V('Ji) imply

where for

we have

D'-n' . [Pr . - D!-.0 p] 1_ = 0, l '0 r Ii

j=0'''''{i-1, i=l, ... ,ro,

or {~2:lii+(Jlr +m+u-jo -1). iEi1

Now it is not difficult to check that

7]+{ = 2: O'i + 2: O'i=n + 1 - O'r

iEio iEil

or

7] + { = 2: Iii + 2: Iii + (lir + m + u - jo - 1) = Ii + m + u - jo - 1. iEio iEil

This combined with i) and ii) implies

7] + {~ n + 1- jo, i.e., {~n - jo - 7] + 1.

The latter, in its turn, yields p. . - Dio P-

r,Jo - nr .

Thus the proof of the Theorem 13.12 is completed. Repeating the proof of Theorem 13.12 for a particular case we get

222 Multivariate Polynomial Interpolation Arising by Hyperplanes (Ch. 13, § 13.2

COROLLARY 13.13. Let hyperplanes of1{. be all finite and contain the origin, i.e., 1{. = 1{.o and let the polynomials ofV(1{.) be all homogeneous:

V(1{.)={Pi,; E 1r~_;(I~1:): j = 0, ... , Jli - 1, i = 1, ... , r}.

Then there exists P E 1r~(~1:) satisfying the conditions (13.2.17) if and only ifV(£) is consistent (which, in this situation, reduces to condition (a) for s = 1, A = ~1: and f3 = 0).

Polynomial P is unique iff Jl ~ n + 1 .

Proof of Theorem 13.11. We will prove the sufficiency by induction on n + k. In the case Jl < n+8, we start by adding a hyperplane L to the collection 1{. and polynomials on manifolds of £'(L, 1{. U {L}) to the sequence V 1t , such that the consistency of the resulting polynomial sequence still holds. We choose the additional hyperplane L such that

i) L does not contain any point from £1:, ii) £1:-1 n L consists of only finite points. In the case 8 = k, we define values and corresponding derivatives of the desired

polynomials on (proper) points of £'(L, 1{. U {L}) arbitrarily. Now let us have

).' E £'(L,1{.U {L})\£', 8~k-1, and IE £,+1 ().',1{.U {L}).

The above assumption then implies that I = ). n )" with ). E £'. We first define polynomials on I as follows

where the coefficients cj ,a are found from the relation

DiJ..(AI) = L cj,aD{J..(A)DfJ..· Hlal=i

(13.2.22)

In the case 8 ~ k - 2, the consistency on A of polynomials just defined clearly follows from the consistency of V 1t , while for 8 = k - 1 the consistency condition (d) coincides with univariate Hermite interpolation by polynomials of degree ~ n with Jl + 1- (k - 1) ~ n parameters. Hence, by induction hypothesis we can define a polynomial PAl,o E 1rn(N) such that

forall IE£'(L,1{.U{L}), i~l£d-s-i.

In its turn this condition ensures (since (13.2.22) holds and I£d - s -1 = 1£ AI- s for above I,).) full consistency along 1 = ). n A and therefore the consistency in general of the resulting class V 1tU{L}'

Hence we can, without loss of generality, restrict ourselves to the case Jl ~ n + s.


Now, on account of Theorem 13.12, to complete the proof it is sufficient to construct a polynomial class V&-l = V~-l such that for each P E 1I"n(I~k) the following conditions are equivalent:

1) D~..LpIA = PA,o for all PA,o E V',

2) D~..LpIA = P A,/3 for all PA,fj E V6 - 1 •

We start the construction with the case 1 ~ 8 ~ k - 1. Let us define PA,fj E V 6 - 1 ,

where A E [,'-1 is proper and 1,81 ~ ICAI- 8 + 1. If Aoo rt c' then we find PA ,/3 E 1I"n-I/3I(A) according to Theorem 13.12 by the conditions

. I /3 q..L(A)PA,/3 A = PA,A,j for all j ~ ICAI- 8 -1,81, A E C'(A). (13.2.23)

In the case Aoo E C6 the relation (13.2.7) on account of Remark 13.6.1 uniquely determines the ICA ... 1-8-1,81+ 1 highest homogeneous components of PA,/3 , whose sum we

denote by P A,/3' After this, we define the second polynomial PA,/3 E 1I"n-ICA ... I+6-b according to Theorem 13.12, by the conditions

. - fj -fJ I q..L(A)PA,/3 = PA,j - P A,/3 A' j ~ ICAI- 8 -1,81, A E C'(A). (13.2.24)

The relations (13.2.23) and (13.2.24) determine PA,/3 and PA,fj uniquely, since in the first case we have

IC6(A)1 ~ I'HI-ICAI + [lCAI- 8 + 1-1,81] = I'HI- 8 -1,81 + 1 ~ -1,81 + 1,

and in the second case

For the polynomial PA,/3 = P A,/3 + PA,/3 we have

for all j ~ ICAI- 8 -1,81, A E C'(A), (13.2.25)

and ptij]=pf ... .i j ~ ICA ... I- 8 -1,81· (13.2.26)

In the case of infinite A E C,-I, 2 ~ 8 ~ k, 1,81 ~ ICAI- 8 + 1 (then all A E C'(A) are infinite too) conditions (13.2.23) uniquely determine a homogeneous polynomial of degree n - 1,81 on A 0 according to Corollary 13.13 since

IC6(A)1 ~ I'HI-ICAI + [lCAI- 8 + 1-1,81] = I'HI- 8 -1,81 + 1 ~ n -1,81 + 1.

In the case 8 = k, besides the above-mentioned polynomials on infinite hyperplanes, the sequence V,-l consists of additional polynomials on finite hyperplanes:

for A E Ck - 1 - proper, ,8 E IZt-1, 1,81 ~ ICA\- k + 1.


Now let us check the consistency conditions for V 8 - 1• Let V EC·-2 be finite. Then for)' E C8 (V) we have an order of consistency

(13.2.27)

since U v = O. For the checking of condition (a), in view of (13.2.4), (13.2.23) and (13.2.27) we take

Q~,,,(t) = L CaP).. ,a , t E ).1. (V),

1011="

where the coefficients COl = Ca(t) are to be found from the relation

D~.J.Dr = L caDr.J.(V). 1011=,,+1.81

The condition (b), in view of (13.2.27), is verified with

). E c"(V) - improper,

where -.8

P" = P).."",,,'

Here we use the relation (13.2.26) in the case 1 ~ s ~ k -1 and (13.2.15) in the case s = k . The condition (c) follows from the relation (13.2.23) and the consistency condition (b) of V' if 1 ~ s ~ k - 1, and from the relations (13.2.13), (13.2.15), if s = k.

In the case of improper V EC"-2, we need only to check the condition (a) for V 8 (VO), which can be done as for the previous case by using the relation (13.2.27). Now the implication 1) => 2) readily follows from the uniqueness of determination of V·- 1 while the opposite implication is ensured from the way V8- 1 is constructed. Indeed, to determine P)..,a E V8, ). E C', lal ~ IC)..I - s, by means of the polynomials V 8 - 1 , we consider the collection of (s - 1)-dimensional manifolds: CAn)...J. in the space ).1., dim).1. = s and the interpolation of degree i ~ ,,(1) - s by traces {D:PI).., t E ).1.} obtained with the help of induced polynomials on lines {CAn)...J.}8-1.

Obviously, the uniqueness condition in Theorem 13.11 is satisfied here, since it transforms to the inequality ,,().) ~ i + (s - 1).

This completes the proof of Theorem 13.11.

§ 13.3. Special Cases and Consequences

13.3.1. Interpolation on the sphere by homogeneous polynomials. The analogue of the considered interpolation on the sphere will be obtained applying Theorem 13.11 for the collection 1£ with 1£ = 1£0 and class V' consisting of homogeneous polynomials. This case is much simpler, since here we deal only with finite objects.

Ch. 13, § 13.3] Special Cases and Consequences 225

Let S be an origin-centered sphere in ~k. A hypersphere of dimension v, 0 ~ v ~ k - 1, is defined as the intersection

I:= LnS with {O} E L, dimL = v + 1.

Let E= {Ill ... ,I,,}.

By .c!, 0 ~ 8 ~ k - 1, we denote the set of all hyperplanes of (k - 1 - 8 )-dimension being intersections of hyperplanes from E. Define

1I'm(I) := {P II: P E 1I'~(L)}.

Let f be defined on S, y be a tangential direction to S, then

DyfO = f(· + ty) - f(·), t

where (. + tY) is the intersection of S and the line between (. + ty) and the origin. For hyperspheres ; c I and multi-index a = (ai, ... , a.) E IZ+, 8 = dimL - dim;,

D C)( f' DC)(l DC)(' f 1.l.(L) .= 't(L) ... I-}(L)·

Suppose we have the polynomial class

The notations and definitions used in this part without especial mention are similar to those used previously despite the simplicity caused by the absence of infinite cases.

DEFINITION 13.14. The class V(E·) is said to be coordinated iff the class V(.ct) is coordinated in A for every

The latter, in the cases 0 ~ 8 ~ k - 2 and 8 = k - 1, means that the conditions (a') and (d') hold, respectively:

(a') for every; E EX+! there exist polynomials

Q111 E 11'~ (/.L (A)) i, . Ifjl -v=O, ... ,vA (I),


forl C Xi, 1 E X n 1.L (A), 0 ~ j ~ min (v, Jli - 1), i = 1, ... , r;


(d') if A E Ck - 2 and I'~ > n + 1 - 1,81 then there exists a polynomial

PX,f3 E 1I"n-If31 (A)

satisfying for i = 1, ... ,q, j = 0, ... ,(I'i - 1 - 1,81)+ the following relation

Dj'APX,f3(ti) = Pi,i·

THEOREM 13.15. Let 1I"(C"), ° ~ s ~ k-1, be the introduced polynomial class. The necessary and sufficient condition for existence of a polynomial P E 1I"n(8) such that

is the coordination of the class 1I"(C"). The polynomial P is unique iff I' ~ n + s.

13.3.2. Hermite interpolation. It is not difficult to get from Theorem 13.11 (with s = k) a pointwise interpolation given by Theorem 13.3, where there does not appear any coordination conditions. Of course, on account of Theorem 13.11 one can easily omit the restriction C is admissible, i.e., every k hyperplanes from 1i have exactly one common point, considering the case of infinite points too.

13.3.3. Tensor-product interpolation. Now we choose the following collection of distinct hyperplanes:

1i = {Li,j: i = 1, ... , k; j = 0, ... , nil,

where Li,j is given by the equation Xi = ai,i. The only infinite points of Ck are I~, where Ii is i-th axis of ~k, I'(/~) = n - ni. It

is not difficult to verify that interpolation conditions of P E 1I"n(~k) on this infinite point gives the coefficients of its monomials from 1I"n(~k)\1I"n(~k), where

1I"n(~k) = { ~ c"x": c" E ~}, "~n

Therefore, taking these conditions to be equal to zero we attain to tenzor-product interpolation:

for an arbitrary real number set

T = {c" E~: Cl' E /Z~, Cl' ~ n}

there exists a unique polynomial P E 1I"n(~k) such that

for all c" E T.

Using Theorem 13.11 one can consider mixtures of considered interpolations too.

Ch. 13, § 13.3) Special Cases and Consequences 227

For example, if n1 + 1 hyperplanes of1l coincide with the above L 1,j, j = 0, ... , nl,

and the collection 1l\{L1,j}j~1 is admissible then we get a correctly defined interpolation with the polynomial class

and the finite points of £k.

13.3.4. Finite element (F-E) interpolations. Here, we will discuss the cases k = 2 (in details) and k = 3 of F-E interpolation, i.e., interpolations on the triangle 'I' and pyramid '.P.

In (n, v) F-E interpolation (n is degree, v is smoothness) the following parameters are given in the way to assure a smoothpasting (belonging to the space CV) of interpolant polynomials along the common side of adjacent elements in a triangulation:

i) values of the polynomial P E 7rn(1~k) and its derivatives (up to order no) at the vertices of 'I'('.P) ,

ii) values of the polynomial and its normal derivatives to the sides of 'I' (faces and sides of '.P),

iii) values of the polynomial and its derivatives (up to order c) at an interior point (center) of'I'('.P).

Let us start with the case k = 2. The above setting requires to consider the interpolation by traces of the polynomial P E 7rn(l~k) and its normal derivatives up to order v on each side of the triangle, i.e., £ consists of three lines: Lo, L1, L 2 , each of them of multiplicity v + 1 and s = 1.

In this case, coordination conditions are reduced to (a) on the vertices of'I'. Since interpolating parameters are arbitrary, we must have full coordination up to the order of derivative:

no ~ 2v (13.3.1)

at each vertex of the triangle. Again, since the values of the parameters are arbitrary, the above coordination will

be guaranteed iff the collection of interpolation parameters includes values of P and its no derivatives at the vertices of '.P. To complete the collection of parameters one must supply to these parameters the values of P and its normal derivatives to the sides of the triangle until the determination of all traces participating in the above interpolation is unique.

It is obvious that we have the following necessary condition too:

2no + 1 ~ n. (13.3.2)

Therefore, (13.3.1) and (13.3.2) imply the following necessary condition for the existence of (n, v) F-E interpolation:

n ~ 4v+ 1. (13.3.3)

Moreover, we obtain the general construction of F-E interpolations in the case k = 2.


Let n = 3(v + 1) + c; c~-I, v~O. (13.3.4)

We put r = n+1-2(no+l), where no satisfies the conditions (13.3.1) and (13.3.2). Then we take the parameters of i), iii) and normal derivatives of order i at i + r

points of each side of triangle, for i = 0, ... , v. One can easily verify, using Lemma 11.16 and the relation (13.3.2), that the

resulting interpolation is correctly-defined if the number of described parameters is equal to dim 1rn(l~k) and that the latter holds iff no = 2v or no = 2v + 1.

Let us note that the interpolating polynomial P E 1rn(JW.k) can be found by the formula

where P,! is determined by the parameters on the sides of '.t according to Theorem 13.12, then Pe E 1re(JW.k) is determined by the Taylor interpolation using its parameters at the center, obtained by the above formula from similar ones of P already given.

Thus we obtain that the condition (13.3.3) is necessary and sufficient for the existence of(n,v) F-E interpolation on '.t.

Some examples are given in Fig. 13.1.

n = 5, v = 1,

no = 2, c = -1 (the Argyris triangle)

Fig. 13.1.

n = 9, v = 2,

no = 4, c = 0

In the above figures we denote the values and derivatives by points and circles, respectively, and (multiple) normal derivatives by (multiple) arrows.

Ch. 13, § 13.3) Special Cases and Consequences 229

In the case k = 3 we consider the similar interpolation by traces on the faces of ~, hence again s = 1. The analogue of (13.3.1) here is:

(13.3.5)

where nl is the derivative order of full coordination at each side of the pyramid, obtained from condition (a).

We must look for a plane (n, nt) F-E interpolation, therefore, the condition (13.3.3): n ~ 4nl + 1 combined with (13.3.5) implies that the condition

n ~ 811 + 1

is a necessary condition for the existence of space (n, II) F-E interpolation. The condition (13.3.4) in this case is replaced by:

n = 4(11 + 1) + c; c ~ 1, II ~ 0,

and the construction of F-E interpolations is carried out as in the plane. Here are some examples.

n = 3, II = 0,

no = 0, c =-1

Fig. 13.2.

n = 9, II = 1,

no = 4, nl = 2c = 1

Here, the parameters at the frontal side and at the center are omitted. Circles (with straight lines) denote derivatives in the faces (in the space) and arrows denote normal derivatives to the sides and faces of~.

230 Multivariate Polynomial Interpolation Arising by Hyperplanes (Ch. 13, § 13.3

Notes and References.

The Lagrange case of Theorem 13.3 (part ii» was proved by Chung and Yao [1977], the Hermite case is due to Hakopian [1984b]. The interpolation by traces on hyperplanes considered in Sections 13.2 and 13.3 were studied by Hakopian and Sahakian [1989b] (see also Zenisek [1974], and Le Mehaute [1984] for F-E interpolation).

Chapter 14

MULTIVARIATE POINTWISE INTERPOLATION


In this chapter, we consider the problem of interpolation of values of a function and its partial derivatives by multivariate polynomials from a certain finite-dimensional space. The interpolation problem consists of the following components:

a) the space of polynomials

1I"(S) = {P: P(x) = P(Xl, ... ,Xk) = L aax~l .. . X~k}, a=(al •... ak)eS

where S C /Z~ is a finite normal set, i.e., a E S, fJ E iZi and fJ ~ a imply fJ E S; b) collection of sets

6

Hv C Rk, 11£1:= LIHvl = lSI = dim1l"(S); (14.1.1) v=l

c) set of nodes

Zv E Rk, zv::j; z/I for v::j; 1', v, I' = 1, ... , s. (14.1.2)

DEFINITION 14.1. The Birkhoff interpolation problem (1l, S, Z) is to find a (unique) polynomial P E 1I"(S) satisfying the conditions

aEHv , v=I, ... ,s, (14.1.3)

for any given collection of values

A = {.~a.v E Rk: a E Hv, v = 1, ... ,s}.

From now on, for the brevity, we will write equalities like (14.1.3) in the form: D 1t Plz = A.

231

232 Multivariate Pointwise Interpolation [Ch. 14, § 14.1

If for every A there exists a unique polynomial P satisfying (14.1.3), then we say that the problem ('H, 8, Z) is regular or the interpolation scheme (1f.,8) is regular in Z, otherwise the problem (1f., 8, Z) is called singular.

If sets H" and 8 are triangles of the form

(14.1.4)

i.e., values of a function and its derivatives up to some order m" at the knot z" are interpolated, we obtain the case of Hermite interpolation and the Lagrange case when m" = 0, 11 = 1, ... ,s.

The conditions (14.1.3) on P can be expressed in the form of a system of 11f.llinear equations with 181 unknowns - coefficients of P (recall that 181 = 11f.1). By

d(z) == d"H,s(Z) == d"H,S(Zl, ... , z.) (14.1.5)

we denote the determinant of this system as function of nodes Zl, •.. , Z. and call it the Vandermonde determinant of the interpolation scheme (1f., S). It is clear that the following conditions are equivalent:

i) (1f.,8) is singular on Z, ii) d(Z) = 0, iii) there exists a polynomial P such that

P E '11"(8), (14.1.6)

Note that the determinant d(Z) is a polynomial of variables z" = (z",l, . .. , Z",k), 11 = 1, ... , s, and therefore, if d(Z) i= 0 for some Z, then d(Z) =F 0 for almost every Z (with respect to the Lebesgue measure in ]Rk.).

DEFINITION 14.2. For a given space '11"(8) and set 1f. ofform (14.1.1) we say that i) (1f., 8,) is regular if(1f., 8, Z) is regular for all Z, ii) (1f., 8) is almost regular if(1f., 8, Z) is regular for almost all Z, iii) (1f.,8) is singular if(1f., 8, Z) is singular for all Z.

Let us consider some examples:

Example 14.2.1 (Taylor interpolation). '11"(8) = 'll"n(]Rk), 1f. = {T:}, Le., the values of a function and its partial derivatives up to the order n are interpolated at one knot with polynomials of degree n. In this case, the interpolation scheme (1f.,8) is regular, since for every Z E ]Rk the Taylor formula

(14.1.7)

gives the unique polynomial P E '11"(8) = 'll"n(]Rk) satisfying conditions (14.1.3).

Example 14.2.2. The Lagrange interpolation in ]Rk with k + 1 knots, i.e., the values of a function are interpolated at k + 1 nodes with polynomials of first degree. It is clear that this scheme is almost regular but not regular. Moreover, the condition VOlk[Z] =F 0 is necessary and sufficient for the regularity of (1f., S, Z).

Ch. 14, § 14.1] Birkhoff Interpolation 233

Example 14.2.3. 1I"(S) = 1I"2(JR2), 1£ = {Tl, Tn, i.e., values of a function and its first partial derivatives are interpolated at two nodes. In this case, the scheme (1£, S) is singular since, for every two nodes Z1, Z2, the polynomial P = pl satisfies the conditions (14.1.6), where P1(x,y)= ax + by + c = 0 is the equation of the line passing through Z1 and Z2.

Exercise 14.2.4. Calculate the Vandermonde determinants of the previous three examples.

The class of regular interpolation schemes (1£, S) is small and is described completely by Theorem 14.6 below. The problem of the description of almost regular interpolation schemes is more complicated and is still open even in the case of Hermite interpolation in JR2. In fact, here we deal with two problems: to determine whether the scheme (1£, S) is almost regular and if so, to describe the set of nodes Z for which (1£, S, Z) is regular. We are interested, in this section, in the first problem (about the second one, see Chapter 12).

DEFINITION 14.3. The scheme (1£, S) is said to satisfy the P6lya conditions iff

• 2:IH"nNI ~ INI (14.1.8) ,,=1

for every normal set N C S.

In the univariate case P6lya conditions are necessary and sufficient for the almost regularity of (1£, S) (see Nemeth [1966], Ferguson [1969]). In the multivariate case, these conditions are only necessary (see Example 14.2.3).

THEOREM 14.4. Every almost regular Birkhoff scheme (1£, S) satisfies the P6lya condition (14.1.8).

For the proof we need the following simple lemma:

LEMMA 14.5. For the arbitrary collection of sets

• H"C]Rk, with 2:IH"I<ISI=dim1l"(S) (14.1.9)

,,=1

and the set of knots Z = {z" }~= 1 C ]R2 there exists a polynomial P1( = P1(.z such that

Proof. Indeed, the statement of the lemma reduces to a linear homogeneous system

equations with lSI unknowns, which by virtue of (14.1.9) has a nontrivial solution.


Proof of the Theorem. Assume that the inverse inequality

, LIHvnNI < INI v=l

holds for some normal set N C S. Then, by Lemma 14.5, for every set of nodes Z = (Zl, . .. ,z,) there exists a polynomial P E 7r(N), P t= 0, such that

for o:EHvnN, v=1, ... ,s.

Hence, on account of the normality of the set N we have

for 0: E S\N,

i.e., the polynomial P satisfies conditions (14.1.6) from what which follows the singularity of (1f., S).

Exercise 14.5.1. Check that the P6lya condition holds for every Hermite interpolation scheme.

In contrast to the univariate case, where the problem of the description of regular matrices up to now is open (see Section 1.4), in the multivariate case the following theorem gives the characterization of all regular schemes:

THEOREM 14.6. The Birkhoff interpolation scheme (1f., S) is regular if and only if the sets H v, v = 1, ... ,s, are disjoint.

Proof. Let us first show that any interpolation scheme (1f., S) with disjoint Hv is regular. In this case

and we order the rows of the Vandermonde determinant d1i,s(Z), i.e., the points 0: = (0:1' •.• ' O:k) of the set S in the following way. First, we order the rows according to increasing 10:1, among those 0: having the same 10:1 - according to increasing 0:1> among those with the same 10:1 and 0:1 - according to increasing 0:2 and so on. With this ordering, the determinant d1i,s(Z) will be an upper triangular with diagonal elements of the form 0:1! •••• O:k! Hence d1i,s(Z) never vanishes and the scheme (1f., S) is regular.

Assume now that (1f., S) is regular. Let us consider the following additive measure for the sets A C Z~:

, v(A) = v1i(A) := L IHv n AI,

v=l

The conclusion of the theorem then can be formulated as

v(N) = INI (14.1.10)

Ch. 14. § 14.1) Bir1chofT Interpolation 235

for any normal set N C S. By Theorem 14.4, the interpolation scheme (1£, S) satisfies the P6lya condition, which means that v(N) ~ INI for any normal set N C S. Therefore, we need only to prove that

v(N) = INI (14.1.11)

for any normal set N C S. Let us prove first that the inequality (14.1.11) holds if N is an arbitrary set in the form:

N = Si('Y) := {a E S : ai ~ 'Yi}, i = 1, ... , k, 'Y = ('Y1, ... , 'Yk) E S.

Suppose the inverse inequality

v(S;('Y» > IS; ('Y)I (14.1.12)

holds for some 'Y E S, j = 1, ... ,k. Then v(S\S;(-y» < IS\S;('Y)I and, by Lemma 14.5, for the arbitrary set of nodes

with xi = 0, v = 1, ... , s

there exists a polynomial P E 7r(S\S;('Y», P 1= 0, such that

aE(Hv\S;('Y»), v=I, ... ,s. (14.1.13)

On the other hand, since P E 7r(S\S;('Y» then

P(X1, ... ,Xk) = x? P1(X1, ... , Xk)

and hence a E (H v n S; (-y») , v = 1, ... , s.

This, combined with (14.1.13), shows that the polynomial P satisfies (14.1.6), i.e., (1£, S, Z) is singular, which is in contradiction to the regularity of (1-1., S).

Let now N1 and N2 be any normal subsets of S satisfying (14.1.10). Then their sum and intersection are also normal subsets of S and, according to the P6lya condition, we have

Therefore

v(N1 u N2) = v(Nt} + v(N2) - v(N1 n N2) ~ IN11 + IN21-IN1 n N21

= IN1 U N21 ~ v(N1 U N2)'

which means that (14.1.10) holds for the sets N1 U N2 and N1 n N2. Now, using the equality

k

N = U n Si('Y), 'YEN i=l

we receive (14.1.10) for an arbitrary normal set N C S.


§ 14.2. Shifts of Sets and Differentiation of the Vandermonde Determinant d1{. (Z)

In this section, we will discuss shifts of sets and their connection to the derivatives of the Vandermonde determinant d1{.(Z). Shifts will be helpful in the investigation of almost regular interpolation schemes.

Throughout this section, k = 2, n E Z~ and

Tn := T; = {a = (al,a2): a E Z!, lal = al +a2 ~ n}.

For the arbitrary set He Tn by Rl(H) := R(H) denote the class of all subsets of Tn, which results from moving a point (i,j) of H to the right, to the position (i + 1,j) provided that (i + 1,j) E Tn \H, and let Ul(H) := U(H) be the class of all subsets of Tn, which results from moving a point (i,j) of H up, to the position (i,j + 1) E Tn \H.

For example, if H = T2 ,n = 3, then R(H) = {Hl,H 2 ,H3 } (see Fig.14.1)

• • • • •

H

Fig. 14.1.

Note that the classes R(H) and U(H) can be empty. Next, we define multiple shifts by induction:

RP(H) = U R(Q), Uq(H) = U U(Q) QERP-l(H) QEUq-1(H)

and mixed shifts UqRP(H) = U uq(Q),

QERP(H)

where p, q = 1,2, ... , RO(H) := UO(H) := {H}. Denote by Y(H,p,q) = Yn(H,p, q) the set of all collections of shifts of H:

Y(H,p,q): = {Q = (Qo, ... ,Qp+q): Qo = H; Qi E R(Qi-l)

for i = 1, ... ,p; Qi E U (Qi-l) for i = p + 1, ... ,p + q}

(14.2.1)

(14.2.2)

(14.2.3)

Ch. 14, § 14.2] Shifts of Sets and Differentiation of Vandermonde Determinant 237

and by Y(H -+ H') - the set of collections which transfer H into H' E UIJ RP(H):

Y(H -+ H') = {Q E Y(H,p,q): QP+IJ = H'}. (14.2.4)

Let the point a E H transfer to the point

7"(a) = (7"1 (a), 7"2(a)) = 7"(a, H, Q) = h(a, H, Q), 7"2(a, H, Q))

of the set QP+IJ after the shifts with respect to the collection Q = (Qo, ... , QP+IJ) E Y(H,p, q).

The mapping 7"( a) has the following properties: for arbitrary a = (i,j), a ' = (i',i') E H,

a) if i<i', j=j', then 7"l(a,) < 7"2(a), b) if 7"l(a) = 7"l(a' ), 7"2(a) < 7"2(a' ), then j <j'.

(14.2.5)

To check this it is sufficient to follow the movement of the point a E H, with respect to the collection Q and note that

c) when the point a moves according to the collection (Qo, ... , Qp) then its second coordinate remains unchanged and the first coordinate of a is constant when it moves according to (Qp, ... , QP+IJ).

Finally, note that for every H' E UIJ RP (H)

where

(XH'YH):= L (i,j). (i,j)EH

(14.2.6)

Let us consider in more detail two particular cases. Let H = T m for some m < n and H' coincide with one of the following triangles (I ~ 0, J ~ 0, see Fig. 14.2).

B1 = {(i +I,j + J): (i,j) E H}, B2 = {(i+j+ I,j + J): (i,j) E H},

B3 = {(i + I, i + j + J) : (i, j) E H}, B4 = {(i + j + I, J - i) : (i, i) E H}, J ~ m.

(I,J) •

H

Fig. 14.2.



LEMMA 14.7. Let Q = (Qo, ... , Q"+9) E Y(T m -+ H'), where p and q are determined by the equality (14.2.6). Then for every a = (i,i) E H

{ (i + I, i + J),

( H Q) _ (i + i + I, i + J), r a, , - (i+I,i+i+J),

(i+i+I,J-i),

if H' = B1, if H' = B2 ,

if H' = Ba, if H' = B4 •

(14.2.7)

Proof. Let H' = B1 and r(a) = r(a,H,Q), where Q = (Qo, ... ,Q"+9) E Y(H-+ B1). Then the property c) of the mapping r(a) implies

I{(i,i) E Q" : i = io}1 = I{(i,i) E B1 : i = io}l,

I { (i, i) E Qp : i = io} I = I { (i, i) E H : i = io} I (14.2.8)

for every (io,io) E z~. On the other hand, the condition (14.2.8) determines uniquely the set Qp (see Fig. 14.3):

Qp = {(i + I,i) : (i,i) E H},

Jo I---~

(HI,;)

Fig. 14.3.

Besides the image of the point a = (i,j) after the shift Qo -+ Q" is the point (i+I,j), while after the shift Qp -+ Qp+9 it is (i+I,i +J), i.e., r(a) = (i+I, i+J).

Ch. 14, § 14.2) Shifts of Sets and Differentiation of Vandermoode Determinant

The remaining equalities in (14.2.7) can be checked similarly.

N ow let us consider the case

H = Lm := {a = (i,j) : a E Z+, lal = m}, H' = {a = (i + I,j + J) : (i,j) ELm},

m~O,

I,J ~ 0,

239

(14.2.9)

and let p = (m + l)I,q = (m + l)J. Then for every Q E Y(H -+ H') the mapping r(a) = r(a,H,Q) can be presented in the form

1 = O, ... ,m, (14.2.10)

where {oq(O), ... ,oq(mn is some permutation of the integers O, ... ,m. (It is not difficult to see that here the image r( a) of the point a is not always uniquely determined (see Fig. 14.4».

Let us put c(O"Q) = 1 if the permutation O"Q is even and c(O"Q) = -1 if it is odd.

LEMMA 14.8. For the arbitrary sets H and H' in the form of(14.2.9), the following inequality holds:

E(H -+ H'):= (14.2.11)

Proof. Let us denote for every permutation 0" = {O"(O), ... , 0"( mn of numbers {O, .. . ,m}:

Y,,(H -+ H'):= {Q E Y(H E H') : O"Q = O"} (14.2.12)

and note that Y,,(H -+ H') =F {3 iff (see (14.2.10»

r(l) := 1+ O"(l) - 1 ~ 0, u(l) := J - 0"(/) + 1 ~ 0, 1 = 0, ... , m. (14.2.13)

After the r(l) right and u(l) upper shifts the point (1, m-l) moves into the position (0"(1) + I, m - O"(l) + J) E H' (see Fig. 14.4 for m = 4), besides

m m

Lr(I)=(m+1)I=p, LU(I) = (m+ l)J = q. 1=0 1=0

240 Multivariate Pointwise Interpolation

I I I I

o

I I + D _________ J I I

!.2--------+-.J I : ---------I--J......J

3 I

2

-4------.J I ---1-----4t - - - - - - - - L--__ _

Fig. 14.4.

(Ch. 14, § 14.2

Then note that (14.2.11) and (14.2.12) imply

E(H ~ H') = Lc(u)IYq(H ~ H')I, (14.2.14) q

where the sum extends over all permutations u of numbers {O, ... , m}, provided that IYI = 0 if Y = 0. Using the property c) of shifts we get

Qp = {(u(/)+I,m-/) :/=O, ... ,m}

for every Q = (Qo, ... ,Qp+q) E Y(H ~ H'). Here (u(/) + I, m -I) is the image of the point (I, m - I) E H after the shift H ~ Qp (see Fig. 14.4). This means that if Y(H ~ H') :f; 0, i.e., (14.2.13) holds, then

IYq(H ~ H')I = IYq(H ~ Qp)I·IYq(Qp ~ H')I = (r(o») . (p ~(~~O») ... (p - reO) - ~(i)- rem - 1») . (u(O») . ( q ~(~~O») ...

( q - u(O) - ... - u(m -1») ... u(m)

_ "TIm 1 _ "TIm 1 - p.q. 1=0 r(/)!u(/)! - P·q·,=o (I + u(1) -/)!(J - u(/) + I)!·

Ch. 14, § 14.2) Shifts of Sets and Differentiation of Vandermonde Determinant 241

Here we assume that

for j = O.

Now, putting N = 1 + J and

for i < 0 or i > N

(this corresponds to the case Yq(H -+ H') = 13) we get

lyq(H -+ H')I = p!q!(N!)-m-l IT (I +:&) -I) /=0

for every permutation (1'. Then according to (14.2.14)

E(H -+ H') = p!q!(N!)-m-l D(N, i),

where

(~) (/~1) (I:m)

D( N, i) = Dm (N, i) = det ( 1 ~ 1) ( ~) ( 1 + ~ - 1) ............................................. .

(/~m) (/-~+1) (~) where N ~ i ~ 0, m ~ O.

It remains to prove that D(N, i) i= O. Summing up the rows of the determinant D(N, i) and using the following formula for combinations

we obtain the following equation

D(N,i) = D+(N,i), (14.2.15)

where

(~) (/~ 1) (I:m)

D+(N, i) = D~(N, i) := det (Nil) (N+l) 1+1 (N+1)

I+m .........................................

(Njm) (N+m) 1+1 (N+m) I+m

242 Multivariate Pointwise Interpolation (Ch. 14, § 14.2

Now, applying the following relation of combinations:

(k) _ ~ (k - 1) 1 -11-1 '

we get

( m+N) D+ (N, i) = N~ ~ + 1) ... ~N + m) D+ (N _ 1, i-I) = m + ~ D+ (N - 1, i-I).

z(z+I) ... (z+m) (m+l) m+l

So one can conclude

( m+N) ... (m+N-i+l) m+l m+l + .

( .) ( ) D (N - 1,0). m+1 m+l

m+l . m+l

On account of (14.2.15)

D+(N - i,O) = D(N - i,O) = 1,

since the second determinant here has an upper triangle form with l's in the main diagonal. Therefore, we finally get:

D( N, i) = ( .) ( ) # O. m+1 m+l m+l . m+l

( m+N) (m+N-i+l) m+l ... m+l

Now let 8

EIHvl = ITnl· (14.2.16) v=l

Consider the Vandermonde determinant d1i(Z) := d1i,T ... (Z) which consists ofthe following rows (see (14.1.5))

(14.2.17)

where Z = {zv = (xv, Yv )}~=1 E ]R28. We assume that its rows are ordered according to increasing i + j and for rows with the same i + j - according to increasing j and for rows with same i and j - according to increasing v.

Let us now fix the number 11, 11 = 1, ... , s, and differentiate d1i (Z) with respect to XJj' Then we get the sum of determinants which result from d1i (Z) after the differentiation of one of these rows of the form (14.2.17) with respect to x JJ • Moreover,

Ch. 14. § 14.2) Shifts of Sets and Differentiation of Vandermonde Determinant 243

the changed row must satisfy the conditions (i, j) E HI" (i + 1, j) E Tn \H 1" in the opposite case the determinant-summand will contain one null or two identical rows. Hence

o oz d'H(Z) = 2: CQd'HI(Z), I' QER(H,.)

where 11.' = {Hl, ... ,HI'-l,Q,HI'+1, ... ,H,} and cQ = (-IY, with T being the number of row changes needed to bring the differentiate determinant into above described order. Similarly

For mixed derivatives we have

(14.2.18)

where the sum is taken over all collections

v = 1, ... , s, (14.2.19)

with (14.2.20)

Notice also that

v = 1, .. . ,s, (14.2.21)

according to (14.2.6).

LEMMA 14.9. The following two conditions are equivalent:

i) d'H(Z) i= 0 for almost all Z = (Zl,Yl, ... ,z"y,) E ~2', (14.2.22)

ii) there exist numbers (Pl, ql, ... , P" q,) c ;:z~ such that

C:= 2: C[Ql •...• Q.] i= 0, [Ql, ... ,Q']

where the sum is taken over all collections (14.2.19) provided

v,p.= 1, ... ,s.

(14.2.23)

(14.2.24)


Proof. Taking on account that d1t(Z) is a polynomial of the variables Xl, Yl, ... ,

Xk, Yk, we get that i) is equivalent to the condition

for some Pl, ql, ••. ,p" q, E iZ~, where '0 is the origin of :rm.2,. Hence, according to (14.2.18)-(14.2.20)

'D = L e[Ql , ... ,Q"]d1t, ('0). [Ql, ... ,Q"]

(14.2.25)

If the condition (14.2.24) is not satisfied, then the determinant d1t'('O) contains two identica] rows and hence d1t, (0) = O. On the other hand, if (14.2.24) holds, then taking on account the equality

(see (14.2.14)), we get

where dn(x, y) is the Vandermonde determinant consisting of the rows

_8.,..i+_j....,. ( 1 n n-l n ) ""'"8 ·8 . X y . .. x x . Y ... Y , x' 11

(i,j) E Tn.

Therefore'D = edn(O, 0). Next, taking on account the above-mentioned ordering of rows of the determinant we have that all numbers under the main diagonal of the determinant dn(x, y) are equal to zero. Hence,

d ( ) II ·, ., -J. 0 n X,y = I.J. T for every (x, y) E :rm.k •

(i,j)eT ..

This means that'D i: 0 is equivalent to e i: O. Lemma 14.9 and (14.1.6) yield:

COROLLARY 14.10. Let, for the collection 1£ ofform (14.2.16), there exist numbers (Pl,ql, ... ,p"q,) C iZ~ such that the shift 1£ -1£' with condition (14.2.24) is unique, i.e., the image r(o:, H v' QV) of point 0: E Hv (v = 1, ... ,8) does not depend on the collection [Ql, ... , Q'] of form (14.2.19). Then the interpolation scheme (1£, Tn) is almast regular.

Ch. 14, § 14.3) Quadratic Transformations

§ 14.3. Quadratic Transformations

Let K be the set of collections r = {rl. r2, r3; r} satisfying

ri + rj ~ r, 1 ~ i < j ~ 3.

Denote

3

I(r) :=r+l- Eri, where _ r(r + 1) r := 1 + ... + r = 2 '

i=l 3

(r, B) := r{3 - E ri{3i i=l

Let us consider the transformation

r* = {rr,r;,ri;r*}:= {r1 - J.l,r2 - J.l,r3 - J.l;r - J.l},

where J.l E 1E1 is determined from the equality

r1 + r2 + r3 = r + J.l.

The following lemma holds:

r EK,

245

(14.3.1 )

(14.3.2)

(14.3.3)

(14.3.4)

(14.3.5)

LEMMA 14.11. For arbitrary r = {r1, r2, r3; r} E IC, B = {{31, {32, (33; {3} the following conditions hold:

i) f* E K,

ii) I(r*) = I(r), iii) (r*)* =r,

iiii) (r, B) = (r*, B*).

Proof. i). From (14.3.1), (14.3.3), (14.3.4) it follows that for 1 ~ i < j ~ 3

r: + r; = ri + rj - (r1 + r2 + r3 - r) - J.l = r - J.l - rl ~ r*

ii). According to (14.3.2)

3

(I = {1,2,3}\{i,j}).

2J(r*) = (r* + l)(r* + 2) - E r:(r: + 1) = (r + 1- J.l)(r + 2 - J.l) i=l

3

- E(ri - J.l)(ri + 1 - J.l) = (r + 1)(r + 2) - (2r + 3)J.l + J.l2 i=l

3 3

- L ri(ri + 1) + L 2(ri + 1)J.l- 3J.l2 i=l i=l

= 2J(r) + 2 (t, ri - r) J.l- 2J.l2 = 2I(f).


iii). According to (14.3.3) and (14.3.4)

hence

1'* := "Yi + "Y2 + "Y; - "Y* = -1',

(r*)* - { * .* * . * } - f - "Yl - p.,"Y2 - p.,"Y3 - p.,"Y - I' - .

iiii). According to (14.3.2) and (14.3.3)

3

{f*, B*} = ("Y - p.)(P - II) - ~)"Yi - p.)(Pi - II) i=1

[Ch. 14, § 14.3

= {f, B} - P("Y1 + "Y2 + "Y3 - "Y) - "Y(PI + P2 + P3 - P) - 21'11 = {f, B}.

Let us consider now an arbitrary homogeneous polynomial

Q(z, y, t) = L aiiziyit' E 1r:'(1R3) i+i+l=n

and denote by nl, n2, n3 the order of zero of Q in the points

VI = (1,0,0), V2 = (0, 1,0), V3 = (0,0,1),

respectively (nl ~ 0, n2 ~ 0, n3 ~ 0), i.e.,

(i,j,l) E Ln ,,-1. II = 1,2,3, (14.3.6)

(nil = 0 means that Q(vII ) =F 0). It is not difficult to see that (14.3.6) holds iff

for (i,j,l) with ai,i,l =F o.

Hence, the polynomial

Q ( t) Q(yt t) '" a'·J·-Hlyi+lti+i E -200n(trn3) 1 Z, y,:= , z , zy = L..J <l7 .. IN. (14.3.7) i+i+l=n

can be presented in the form

(14.3.8)

where Q* E 1r~_1'(1R3) with I' being determined from (14.3.5).

DEFINITION 14.12. The polynomial Q* is called the quadratic transformation ofQ.


THEOREM 14.13. a) Let the polynomial Q E 7r~(~3) have a zero of order nv at the point vv, v = 1,2,3, with {nl,n2,n3;n} E K. Then its quadratic transformation Q* E 7r~(~3) has a zero of order n~ in vv, v= 1,2,3, where {ni,n;,nj;n*} = {nl, n2, n3; nr (see (14.3.4)).

b) For the arbitrary point Vo = (xo, Yo, to) with xoYoto =P 0 the orders of zeros of the polynomials Q at the point Vo and Q* at

v~ := (~, ~)-) Xo Yo to

are the same.

Proof. According to (14.3.7) and (14.3.8)

Q*( t) ~ a'J·xi+l-n'yi+l-nlti+i-n3. X,Y, = L...J • i+i+l=n

On the other hand, by (14.3.5) we have that

(j + 1- nd + (i + j - n3) = j + n2 - J.l ~ n2 - J.l = n;,

(j + 1- nd + (i + 1- n2) = 1 + n3 - J.l ~ n3 - J.l = nj,

hence the order of zero of Q* at Vv is n~, v = 1,2,3. The second statement of the theorem follows from the following equation:


In this section, we consider the Hermite interpolation in ~2 where interpolation parameters are the values of a function and its partial derivatives up to some order ni at the points Zi, i.e., the sets Sand H v are triangles:

S=Tn , v = 1, .. . ,s.

Here the interpolation scheme !Jl = {nl,"" n.; n} is a collection of non-negative numbers such that (see (14.1.1))

• 2:nv = n+ 1, v=1

where m = 1 + ... + m = ITm-11 = dim 7rm_I(~2). (14.4.1)

By n, we denote the set of such collections 91.


DEFINITION 14.14. For the collection IJl = {nl, ... , n,; n} E 0 and knot set Z = {zv = (xv,Yv)}~=l C ~2 the Hermite interpolation problem (IJl,Z) is to find a (unique) polynomial P E 1I"n(~2) satisfying

8i+i.P( z. ) I 8 8 = Ao,v,

x' 11 z=z" a E Tn,,_l, II = 1, ... ,s, (14.4.2)

for given collection of values

A={AO,v:aETn_1, lI=l, ... ,s}.

In the following, we will write equalities such as (14.4.2) briefly as: Dm Plz = A. It is convenient also to accept that nv can also take zero value. This will mean that at the corresponding point Zv there is no interpolation condition (14.4.2).

Denote by dm(Z) = dm(xl, Yl, ... , x" y.)

the determinant of the linear system (14.4.2) which consists of rows

(ji+i n-l n) .. (1 Xv Yv ... x~ Xv ·Yv.·.Yv ; 8x~811v (14.4.3)

(i,j) E Tn,,-l. II = 1, ... ,So

As was pointed out in Section 14.1, we obtain the equivalence of the following statements:

i) (1Jl, Z) is singular, ii) dm( Z) = 0, iii) there exists a polynomial P such that

(14.4.4)

Denote by RO, ARO and SO, the classes of regular, almost regular and singular collections 1Jl, respectively (see Definitions 14.2 and 14.14).

Theorem 14.6 shows that the class RO consists only of collections IJl with s = 1, i.e., Taylor interpolations, when values of a function and its derivatives at one point are interpolated (see Example 14.2.1). This can be checked directly too, without applying Theorem 14.6. Indeed, if s > 1, then by virtue of (14.4.1) nv ~ n for all II and putting points Zl, ... ,Z. on an arbitrary line with equation ax+by+c = 0, lal + Ibl >0, we obtain a singular problem (1Jl, Z) since the polynomial P(x, y) = (ax+by+ct satisfies the conditions (14.4.4). The problem of a full description of almost regular and singular Hermite interpolation schemes up to now remains open. Below, we bring some results in this direction. Meanwhile, we consider the relation between the bivariate Hermite interpolation and the interpolation with homogeneous polynomials in ~3 which we will need in the following.

Let us put for IJl E 0

.em = {Ln,,-d~=l' where L~ = {(i, j, 1) E Z! : i + j + 1 = m}, (14.4.5)

Ch. 14, § 14.4] Hermite Interpolation 249

and consider the interpolation problem (.em, L~, V) (see Definition 14.1) for the knot set

Vv -:/; AVV" for II-:/; II', A E IR\{O}. (14.4.6)

Here, interpolation parameters for homogeneous polynomials from the space 1r(L~) = 1r~(IR3) are derivatives of order "v - 1 at points Vv :

(i,j,/)ELn ,,_l, 11=1, ... ,8.


THEOREM 14.15. The regularity of (.em, L~, V), for almost all V of the form (14.3.6) (with respect to the Lebesgue measure in IR3.) is equivalent to '.Yt E ARO.

Proof. Let us consider the pair of associated polynomials:

P(x,y) = L: aiixiyi E 1rn(IR2), i+i~n

Q(x,y,t) = L: aiixiyit' E 1r~(IR3). i+i+l=n

It is not difficult to check that

8i+i P(x, y) _ 8i+i Q(x,y, 1) 8xi8yi 8xi8yi (i,j) E IE~.

This implies that for the arbitrary '.Yt = {n1' ... , ",; n} and the sets of nodes

V = {vv = (xv, yv, 1)}~=1 C IR3

the conditions Dmplz = 0 and

i + j + 1= nv - 1, II = 1, ... ,8,

are equivalent. On the other hand, the last condition is equivalent to

i + j + 1= nv - 1, II = 1, ... ,8, (14.4.7)

according to the Euler formula:

Thus, conditions Dmplz = 0 and (14.4.7) are equivalent which, in view of (14.1.6), completes the proof.


THEOREM 14.16. Let 91 = {n1, ... ,n.;n}, n1 ~ n2 ~ n3, n2:f. 0. Then 1) if n1 + n2 = n + 1 then the interpolation schemes 91 and

are equivalent, i.e., 'Jt E SO iff91i E SO. 2) if n1 + n2 ~ n, ni + n2 + n3 = n + /l, /l E ;:;zi, then 91 and

are equivalent.

(14.4.8)

(14.4.9)

Proof. According to Lemma 14.11, the collections 911 and 91* satisfy the equality (14.4.1), i.e., 911, 91" EO.

Let Z = {Z/I }~= 1 be an arbitrary set of nodes and l( z) = ° be the equation of the line passing through Zl and Z2. Note that the set of collections Z in which three nodes Z/I belong to a line has a zero measure (in ~2.). Thus, without loss of generality, we can assume that there is no another knot of Z on the line l(z) = 0.

If Ni E SO then, by virtue of (14.4.4) there exists a polynomial Pi E 7rn_1(~2), P1 1= 0, such that

D'JhP1\Z = 0. (14.4.10)

Now it is clear that the polynomial P(z) = P1(z)l(z) satisfies the condition (14.4.4), i.e., 91 ESQ.

Inversely, if 91 E SO, then there exists a polynomial P with (14.4.4) and the condition n1 + n2 = n + 1 implies that P(z) = ° on the line l(z) = 0. Then in view of Lemma 11.16 we have that

P(z) = P1(z)l(z),

which, in turn, implies that P1 satisfies (14.4.10) since the line l(z) = ° does not contain any knot of Z different from zland Z2. Thus we get 911 E SO.

Let us prove now the second statement of Theorem 14.16. Let V be an arbitrary collection in the form of (14.4.6) such that

V1 = (1,0,0), V2 = (0,1,0),

V3 = (0,0,1), X/IY/It/l:f. ° for v = 4, ... , s, (14.4.11)

and let V" = {v~}~=l' where

v: = Vv if v = 1,2,3; v: = (l/x/I' I/Y/I' l/t/l) if v = 4, ... , s. (14.4.12)

Assume that 91 E SQ, then, according to Theorem 14.15, the interpolation scheme (.em, L~, V) is singular and hence there exists a polynomial

Q(X,y,t) = Q 1= 0, i+i+l=n

Ch. 14, § 14.4] Hermite Interpolation 251

such that

an .. -1Q(v) I . . I =0, ax' aY' at tI=tI ..

i + j + 1= nv - 1, v = 1, ... , s . (14.4.13)

Let Q"(x, y, t) be the quadratic transformation of the polynomial Q (see Definition 14.12). Then by Theorem 14.13 we have

Q" "I- 0, (14.4.14)

Thus in view of (14.1.6) the interpolation scheme (.cm*, L!, V") is singular for every set of nodes V" in the form of (14.4.12). Let us prove that from this follows the singularity of the scheme (.cm*,L!,V") for almost all V ...

Denote for a given matrix r = hi,j,l}~,j,l=l with det r :I 0

QHx,y,t) = Q"(r-l(x,y,t)), rv" = {rvi, ... , rv:}.

We get from (14.4.14) that

Qr "I- 0,

which means the singularity of the scheme (.cm*, L!, V"). On the other hand, the set of collections U = (Ul, ••. , u.) E jR3. which cannot be presented in the form U = rv", with V" is defined in (14.4.12) and V satisfies the conditions (14.4.6) and (14.4.11), has zero measure (in jR3.). Therefore, the scheme (.cm*,L!) is singular and on account of Theorem 14.15 we get that 'Jl" E So..

To complete the proof of Theorem 14.16 we need to show that 'Jl" E So. implies 'Jl E So.. This follows from the proved part of the theorem and the fact that if we apply the transformation 'Jl -+ 'Jl" to the scheme 'Jl* then we get 'Jl (see Lemma 14.11, iii)).

From now on, it is convenient to assume that the numbers nv in the scheme 'Jl E 0. are decreasing: nl ~ n2 ~ ... ~ n •. Theorem 14.16 shows that the investigation of an arbitrary interpolation scheme reduces to the following three cases:

1) s = 1, nl = n + 1, 2) nl + n2 > n + 1, 3) nl + n2 + n3 ~ n.

In the first case, as we already know from Example 14.2.1, 'Jl ERn. It is not difficult to prove (see Theorem 14.17 below) that, in the second case, 'Jl E So.. There is a hypothesis (see Hakopian, Gevorgian, and Sahakian [1990]), that, in the third case, 'Jl EARn, nevertheless it was only proved under the restriction nlO ~ 1, when the number of nodes, at which both the values of the function and its derivatives which are interpolated, is less than 10 (see Theorem 14.18 below).

THEOREM 14.17. If'Jl = {nl, ... ,n.jn} E 0. and nl + n2 > n + 1, then 'Jl E So..

252 Multivariate Pointwise Interpolation [Ch. 14. § 14.4

Proof. Let !Jl E 0, nl + n2 > n + 1 and Z = {zv }~=l be a set of nodes. Without loss of generality, one can assume that the line l(z) = 0, passing through Zl and Z2

does not contain any other node from Z.

Since

nl - 1 + n2 - 1 + n3 + ... + n • •

= 2:nv - (nl + n2) < n + 1 ~ (n + 1) = n, (14.4.15)

v=l

we can apply Lemma 14.5 to find a polynomial PI E 1I"n_I(~2), PI t 0, such that D~hp,nllz = O. Now the polynomial P(z) = P1(z)1(z) clearly satisfies the condition (14.4.4), hence !Jl E SO.

THEOREM 14.18. If!Jl= {nt, ... ,n.jn} EO, nl0 ~ 1 and

(14.4.16)

then !Jl E ARO.

Proof. We will prove the almost regularity of!Jl using Corollary 14.10. For this purpose, we need some notations:

Hv = Tn ,,-I, v = 1, ... ,8, II = 12 = 13 = Is = 0,

14 = nt, 15 = n - n5 + 1,

16 = Ir = n - n5 + n6 + 1, 19 = ns - n9 + 1,

J1 = n - nt + 1, J2 = n - nl-n2 + 1,

J3 = n - nt - n2, J4 = n - nt - n4 + 1,

J5 = J6 = Js = 0, J7 = 1, J9 = n9 - 1.

Let us define the following triangles (see Fig. 14.5):

Gv = {(i + Iv,i + Jv): (i,i) E Hv}, v = 1,4,5,8,

Gv={(i+i+lv,i+Jv):(i,i)EHv}, v=3,6,

Gv={(i+Iv,i+i+Jv):(i,i)EHv}, v=2,7,

Gv={(i+i+Iv,Jv-i):(i,i)EHv}, v=9.


Fig. 14.5.

The condition (14.4.16) implies that the sets Gil' II = 1, ... ,8, belong to Tn and are disjoint. The remaining points of Tn,· i.e., the points of the set

(it can be seen from the Fig. 14.5 that it is not empty) denote by Glo, ... ,G$ in an arbitrary way. Now let us put

11= 1, ... ,8, (14.4.17)

and prove that the shift defined by the numbers (14.4.17) is unique, i.e., if 1i' = {Hf, ... , H;} is in the form of (14.2.20) and

for 1I:f Il, (14.4.18)

then the image of an arbitrary point a E H II under the shift 1i --+ 1i' (II = 1, ... ,8) -1 -$

does not depend on the collection [Q , ... , Q ] of the form (14.2.19). In order to do this, by virtue of Lemma 14.7, it is sufficient to prove that 1i~ =

G~, II = 1, ... ,8, which is done in the following:

LEMMA 14.19. Let 1i~, II = 1, ... ,8, be arbitrary sets satisfying the following conditions:


a)H~CTn, H~nH~=0 for 1Ii=J1., 1I,J1.=I, ... ,s, b)IH~I=IG~I, 1I=1, ... ,s, c)xa,,=xH~' Ya,,=YH~' 1I=1, ... ,s. Then

11 = 1, ... , s.

[Ch. 14, § 14.4

(14.4.19)

Proof. Clearly, if IGvl = 1 for some 11 then b) and c) imply that H~ = Gv , i.e., we get (14.4.19) for 11 > 9. Further, if H~ i= G1 , then Hf C Tn implies that YH~ < Ya 1 ,

which contradicts the condition c), therefore Hf = G1• Similarly H~ = Gs , since in the opposite case xH~ < xas ' Next, if we assume that Gv i= H~ for some 11 E {2,6,8} then taking into account that

we get either xH' i= xa or YH' i= Ya . It remains to prove (14.4.19) for 11"= 3,4,7,9. Conditions a) and c) imply that

H~ U H~ U H~ U H~ = G3 U G4 U G7 U G9 and

(14.4.20)

If we suppose that there exists Q' E (H~ U H~)\(G3 U G4 ), then Q' E G7 U G9 which contradicts the equalities (14.4.20). Hence in view of b) we get

(14.4.21)

Now assume that H~ i= G~, i.e., there exists a point Q' E H~\G3. Then, by virtue of (14.4.21), Q' E G4 and the number XH~, clearly, is greater than Xa3' which contradicts c). This means that H~ = G3 and H~ = G4 .

The proof of (14.4.19) for 11 = 7,9 is similar. Lemmas 14.19 and 14.7 imply, as was pointed out above, that the shift 1{ ~ 1{'

defined by the numbers (14.4.17) with the condition (14.4.18) is unique. Therefore, by virtue of Corollary 14.10, we get that !Jl E ARO.

Now we are going to present another aspect of the investigation of bivariate Hermite interpolation.

Suppose for the collections !)Jtj in the following form

• L:mt < mi + 1, (14.4.22) v=l

and for the interpolation scheme !Jl = {n1, ... , n.; n} E 0 the following decomposition holds

which means that

q

nv = L:mt, j=l

1I=1, ... ,s; q

n= L:mv. j=l

(14.4.23)


It is not difficult to check that then '.ll E SO. Indeed, for any set of nodes Z = {z',}!=l' by Lemma 14.5 there exists polynomials Pj such that

Pj 1= 0, rot; I D Pj z = o.

Now because of (14.4.23) we will have (14.4 4) for the polynomial

I}

P= I1Pj, j=l

therefore '.ll E SO. For example, for the singular scheme {2, 2; 2}, we have the decomposition:

{2, 2; 2} = {I, 1; I} + {I, 1; I}. (14.4.24)

It is very likely that all the singular schemes can be presented in the form of (14.4.23). Unfortunately, as in Theorem 14.18, this is still only proven under the restriction n10 ~ 1.

THEOREM 14.20. The interpolation scheme 1)1 = {n1,' .. , n.; n} E 0, with n10 ~ 1 is singular iff it has a decomposition of the form (14.4.23).

Proof. We only need to find a decomposition (14.4.23) for the singular scheme 1)1.

This will be done by induction on n. For n = 2 see (14.4.24) (there is no other singular scheme '.ll with n ~ 2). Now let 1)1 E SO and n10 ~ 1. By Theorem 14.18, the following three cases are possible:

1) n1 + n2 > n + 1. Then '.ll has the decomposition (see (14.4.15»:

'.ll = {I, 1,0, ... ,0; I} + 'Jh, (14.4.25)

where '.ll1 = {n1-1, n2-1, n3, . .. ,n.; n}. 2) n1 +n2 = n+1. Then by Theorem 14.16, the interpolation scheme 1)11 in (14.4.25)

is singular and using the induction hypothesis we get for '.ll1 the decomposition as

in (14.4.23), which, with (14.4.25), gives the desired decomposition for 1)1.

3) n1 + n2 ~ n + 1, n1 + n2 + n3 = n + 1', I' > O. Then by Theorem 14.15 the interp olation scheme

is singular and by the induction hypothesis

1)1" = rot1 + ... + rotl}, (14.4.26)

with rotj in the form of (14.4.22). By Lemma 14.11, if rot = {m1,'" ,m.;m} C Z~ is an arbitrary collection with

mi + m, ~ m, 1 ~ i < I ~3, (14.4.27)


then the collection

where ml + m2 + m3 = m + I is of the form (14.4.27) too. Therefore, we will get a decomposition for '1l:

provided that the following inequalities hold for rolj ,j = 1, ... ,q:

j+ j/ j m i m,:::::: m , 1 :::; i < I :::;3. (14.4.28)

Hence, we only need to prove that there exists such a decomposition (14.4.26) in which the inequalities (14.4.28) hold.

It is not difficult to check that every collection rol in the form of (14.4.27) with ml +m2 > m > 1 can be presented as a sum of two collections of the form (14.4.27):

rol = {I, 1,0, ... ,0; I} + {ml - 1, m2 - 1, m3,"" rn,; m - I}.

This means that after a finite number of steps, we get a new decomposition of the form (14.4.26), where, for each j = 1, ... , q, either m{ + m~ :::; rnj or

rolj = rol° := {I, 1,0, ... ,0; I}. (14.4.29)

Assume that there exists a collection rol° in (14.4.26). Then for some jo E {I, ... , q} we must have that m{o + m~o < mjo , since in the opposite case

q q

(nl -I') + (n2 -I') = ~)m{ + m{) > E mj = n -1', I' > 0, j=l j=l

which contradicts the condition nl + n2 :::; n. Let us now replace the collections rol° and roljo in (14.4.26) by their sum

{ jo + 1 jo + 1 jo jo. jo + I} m1 ,m2 ,m3 , ... ,m, ,m

which, as can easily be checked, satisfies the condition (14.4.28) (for i = 1, I = 2) and is a collection of the form (14.4.27). Therefore, after a finite number of steps in the decomposition (14.4.26), we will have (14.4.28) for all j (for i = 1, I = 2). The proof of (14.4.28) for the remaining i,l, is similar.

It can be seen from this proof, that if Theorem 14.18 is true without the restriction nlO:::; 1, the same can be said about Theorem 14.20. Conversely, one can prove that if Theorem 14.20 is true without the restriction nlO :::; 1, then such is Theorem 14.18.

THEOREM 14.21. Let the scheme '1l = {nl,"" n,; n} E f! be such that for some r, 1 < r < s, and m E ~~

'1l2 = {m, nr+l,' .. ,nr ; n} E ARf!. (14.4.30)

Then '1l E ARf!.

Ch. 14. § 14.4) Hermite Interpolation 257

Proof. Let

(zo = (XO, Yo)),

be the determinants of the interpolation schemes 91,911 and 912, respectively. Since 911 EARn we have that dlJtl (21) I 0 for almost every 21 E IR2,. and by Lemma 14.9 there exist numbers p!, q!, ... ,p,., q,. such that

e := L e['Ql •...• 'Q.] I 0, ['Ql •...• 'Q.]

where the sum is taken over all collections

provided

(14.4.31 )

H~ n H~ = 0, v I 1', v,j.t = 1, ... ,r. (14.4.33)

Now consider the expression

According to (14.2.18) we have

(14.4.34)

where the sum extends over all collections

v = 1, ... , r, (14.4.35)

provided

H~ nH~ = 0, viI', v,j.t = 1, ... ,r, (14.4.36)

According to (14.2.6) for both the cases (14.4.33) and (14.4.36) we have

v = 1, ... ,r. (14.4.37)

Consider the set ,. H' = U H~.

v=1


Since the sets H~ are disjoint (which holds both for (14.4.33) and (14.4.36)), we have that IH'I = ITm-d and

XH, = tXH~ = t (XTftV_1 +pv), v=l v=l

YH' = t YH~ = t (YTftV - 1 + qv). v=l v=l

(14.4.38)

In the case where H~ are defined in (14.4.32)-(14.4.33), clearly we have that H' = Tm - 1 and

(14.4.39)

Then, by (14.4.38) we have the equalities (14.4.39) also in the case when H~ are defined in (14.4.36)-(14.4.37). On the other hand, it is clear that for an arbitrary set H' C Z~ with IH'I = ITm-11 the equalities (14.4.39) imply that H' = Tm- 1 •

Therefore, in (14.4.34), we will have

dw (Z)lz 1 =Z2= ... =zr=zo = d~h(Z2)' Moreover, in (14.4.35) and (14.4.36) n can be replaced by m - 1, i.e., the sum in (14.4.34) can be taken over all collections of the form (14.4.32)-(14.4.33). Then in view of (14.4.31) we get

Now since ~h E ARQ (i.e., d~h(Z2) ~ 0), we get I(Z2) ~ 0 and therefore dm(Z) ~ 0, which means that the interpolation scheme 91 is almost regular.

§ 14.5. The Birkhoff Diagonal Interpolation

In this section, we present one particular case of Birkhoff interpolation when at different nodes fixed (depending on the node) order derivatives are interpolated, i.e., the interpolation scheme ('H., Tn) is considered, with (see Definition 14.1)

Hv = {(i,i) E Z~: i+i = nv -1}, 1/ = 1, ... ,8. (14.5.1)

The non-negative integers nv satisfy the following condition (see (14.1.1):

(14.5.2)

This interpolation is called the Birkhoff diagonal interpolation. It is not difficult to check that the P6lya condition in this case is reduced to

L nv ~], i = 1, ... ,no (14.5.3) v:nv~j

The following theorem holds:

Ch. 14, § 14.5) The Birkhoff Diagonal Interpolation 259

THEOREM 14.22. The Birkhoff diagonal interpolation scheme (1{, Tn) is almost regular if and only if the P6lya condition (14.5.3) holds.

Proof. According to Theorem 14.4, we only need to prove that the P6lya condition is sufficient for the almost regularity.

Let us order the points of the triangle

such that J.l' < J.l iff either 1'1'1 < 1'1',1 or 1'1'1 = hI"I with jl' < jl'" i.e., 11 (0, n), 12 = (1, n - 1), ... , In+1 = (n,O), In+2 = (0, n - 1), ... , InH = (0.0). Denote

II where J.lo = 0, J.lII = 2: n" II = 1, ... , s, (14.5.4)

'=1 and

1I=1, ... ,s. (14.5.5)

Since nil are decreasing, the P6lya condition (14.5.3) implies that the following two cases are possible for every II = 1, ... , s:

a) Gil = {(i + III,j + JII ) : (i,j) E HII },

where III ~ 0, JII ~ 0;

b) Gil = {(i+III,j): (i,j) EHII ,j=O, ... ,I\:II}U{(i,j+J,,) : (i,j) EHII ,

j = 1\:11+1> ••• n" -I}, p" = (I\:" + 1), q" = (n" - 1\:" - 1),

where I" ~ 0, J" = I" - 1, 0 < 1\:" < nIl - 1. On account of Lemma 14.9, to prove the almost regularity of (1{, Tn) we need only

to check that

C := 2: C[Q"l, ... ,Q"') f 0, [Q"l, ... ,Q"')

where the sum is taken over all collections of the form (14.2.19) provided

H~ n H~ = 0, II f J.l, II, J.l = 1, ... , s.

By the properties (14.2.5) of shifts from (14.5.5) we get

1I=1, ... ,s.

(14.5.6)

(14.5.7)

(14.5.8)

It is not difficult to check (by induction on II), that (14.5.7) and (14.5.8) imply

1I=I, ... ,s.


This means that in (14.5.6) we have (see (14.2.4»

Q"EY(H"-+G"), v=I, ... ,s,

and

e = L: L: e[Ql •...• Q•r (14.5.9) Ql eY(H1 ..... G1 ) Q·eY(H1 ..... G1 )

Clearly, in the case when the set Gil is oftype b), the shift HII --+ Gil is unique, i.e., the image r(a) = r(a,H",Q") ofthe point a = (i,j) E HII is uniquely defined:

( ) _ { (i + III ,j), r a - (.. J) l,J + II,

This, and (14.5.9), imply

•

if j=O, ... ,KII ,

if j = KII + 1, ... nil - l.

e = 6 II E(HII -+ Gil), 161 = 1, 11-1

(14.5.10)

where the quantity E(HII -+ Gil) is defined in (14.2.11), if Gil is a set of type a) and E(HII --+ Gil) = IY(HII --+ GII)I, if Gil is a set of type b). Now (14.5.10) and Lemma 14.8 yield e =f O. The condition (14.5.6), and therefore Theorem 14.22 are proved.

§ 14.6. Uniform Hermite Interpolation

In this section, we discuss a particular case of Hermite interpolation, when at each node ZII are interpolated values of a function and its derivatives up to some fixed order 1-1 (the same for all nodes), i.e., nl = n2 = ... = n. = I. For this uniform interpolation scheme, we use the notation: (I, n)-interpolation. The condition (14.4.1) here looks as

sl(1 + 1) = (n + 1)(n + 2).

This means that the (I, n )-interpolation is defined if

(n + 1)(n + 2) = 0 (modr), where r = 1(1 + 1). (14.6.1)

Note that if the pair (I, n) defines an interpolation scheme, then so do also pairs (I, n + kl*), k = 1,2, .... In particular, (14.6.1) holds if

n E {I - 1 + kr, 12 - 2 + kl", r - 2 + kl", r - 1 + kl" }k°=o.


THEOREM 14.23. If (l,r - 2) and (I, r - 1) are almost regular, then so are the interpolation schemes

(/,I-l+kr), (/,r-2+kr), (/,r-l+kr),

If, in addition, (/,12 - 2) is almost regular, then so are

(/,12 - 2 + kl"), k = 0,1 ....

k = 0,1,. . .. (14.6.2)

(14.6.3)

Ch. 14, § 14.6) Uniform Hermite Interpolation 261

Proof. Let us prove the almost regularity of (I, 1 - 1 + kl*). We will use induction on k. The case k = 1 is obvious. Suppose that (1,1 + (k - 1)l*) is almost regular. It is not difficult to check that

Now, on the account of Theorem 14.21, it is enough to prove the almost regularity of the following Hermite interpolation scheme:

~ = {nl> ... ,n6;n} :=

: = {I + (k - 1)/*, /*, /*, ... , /*, /* - 1, ... , /* - 1, 1, ... ,1; 1 - 1 + k[*}. ______ ' 'V' ' ~

1: 1:-1 2(1+1)

The latter, in turn, follows from Corollary 14.10. The unique shift is illustrated in Fig. 14.6, where the numbers m denote the shifted triangles Tm.

k-l

~ ~ ~ ••••••••••

lSl

n = 1- 1 + kz*

1 - 1 + (k + 1 )1*

1 ~ [§J : [§J L-____ -->o.

'----v--'

1 + 1

Fig. 14.6.

The uniqueness of this shift follows from Lemma 14.7 similarly to the uniqueness of shift in the proof of Theorem 14.18.


The almost regularity of the remaining schemes in (14.6.2) and (14.6.3) follows in an analogous way, by using the following equalities:

ITz·-2+k/" 1 = 111·-2+(/:-1)1·1 + kITz·-11 + (k + 1)111·-21,

rr,·-l+kI·1 = 111·-1+(/:-1)1·1 + (k + 1)111·-11 + klTz·-21,

ITz~-2+k1·1 = ITz~-2+(/:-1)1·1 + kITj·-11 + (k - 1)1Tz·-21

+ 2ITj~-21 + 2(/- 1)ITz-d.

The corresponding shifts are presented below in Fig. 14.7.

~ ~ /·-1

k-l .........

tsJ ~ /·-2

n = 1* - 2 + kl*

k .........

l*-2+(k-l)l· tsJ

~ n = 1* - 2 + k1* ~

k-l

~ ~

Fig. 14.7.

n = 1* -1 + kl*

THEOREM 14.24. For 1 = 2,3,4 the interpolation schemes (I, n) are almost regular except for the two cases (2,2) and (2,4).

Ch. 14, § 14.6) Uniform Hermite Interpolation 263

Proof. We will consider the proof of theorem for the case I = 3 only. The proof for the case 1= 4 is similar. The proof presented below cannot be applied for the case I = 2, since the schemes (I, I· - 2) and (1,12 - 2) here are singular and we can not use Theorem 14.23.

So, let 1= 3, I· = 12 and (n + 1)(n + 2) = 0 (mod r). Then it is not difficult to check that n can be presented in one of the following ways:

n = 2 + 12k = 1+ kr, n = 7 + 12k = 1(1 + 2) - 1 + kl.,

n = 10 + 12k = r - 2 + kl·, n = 11 + 12k = r - 1 + kr.

Therefore, in view of Theorem 14.23 it is sufficient to show that the interpolation schemes (3.7), (3,10) and (3,11) are almost regular.

The almost regularity of the scheme (3,7) can be easily checked using Theorems 14.16 and 14.18. In the case of schemes (3,10) and (3,11) it follows from Corollary 14.10. The corresponding unique shifts are presented in Fig. 14.8 below (the numbers v = 1, ... , s denote the images of the points from the v-th triangle 11):

8 8 7 985 9 8 64 9 8 754

10 9 7542 10 9 76532 10 10 865322 11 10 9 6 5 4 3 2 1 11 11 107643 2 1 1 11 11 11 7 6 4 3 3 1 1 1

(2,10)


10 10 8 10 9 7 10 9 85 11 10 8 64 1110 9754 12 11 97542 12 11 9 76 5 3 2 12 12 10 8 6 5 3 2 2 13 12 11 8 6 5 4 3 2 1 13 13 12 8 7 6 4 3 2 1 1 13 13 13 9 7 6 4 3 3 1 1 1

(2,11)

Fig. 14.8.

The necessity of the P6lya condition for almost regularity of the Birkhoff interpolation (Theorem 14.4) was proved by Lorentz G. G. and Lorentz R. A. [1984] as well as Theorem 14.6 for k = 2. For arbitrary k, Theorem 14.6 was proved by Lorentz G. G. [1989]. Jia and Sharma [1990] have generalized it to the case when S is not necessarily a normal set. The results of Section 14.2 are due to Hakopian, Gevorgian, and Sahakian [1990], as well as Theorems 14.15-14.20 and 14.22. Similar arguments (shift of sets and differentiation of the Vandermonde determinant) were used in Lorentz G. G. and Lorentz R. A. [1987], [1990], and in Lorentz R. A. [1989]


to prove Theorems 14.21, 14.23 and 14.24. Earlier, Theorem 14.24 with I = 2,3 was proved by Hirshovitz [1985], using completely different techniques (methods of algebraic geometry).

265

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Zavialov J. S., Kvasov B. I., and Miroshnichenko V. L. [1980], Spline Function Methods, Nauka, Moscow, 1980.

Zensykbaev A. A. [1974], Approximation of certain classes of differentiable periodic functions by interpolating splines on uniform grid, Mat. Zametki, 15, 6 (1974), pp. 955-966 (in Russian).

Zensykbaev A. A. [1981]' Monosplines of minimal norm and best quadrature formu-las, Uspehi Mat. Nauk, 36,4 (1981), pp. 107-159 (in Russian).

Zensykbaev A. A. [1989], The fundamental theorem of algebra for monosplines with _ multiple nodes, J. Approx. Theory, 56, 2 (1989), pp. 121-133. ZeniSek A. [1974], A general theorem on triangular CM elements, RAIRO, Ser.

Rouge, Anal. Numer., R-2 (1974), pp. 119-127.

INDEX

Barycentric coordinates 145 Best method 75, 102

Bernoulli polynomials 119

Block: even 13

Hermitian 15

odd 13 supported 13

Box splines:

definition of 163

differentiation of 165 geometric interpretation of 164

independence of translates of 188 interpolation by 192

recurrence relations for 165

translates of 168 B-splines:

contour integral representation of 40, 41

definition of 30, 55, 124, 137

differentiation of 38, 144 geometric interpretation of 134 linear independence of 35, 118 normalized 31, 37

perfect 83, 89 periodic 118 recurrence relations for 36, 144

with Birkhoff knots 53

Coalescence 63 Complete spline interpolation 73

Divided difference 1,4,5,7,53, 158

273

Formula:

Frobenius 8

Hermite-Genocchi's 9

Newton 5

Popovichiu's 10

Tschakaloff's 39, 40, 43

Free knots 21

Fundamental theorem of algebra:

for monosplines 114

for perfect splines 86

for periodic perfect splines 130

Interlacing condition 88

Interpolation:

at the extremal points 99

by spline functions 45, 60, 124

Birkhoff 13, 52, 60

Favard's 82, 128

Hermite 2,49,68,69, 124

Lagrange 1

Interpolation scheme:

almost regular 232

regular 232

singular 232

Knot set:

degeneration of 145

b-regions of 145

in a general position 145

unimodular 188

Louboutin's problem 82

274

Matrix: conservative 13, 53

incidence 12 normal 12

poised 13 Monosplines:

definition of 109 271'-periodic 120, 122 zeros of 111, 112

Multivariate interpolation: Birkhoff 231 Birkhoff diagonal 258

Chung-Yao 207 finite element 227 Hermite 207, 226, 232, 247 Kergin 203 Lagrange 198, 232 mean value 198 on the sphere 224 pointwise 206 Taylor 232 tensor-product 226 uniforme Hermite 260

Optimal nodes 104, 105 Oscillating perfect splines 90

(r + I)-Partition: definition of 57 s-regular 58

Partition of unity 31 Peano's kernel 28

Perfect B-spline 83, 89 P6lya condition 12, 53

Quadratic transformation 245 Recovery:

best method of 75, 102 optimal 102

Regular pair 53, 54

Index

Refinement 41

Ridge functions 147

Set of hyperplanes:

admissible 206

in general position 206

Sign changes 15, 42, 60

Smolyak lemma 75

Spline functions 149

Splines:

definition of 19

natural 67, 78

perfect 82,83,86,87,90,100,106, 128

periodic 117

polyhedral 159

representation of 20, 26 117, 122

simplex 159

closure of 21, 25

with multiple knots 25

with knots (:I:, E) 53

Steffenson's rule 8 Theorem:

Atkinson-Sharma 13

Borsuk's 84

Budan-Fourier 14, 17

Holladay's 72, 73

Peano's 28

Total positivity 45, 63, 66

Totally positive:

kernel 45, 46

matrix 47

Triangulation 159 Truncated power function 20, 63, 139

Vandermonde determinant 232

Variation diminishing property 41, 42, 44, 47

NOTATION

Rk k-dimensional real vector space (R := R1),

{e1, ... ,ek} standard basis for Rk, i.e. (ei)j = Oi,j,

R~ := {x = (xl' ... ,Xk) E Rk: Xi ~ 0, i = 1, ... ,k},

X~Y meansXi~Yi, i=I, ... ,k,forx=(x1, ... ,xk), Y=(Y1'···'Yk)ERk,

inner product in Rk,

XI1IIk := (xl' ... ,xk) for x = (x 1, ... ,xn ) ERn, n ~ k,

7lk:= {a= (a 1, ... ,ak) ERk: a i are integers} (7l:=7l1, 7l~ :=7lknR~),

lal := a 1 + ... + ak'

",I .- '" 1 '" 1 (0 1 .- 1) ..... - "'1· ··· ... k· .. -,

",01 ._ ",011 OI k ""' .- ""'1 • •• Xk '

Sk := {Po, ... , Ak}: Ao + ... + Ak = 1, Ai ~ 0, i = 0, ... , k}, For a set A C Rk :

volk(A) = mk(A) k-dimensional Lebesgue measure of A,

(A) linear span of A,

[A] := { L Aaa: L Aa = 1, 0 ~ Aa ~ 1 } aEA aEA

convex span of A,

IAI cardinality of A,

A(v):= {B C A: IBI = II}, XA (x) characteristic function of A,

7rn (Rk) := { L aOlxOI : aOl E R}, OIEz~,IOII~n

00

7r(Rk) := U 7rn (Rk), n=O

7r~(Rk) := { L aOlxOI : aOl E R}, OIEz~,IOII=n

C(O) space of continuous functions on 0,

275

276 Notation

space of functions with continuous n-th derivative on n, space of continuous functions on IRk with compact support,

W;[a,b] := {f E C,.-l[a,b]:/(,.-l) abs. cont., 1I/(")lIp < co},

{ 6 }l/P II/lIp:= J I/(t)IP dt ,

o

11/1100 := supvrai I/(t)l, fE[o,6]

1 ~ p < co,

1I"n the set of all algebraic polynomials of one variable of degree ~ n,

supp := {z : I(z) i= O}.

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Spline Functions and Multivariate Interpolations