rpruJ~ ( ('1'\ON,C~,t"\c.I ",,-JH\J '- rMOMENT PROBLEMS AND ORTHOGONAL FUNCTIONS

A. Bultheel

Department of Computer Science, K.U. Leuven,

B-~1 Leuven, Belgium.3oo-t

P. Gonzalez-Vera

Facultad de Mathematicas, Universidad de La Laguna

La Laguna, Tenerife, Canary Islands, Spain.

E. Hendriksen

University of Amsterdam, Department of Mathematics,

Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.

O. Njastad

Department of Mathematics, University of Trondheim-NTH,

N-7034 Trondheim, Norway.

Abstract

The Caratheodory coefficient problem for an infinte sequence {Cn} can be formulated as follows: Find a

Caratheodory function F(z) (i.e. an analytic function mapping the unit disk D = {z :1 z 1< I} into the

right half plane P = {w : Rew > O}) whose Taylor coefficients at a fixed point in D (for convenience the

origin) are the {cn}. This problem is equivalent to the Trigonometric moment problem: Find a measure

J.l(B) on [-71",71"] such that f~oo e-in8 dJ.l(B) = Cn for n = 0, 1,2, ... These problems are closely related to the

theory of Szego polynomials, i.e. orthogonal polynomials on the unit circle T = {z :1 z 1= I}.

Let {an} be an arbitrary given sequence of not necessarily distinct points in D, and let {wn} be a

given sequence of numbers. The Nevanlinna-Pick problem for this situation is to find a Caratheodory

function F(z) with the interpolation property F(an) = Wn for all n. (When points an are repeated,

the interpolation property involves the appropriate number of Taylor coefficients.) Also this problem is

connected with the problem of finding a measure generating "moments". The Nevanlinna-Pick problem is

related to a theory of rational functions which are orthogonal on T (with poles at the points 1/&n). This

relationship is analogous to and generalizes the relationship between the Caratheodory coefficient problem

(or Trigonometric moment problem) and polynomials orthogonal on T.

The situation when points {an} are given on the boundary T of D is also briefly discussed.

Keywords: Caratheodory coefficient problem, Nevanlinna-Pick interpolation problem, Moment problems,

Orthogonal polynomials and orthogonal rational functions on the unit circle.

1

1 Introduction

In this paper we shall use the following notations: D = {z E C :1 z 1< I}, E = {z E C :I z I> I}, T = {z E C :1 z 1= I}, H = {z E C; Imz > 0, P = {z E C : Rez >O}, R = {z E C : I mz = O}. By a Caratheodory function we mean an analytic function

F(z) on D mapping D into P. By a Schur function we mean an analytic function S(z)

on D mapping D into D. By a Nevanlinna function we mean an analytic function G(z)

on H mapping H into H.

The Caratheodory Coefficient Problem (CCP) for an infinite sequence {en : n = 0,1,2, ... }

can be formulated as follows: Find a Caratheodory function whose Taylor coefficients at

the origin are the {en}. This problem is equivalent to the Trigonometric Momen t Problem

(TMP): Find a measure fl on [-11",11"] such that

[1<1< e-inOdfl(O) = Cn for n = 0,1,2, ....

(In the following we shall by a measure mean a finite positive measure.) These problems

are closely related to the theory of Szego polynomials, i.e. orthogonal polynomials on the

circle T. They are also related to the theory of Schur functions and various variants of

the famous Schur algorithm. Among books and articles where these subjects and their

relationships are treated, we mention [1, 12, 13, 14, 22, 26,31, 34, 35, 36, 40, 58, 62, 63].

A survey of applications in digital signal processing with references to other work in the

same direction can be found in [18]. See also [23].

Let {an: n = 0,1,2, ... } be an arbitrary sequence of not necessarily distinct points in D,

and let {wn : n = 0,1,2, ... } be a given sequence of complex numbers. The Nevanlinna­

Pick Interpolation Problem (NPIP) for this situation is to find a Caratheodory function

F(z) with the interpolation property F(an) = Wn for all n. (When points an are repeated,

the interpolation requirement involves the appropriate number of Taylor coefficients.) See

[44, 45, 54, 55]. Also this problem is connected with the problem of finding a measure

fl generating generalized moments appearing as coefficients in series expansions for F(z).

These problems are related to a theory of orthogonal rational functions on T (with poles

at the points l/iin). This relationship generalizes the relationship between CCP (or TMP)

and the theory of orthogonal polynomials on T. This approach to the Nevanlinna-Pick

theory from the point of view of orthogonal functions has not fully been put forward in

the past. From a purely mathematical point of view the theory of such orthogonal rational

functions as far as we know was initiated by Djrbashian about 1960 (see the survey paper

(20)).

2

Independently, partly from an applied point of view, the same constructions were used by

Bultheel, Bultheel and Dewilde, Dewilde and Dym about 1980 (see [2, 3, 4, 19]). Special

cases involving cyclic repetition of a finite number of points have later been discussed in

detail (see [6, 10, 25]). The groundwork for a systematic general treatment can be found

in [5], see also [7, 8, 9]. - A survey of applications of the Nevanlinna-Pick theory in circuit

and other work in the same direction, is given in [17].

When interpolation points are situated on the boundary T of D, the interpolation condition

for F( z) must be formulated in terms of limits and asymptotic expansions in angular

regions at the interpolation points. In such situations the theory of the orthogonal rational

functions partly change character.

The Nevanlinna-Pick problem for H is concerned with finding a Nevanlinna function with

interpolation properties analogous to those above. By using the Cayley transform, a theory

of this half-plane situation can be obtained from the disk situation above. To the theory

of orthogonal rational functions on T there corresponds a theory of orthogonal rational

functions on R. When all the interpolation points coalesce at the (boundary) point at

infinity, the corresponding interpolation problem (in this situation asymptotic expansion

problem) is equivalent to the classical Power Moment Problem (PMP). The correpsonding

orthogonal rational functions are orthogonal polynomials. (See[l, 28, 39, 40, 56, 58, 62].)

The fact that the interpolation point corresponding to orthogonal polynomials on T is an

interior point while the interpolation point corresponding to orthogonal polynomials on Ris a boundary point gives rise to differences in behavior between these classes of orthogonal

polynomials.

When the interpolation points arise by cyclic repetition of 0 and 00, the interpolation

problem is equivalent to the so-called strong or two-point moment problem, and orthogonal

rational functions are orthogonal Laurent polynomials. (See [33, 34, 36, 37, 38, 50].) The

situation when a finite number of points on the boundary R are repeated cyclically has

been treated in [26, 29, 32, 46, 47, 48, 49].

2 The Caratheodory Coefficient Problem

We recall that the CCP is the following: For a given sequence {cn : n = 0,1,2, ... } of

complex numbers, find a Caratheodory function F( z) having the Taylor series expansion

F(z) = I::~=oCnzn at the origin. For converience we shall in the following set J.lo = Co, J.ln =~Cn for n = 1,2, .... We further recall that the TMP is the following: For a given sequence

3

{fln : n = 0, 1,2, ... } of complex numbers, find a measure fl such that fln = J~",e-inB dfl( B)

for n = 0,1,2, .... The connection between CCP and TMP can e.g. be described along the

following lines. (There is a vast litterature dealing with these subjects. For more details,

see references in Section 1.)

The class of Caratheodory functions is characterized by an integral representation formula

of the form

where

D(t,z)=t+z t - z'

fl is a measure and v is a real constant. See e.g. [1, 30]

(2.1 )

(2.2)

We shall in the following only consider CCP for normalized functions, i.e. functions where

F(O) > 0 or equivalently where v = O.

By expanding D( eiB, z) in powers of z and integrating with respect to a measure fl weobtain the formula

Thus the coefficients in the Taylor series expansion

00

FJj(z) = flo + 2 L flnznn=l

for the Caratheodory function FJj(z) are given by the moments

fln= 1'" e-inBdfl(B), n=0,1,2, ....-1r

(2.3)

(2.4)

(2.5)

On the other hand, if there exists a measure fl such that the moments (2.5) are the

coefficients in the Taylor series expansion (2.4) with F(z) for FJj(z), then F(z) = FJj(z)

and so F( z) is a Caratheodory function. These connections demonstrate the equivalencebetween CCP and TMP.

In the following it will be convenient also to make use of the "negative moments" fln defined

by

fln = H_n for n = -1 -2r , , ....

4

(2.6)

It is well known that a necessary and sufficient condition for TMP to have a solution (with

infinite spectrum) is that the sequence {{In : n = 0, ±1, ±2, ... } is positive definite, i.e.

that the forms Lk,m=-n {lk-mXkXn are positive definite for all n. An equivalent condition

is that all the Toeplitz determinants ~n = det({l_k+m : k, m = 0,1, ... , n) are positive. The

solution is unique.

The solution {l can be obtained in various ways. A fruitful approach is to utilize the

connections between TMP (or CCP) and the theory of orthogonal polynomials on the unit

circle T (Szego polynomials).

For a given function J(z) defined on C we define

J.(z) = J(l/z). (2.7a)

For a function given by an analytic expression (rational function, power series expansion)this can also be written as

J.(z) = J(l/z)

where J denote complex conjugation of the coefficients.

(2. 7b)

Let IIn denote the space of all polynomials of degree at most n, and IIn. = {J. : f E IIn}.

It is easily seen that IIn. equals the space of all functions in the linear span of {l,~, ...,~ }Z ~nand that IIn + IIn. equals the space A-n,n of all Laurent polynomials of the form L( z) =

Cnn + ... + cnzn. We write II = U;::'=oITn, IT. = {J. : J E II} = U~=OITM' A = U;::'=oA-n,n =zII + II •. We define the linear functional M on A by

M(zn) = {l-n, n = 0, ±1, ±2, .... (2.8)

Thus M is defined primarily by the given moments on II. and then by (2.6) also on II.

The functional M defines an inner product < , > on II x IT defined by

< P, Q >= M(P(z)Q.(z)).

To solve the TMP now means to find a measure {l such that

(2.9a)

(2.9b)

The monic orthogonal polynomials (with respect to the inner product (2.9 a)) are called

(monic) Szego polynomials. The reciprocal polynomials p~(z) are defined by

(2.10)

5

A polynomial of the form

Kn(z, w) = Pn(z) + wp~(z) (2.11)

is called a paraorthogonal polynomial of order n, when I w 1= 1. It is seen that Kn( Z, w)

is orthogonal to all powers zm, m = 1,2, ... , n -1. While the zeros of Pn(z) and p~(z)

are contained in D and E, respectively, the zeros of Kn(z, w) lie on T and give rise to a

quadrature formula.

Theorem 2.1 The zeros of Kn(z, w) for Iwl = 1 are all simple and contained in T. Let

the zeros be denoted by dn)(w) = eiO~n)(w), k = 1,2, ... , n, and set >'1n)(w) = M(L1n)(z, w)),

where L1n)(z, w) denote the fundamental interpolation polynomials determined by the zeros.

Then >'1n)(w) > 0, k = 1,2, .." n, and the quadrature formula

n

M(P) = L >.~n)(w)p(dn)(w))k=l

is valid for all P E A-(n-l),n-l'

For proof see e.g. [36].

In particular we get from (2.12) the formulan

Jim = M(z-m) = L >'1n)(w)[(kn)(w)tm, Iml < n.k=l

(2.12)

(2.13)

By introducing the measure Jin(O,w) on [-71",71"] defined by having a point mass >'1n)(w) at

each of the points 01n)(w), we may write (2.13) as

(2.14)

The solution of the TMP, or equivalently the CCP, may then be obtained as follows: The

family {Jin(O, w)} is uniformly bounded sincen

o S; Jin(E, w) S; Jin([-71", 71"],w) = L>'1n)(w) = M(l) for E C [-71",71"].k=l

(2.15)

Hence by using compactness arguments (Helly's theorems) together with (2.14) we obtain

a measure Ji such that

Jim = J~e-imOdJi(O), m = O,±1,±2, ....

6

(2.16)

\Vhen J.l is known, also the function

(2.17)

is known in princi pIe. We shall discuss how the theory of orthogonal functions also provides

more direct connections with the function FI'(z), which solves the CCP.

The functions Pn(z), p~(z) are needed both for obtaining J.l and FI'(z), We first describe

how these functions can be calculated recursively. In the following we set

Dn = pn (0) = _ < Z Pn-l, 1 >< p~_I,l > .

These parameters Dn are called Szego parameters or reflection coefficients.

(2.18)

Theorem 2.2 The orthogonal polynomials Pn(z) and their reciprocals p~(z) satisfy the

following recurrence relation:

Po = P~ = 1.

For proof, see e.g. [22, 36].

Polynomials O'n(z) of the second kind are defined by

zmO'n(z) = M {D(t, z)[Pn(z) - -Pn(t)]}, n = 1,2, ... m = 0,1, ...n - 1.

tm

(2.19a)

(2.19b)

(2.20a)

(2.20b)

The value is independent of m. It is easily verified that O'n(z) E IIn. (Here the functional

M is understood to operate on the argument as a function of t.)

The reciprocal of an arbitrary polynomial of degree n can be defined analogously to p~(z)

in (2.10). It can then be verified that the following formulas are valid:

(2.21a)

n-mO'~(z) = M {D(t, z)[_z -p~(t) - p~(z)]}, n = 1,2, ... m = 0,1, ... (n - 1). (2.21b)tn-m

(See [36].)

7

Theorem 2.3 The polynomials of the second kind O"n(z), O"~(z) satisfy the following re­currence relation:

(2.22a)

(2.22b)

For proof, see [36].

We define polynomials of the second kind associated with the paraorthogonal polynomials

by

Further we define the rational interpolants Sn (z, w) by

Ln(z, w) _ O"n(Z) - wO"~(z) for wEe.Sn(Z, w) = ]{n(Z, w) - Pn(Z) + wp~(z)

Note that in particular

O"n(Z) O"~(Z)

Sn(Z,O) = -(-)' Sn(Z, (0) = -~.pn Z Pn Z

(2.23)

(2.24 )

(2.25 )

follows:

Below we give interpolation and convergence results for these functions with respect to

Fp.(z) for Z E C and Z E E.

Theorem 2.4 The functions Sn(z, w), O"nizj, - O"~izj interpolate the function FJ1.(z) aspn Z Pn Z

z-n[FI'(z) - Sn(Z, w)] = 0(1), zED, I w 1= 1,

zn[Fp.(z) - Sn(z, w)] = 0(1), Z E E, I w 1= 1,0"* ( Z )

z-(n+1)[F (z) + _n_] = 0(1) zEDp. P~ ( Z ) ,

zntl[F (z) - O"n(Z)] = 0(1), Z E EI' Pn(Z)

z-n[FI'(z)Pn(z) - O"n(z)] = 0(1), zED

zn[Fp.(z)Pn(z)* + O"~(z)] = 0(1), z E E.

8

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31 )

For proof, see [34, 35,36]. It follows from these results that O"n~ Z ~ is a weak (n, n) two-pointpn Z

Pade approximant to FJ'(z) of order (n, n + 1), while - O"~~Z~is a weak (n, n) two-pointp~ Z

Pade approximant to FJ'(z) of order (n + 1,n). If 8n =1= 0, then O"niz~ and - O"~~Z~are alsoPn Z P~ Z

strong two-point Pade approximants. See [34, 35].

Let 1 w 1= 1. It can be seen from (2.20) - (2.21) that we may write Ln(z, w) = M {D(t, z)[I<n(z, tv)­

~J{n(t, tv)]} (see [36]). Here D(t, z)[I<n(Z, w) - ~I<n(t, w)] E A-(n-1),n-1, hence by (2.12)we get

n

Ln(z,w) = L ).~n)(w)D(dn)(w),Z)I<n(Z,W).k=l

Thus we have proved:

Theorem 2.5 For 1 w 1= 1 the following partial fraction decomposition is valid:

We note that (2.33) may be written as

j1r eiO + ZSn(z, w) = '0 dlln(O, w).-1r e' - Z

Theorem 2.6 The following convergence results hold:

lim Sn(z, wn) = FJ'(z), locally uniformly on DUE,n-+oo

where Wn are arbitrary points on T

lim - O"~~Z~= FJ'(z), locally uniformly on Dn-+oo P~ Z

1· 0" n(z ) ( ) 11' I1m -(-) = FJ' z, loca y uniform yon E.n-+oo pn Z

(2.32)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

For proof of (2.36) and (2.37), see [34, 35]. Formula (2.35) easily follows from Theorem 2.5

together with Formula (2.34).

3 The Nevanlinna-Pick Interpolation Problem in D.

We recall that the NPIP in D may be formulated as follows: For a given sequence {on

n = 0,1,2, ... } of not necessarily distinct points in D and a given sequence {wn : n =

9

0,1,2, ... } of complex numbers, find a Caratheodory function F(z) which interpolates

{wn} at {an}. This means that F(an) = Wn for all n, with appropriate modifications

(Hermitean interpolation) when repetitions of points occur. Thus if e.g. an = a for an

infinite subsequence {an(k)} of {an} and Ck = t Wn(k), the interpolation requirement isF(z) = L:~oCk(Z - a)k. If in particular an = 0 for all n, then NPIP reduces to CCP.

We shall briefly discuss the connection of this problem with certain generalized moment

problems, and describe the relationship with the theory of certain orthogonal rational func­

tions generalizing the relationship between CCP- TMP and the theory of Szego polynomi­

als. We shall also here consider only the situation where the Caratheodory function is

normalized, so that the relevant functions are again characterized by the formula

(3.1 )

We shall for convenience assume that ao = 0 (which can always be obtained by a simple

substitution) .

In the general situation an appropriate series expansion of an interpolating function is a

Newton series. (See [16, 21, 64].) We shall use the notation

n

wo(z) == 1, wn(z) = II(z-ak) for n = 1,2, ....k=l

The function D( t, z) has a formal series expansion of the form

00

D(t, z) = 1 + 2 L An(t)ZWn-l(Z),n=l

where it can be seen by induction that

1

An(t) = -() , n = 1,2, ....Wn t

(3.2)

(3.3)

(3.4)

(Cf. [21].) By formally integrating with respect to a measure J.l we obtain the formula

(3.5)

We note that if all the interpolation points an are contained in a disk Dr = {z E C : Izi ::::;

r} with r < 1, the series (3.3) converges uniformly to D(t, z) for It I ::::;1. (See e.g. [64].)The series on the right in (3.5) then converges to the integral on the left. Thus at least in

this situation Equation (3.5) expresses a real equality.

10

The coefficien ts in the formal Newton series expansion

00

FI'(z) = po + 2 L pn ZWn-l(Z)n=l

for the function FI'( z) are given by the" general moments"

(3.6)

(3.7)

Generalized moments of a measure p were as far as we know first introduced and used in

[41].

In special situations the problem may be approached from slightly different angles. We

shall look closer at what we may call the finite situation, where the sequence {an} consists

of ao = 0 and a finite number of points which are each repeated an infinite number of

times in some order. (Note that this case falls under the situation commented on above,

where the formal Newton series actually converges to the function.) Let a be one of these

points. The interpolation condition at this point is then that the Taylor coefficients at a

of the desired function take given values:00

F(z) = mo(a) + 2 L mn(a)(z - at .n=l

(3.8)

By expanding D(eiB, z) in powers of (z - a) and integrating with respect to a measure pwe obtain

J:1r D(eiB, z)dp(O) = i1r1r::: ~: dp(O) + 2 E(z - at i1r1r' ~~Bdp(O)(3.9)

We thus see that a necessary and sufficient condition for a solution of NPIP in this situation

to exist is that there exists a measure p with the following properties: For each of the

interpolation points a, the coefficients in the series expansion (3.8) are given by

and in addition

j1r eiBmn(a) = ('B \ 1dp(O)forn=1,2, ...-1r et - a n+

F(O) = i:dp(O).

11

(3.10a)

(3.10b)

(3.11)

The above considerations indicate the relationship between NPIP and various generalized

moment problems. In the case that all an are distinct, a necessary and sufficient condition

for the NPIP to have a solution is that all the forms

(3.12)

are positive definite. (See [39].) For the case that repetition of an's occur, the relevant

forms are more complicated. (See [5].) A sufficien t condition for uniqueness involving only

the interpolation points {an} is that00

L::(1-lanl) = 00.n=l

(3.13)

(3.14)

Various weaker sufficient conditions also involving the values {wn} (Carleman-type condi­

tions) can be obtained. (For an approach to uniqueness problems, see in particular [24,

41].)

We shall now consider connections between general moment problems (or NPIP) and the

theory of orthogonal rational functions associated with it. For earlier work dealing with

these functions or this approach to Nevanlinna-Pick theory, see [2, 3, 4, 5, 6, 7, 8, 9, 19,

20]. Cf. also [51, 52].

Let {an : n = 0, 1,2, ... } be an arbitrary sequence in D, with 0'0 = O. We introduce the

Blaschke factors (k(Z) defined by

ak (ak - z)

(k(Z) = -I -, . ( _ \ , k = 1,2, ....ak 1- akz

We assume by convention that ~ = -1 when ak = 0, so that (k(Z) = Z when ak = O. Weemphasize that in the following, all results reduce to results dealing with Szego polynomials

and CCP-TMP when an = 0 for all n.

We define the Blaschke products En byn

Eo = 1, En = II(k(Z) for k = 1,2, ....k=l

We also introduce polynomials 1rn(z) defined byn

1ro(z) = 1, 1rn(z) = II(l-akz) for n = 1,2, ....k=l

Note that

12

(3.15)

(3.16)

(3.17)

where wn(z) is defined by (3.2).

Vve consider the spaces £n and £ given by £n = Span{Eo, El"'" En} and £ = U~=1£n'

Note that when all the points an are distinct we have also £n = Span {1, 1 , ... , 1 },z - al z - an

1 1and when ak i= 0 for k = 1,2, ... , n, we have £n = Span{1, -(-)' ... , -(-)}. The space11"1 z 11" n Z

£n consists of rational functions whose poles (in C) are contained in the set {1/ ak : k =1, ... , n}. These poles are contained in E. We observe that if an = 0 for all n, then £n

is the space I1n of polynomials of degree at most n. In general a function R( z) in £n is a

function that may be written as

q(z)

R(z) = -(-) , where q E I1n.1I"n Z(3.18)

For a function J(z) given on C the substar transform J.(z) is defined by (2.7) as before.

For a function R(z) = q([\ E £n - £n-l we define the superstar transform R·(z) by1I"n Z

(3.19)

We define the spaces £n. and £. as the spaces consisting of all functions of the form R. (z ),

where R(z) E £n and R(z) E £, respectively. Note that R.(z) E £M and R·(z) E £n when

R(z) E £n.

Let M be a given linear functional on £•. It may arise from a NPIP through the formulas

1M(-(-)) = /1n , n = 0,1,2, ... ,Wn Z

{/1n} being the coefficients in the formal Newton series expansion

00

F (z) = /10 + 2 L /1n Z Wn-l ( Z )n=1

(3.20)

(3.21 )

(d. (3.6)). (Note that {1, -h,...,-n,...}is a base for £•.) In the finite situation,WI Z Wn Z

i.e. where {an} consists of ao plus a finite number of points each repeated an infinite

number of times, the functional M may e.g. be defined from the coefficients mn(a) by

z+a z

M(1) = F(O), M(--) = mo(a), M( ( ~ 1) = mn(a), n = 1,2, ... , (3.22)z - a z - a n+

z+a z .(d. (3.8)-(3.10)). (Note that {1, --, ( \?" •• } IS a base for £. when a i= 0.)z-a z-a·

13

By defining

(3.23)

we obtain by linearity a linear functional M on ,c + ,c., and the form (, ) defined by

(R, S) = M(R(z)S.(z)) , R, S E ,c (3.24)

is Hermitean on ,c x,c. (Note that R(z)S.(z) E ,c+,c. when R, S E ,c, by partial fraction

decomposition.) (See also [25].) We further assume that (, ) is positive definite on ,c x ,c

(which will be the case if M is obtained from a NPIP). It then follows (e.g. by using results

in [39]) that there exists a measure J-l such that

(3.25 )

Vie shall briefly describe how the measures J-l and the functions FJL(z) are connected with

the orthogonal functions obtained from the bases {Bn}.

Let {tpn} denote the sequence of monic orthogonal functions obtained by orthogonalization

of {Bn} with respect to (,). (Monic here means that bn = 1 in the expansion tpn(z) =Lk=O bkBk(z).) It follows from (3.17)-(3.19) that we may write

tpn(z) = qn(Z)7rn(z) , qn E I1n

(3.26a)

(3.26b)

(3.26c)

m

We note that since tpn(z) is orthogonal to all the functions z ( \' m = 0,1, ... , n - 1,7rn-l Zwe may write

(3.27)

where

(3.28)

14

Thus the sequence {qn} of polynomials are orthogonal with respect to the sequence of

varying complex measures {Jin}. (Cf. [42,43,59,60].)

As in the polynomial case, the zeros of ~n(z) and ~~(z) are contained in D and E,respectively, and may be multiple. See [5].

We introduce functions Qn(z, w) in Ln which may be called paraorthogonal functions when

Iwl = 1. They are defined by

(3.29)

The function Qn(z, w) is orthogonal to Ln-l n (nLn-l' i.e. to all functions R(z) which are

both in Ln-l and of the form (n(z)L(z) with L(z) E Ln-l'

Theorem 3.1 The zeros ofQn(z,w) for Iwl = 1 are all simple and contained in T. Let

the zeros be denoted by (t\w) = eiO~n)(w), k = 1, ... , n. There exist positive constants

A~n)(W) such that the quadrature formula

n

M(R) = L A~n)(w)R(dn)(w))k=l

is valid for R E Ln-l + L(n-l).'

A proof can be given along the same lines as in the polynomial case. See [5].

It follows from (3.30) that

1 _ ~ (n) 1 _M( ())-L.-JAk (w) (n) , m-O, ... ,(n-l).Wm Z k=l wm((k (w))

(3.30)

(3.31 )

(3.32)

By introducing finite point mass measures Jin(f), w) defined in the same way as in Section

2 we may write (3.30) as

1 111"dJim(B, w)M(-(-)) = (iO) , m = 0, ... , n - 1.Wm Z -11" Wm e

As in the polynomial case we may then obtain one or more measures Ji as limits of subse­

quences of {Jin (B, w)} which are solutions of the general moment problem in the sensethat

M( _1_)= 111" dJi(O)wm(z) -11" wm(O)' m = 0,1,2, ....(3.33)

As in the polynomial case the functions ~n(Z), ~~(z) can be calculated recursively by arecurrence relation.

15

Theorem 3.2 The orthogonal functions <Pn(z) and their superstars <p~(z) satisfy a re­

currence relation of the following form (n = 1,2, .. .):

For proof see [5, 7].

It will here be convenient to introduce the kernel E(t, z) defined by

2tE(t,z) = 1+ D(t,z) = -.t-z

Rational functions 'l/;n (z) of the second kind are defined as follows:

'1/;0 = -f.lo

Bm(z)

M {D(t, z)<Pn(z) - E(t, z) Bm(t) <Pn(t)}, n = 1,2, ...m = 0,1, ... n - 1.

(3.35)

(3.36a)

(3.36b)

It is easily seen that 'l/;n(z) E Ln-l' and it follows by orthogonality properties that the

value is independent of m. See [5, 7].

The superstar functions 'I/;~(z) may then be written as

'I/;~(z )

(3.37a)

(3.37b)

Theorem 3.3 The functions of the second kind 'l/;n(Z) and 'I/;~(z) satisfy the following

recurrence relation (n = 1,2, .. .):

The constants are the same as in Theorem 3.2.

16

For proof, see [5, 7].

We introduce what may (when Iwl = 1) be called paraorthogonal rational functions of thesecond kind

(3.39)

(3.40 )

and rational interpolants

Rn(Z,w) = Pn(z,w) = 1fn(z) - w1/J~(z) .Qn(z, w) 'Pn(z) + w'P~(z)

N h' . 1 R ( 0) 1/Jn(z) Q ( ) 1/J~(z)ote t at In partlcu ar n Z, = --( -) , .lLn Z,OO = --(-)''Pn Z 'P~ Z

We shall present interpolation and convergence results connecting these functions with

FJ-L(z),

Theorem 3.4 The functions 1/Jn~Z~, - 1/J:~Z~ interpolate the function FIJ(z) as follows:'Pn Z 'Pn Z

(3.41a)

(3.41 b)

(3.41c)

(3.41d)

Here g~O), hhO) are analytic functions for zED with g~O)(O)= hhO)(O) = 0, while g~cc), hhcc)

are analytic functions for Z E E with g~oo)((0) = hhOO) ( 00) = O.

These results can be obtained from [5, 7].

Theorem 3.5 The following convergence results hold ifE~=l (1- I an I) = 00:

lim (- ~~~Z~) = FIJ(z), locally uniformly on Dn-+oo 'f/~ Z

lim (~n~z~) = FIJ(z), locally uniformly on En-+cc 'f/n Z

17

(3.42a)

(3.42b)

The proof of (3.42a) can be found in [5], and then also (3.42b) follows.

Note that it follows from this result that the solution J.L is unique when L~=l (1- 1 an I) =(X).

We now consider the cyclic situation, when {an: n = 1,2, ... } consists of a finite number

p of points cyclically repeated.

Theorem 3.6 In the 0J.EJic situation the function Rn(z, w) for n > p interpolates the

function FJ1.(z) as follows:

1[FJ1.(z) - Rn(z, w)] = .r;:(z), I w 1= 1, (3.43a)n-ll z'

~[FJ1.(Z) - Rn(z, w)] = f:(z), I w 1= 1. (3.43b)n-l Z

Here fAO) is an analytic function for zED with fAO)(O) = 0, and fAoo) is an analytic

function for z E E with fAOO) ( 00) = O.

The proof is analogous to the proof in the polynomial case (see [36]), it being taken into

account that (n(z) is orthogonal to <Pn(z) for n > p.

By using similar orthogonal properties it can be verified that Pn(z, w) may be written as

(n(z)

Pn(z, w) = M {D(t, z)[Qn(Z, w) - -( ) Qn(t, w)]}. (3.44)(n t

Here the function D(t,z)[Qn(z,w) - 2:~~~Qn(t,W)] can be seen to belong to Ln-l +L(n-l) •.when n > p. By the quadrature formula (3.30) (with I w 1= 1) we obtain

n

Pn(z,w) = L A~n)(W) D(dn)(w),Z)Qn(Z,W)k=l

(3.45 )

since Qn(dn)(w), w) = O. Thus we have proved the partial fraction decomposition formula

n (n) dn)(w)+z j7r eiO+zRn(z,w) = L Ak (w) ( ) = "0 dJ1.n(B,w). (3.46)k=l (k n (w) - Z -7r ea - z

Theorem 3.7 In the cyclic situation the following convergence results holds ifo;.--~--- ,-~-

L~=l(1- I an I) = 00:

lim Rn(z, wn) = FJ1.(z), locally uniformly on DUE when Wnn-+oo

are arbitrary points on T.

18

(3.47)

The proof is a standard argument making use of the fact that {ltn(B, wn)} converges to theunique solution It.

4 Interpolation points on the boundary

We now consider the situation when some or all of the interpolation points an are allowed

to belong to the boundary T of D. The interpolation condition is then to be understood

in the following sense: limz-+an F( z) = Wn, where the limit is required to exist in every

angular region at an with opening less than 7r, centered on the normal. If an interpolation

point is repeated, the interpolation conditions involve the appropriate number of deriva­

tives. If an interpolation point is repeated an infinite number of times, the interpolation

condition is that F( z) has a given power series as asymptotic expansion in the angularregions described above.

Some of the properties concerning solution of the NPIP or the general moment problem

carryover to this situation. We shall briefly describe the situation when all the interpola­

tion point belong to T. (Except that we shall also here assume 0'0 = 0.) For more details,see [11].

We define the spaces .Ln by

II}S {I --- ... , ~( ) ,.Ln = pan , 7rI(Z)' 7rn Z

(4.1)

and .L = U~=o.Ln' The singularities of the functions in .L are among the interpolation

points, since Iliin = an· Thus.Ln = .LM = Span {I, -n,...,---T1} and .L. = .c. TheWI Z Wn Z

original functional M on .L = .L. in this situation has to be extended to the space .L . .L

consisting of all functions of the form R(z)S(z), where R, S E .L. (Note that in the finite

situation, or generally when all the interpolation points are repeated infinitely many times,

then .L . .L = .L.) We thus assume that a positive definite Hermitian functional on .L . .c is

given. Let {<Pn} be orthogonal functions obtained from the bases .Ln.

We easily verify that

<Pn.(Z) = Cn<Pn(Z), Cn a constant. (4.2)

It can be seen that if Zo is a zero of the numerator polynomial qn(z) of <Pn(z) (see (3.26))

then Ilio is also a zero (Cf. [11]). A standard argument then shows that the zeros of

qn(z) are simple and lie on T. Some of the zeros of qn(z) may be among the singularities

19

(interpolation points) aI, ..., an-I, since these are now on T. (However, all zeros of qn(z)

are different from an.) The degree of qn(z) may also be less than n. It follows that CPn(z)

may have fewer zeros than n. The system {CPn} is called regular if CPn(an-d =I- 0 for all n.When this is the case, the quasiorthogonal functions

Q () ( ) 1+ an z - an-ln z,r = cpn Z + r 1 ---~CPn-l(Z), r E R+ an-l Z - an (4.3)

have n simple zeros on T, for all but at most n + 1 values of r. Quadrature formulas

associated with these zeros valid for .en-I' .en-1 can be obtained so that existence of

a measure J.l and a function F/l.(z) solving the moment or interpolation problem can be

proved. (In the case when CPn( z) itself has n zeros, the quadrature formula is valid for

.en . .en-I.) Note that in the boundary situation, the paraorthogonal functions are just the

orthogonal functions, cf. (4.2).

The recurrence formulas (3.34) and (3.38) and the argument establishing them break down

in the boundary situation. (Recall that in the formulas the expression 1- I an-l 12 occurs

in the denominator.) However, when the system {CPn} is regular, a three-term recurrence

relation connecting three consecutive orthogonal functions can be obtained. (Cf. [11].)

Interpolation problems and moment problems associated with the half plane H and the

real line R can be related to the problems for D and T through the Cayley transform,

given by

With the definition

Z-i 1+Z=-- Z . ZZ+" =Z--, zED ZEHZ 1- z ' .

G(Z) = iF(z)

(4.4 )

(4.5)

we thus have that G(Z) is a Nevanlinna function when F(z) is a Caratheodory function,

and vice versa. To the functions (n (z) correspond the functions

Z -AnOn(Z) = Z - An (4.6)

where An is the interpolation point whose corresponding z-value is an = ~:+~.A base forthe space Mn corresponding to .en consists of {Eo, E1, ••• , En}, where

k

Eo = 1, Ek(Z) = IIOm(Z) for k = 1,2, ...m=l

20

(4.7)

when the interpolation points are in H, and of {To, T1, ••• Tn}, where

k 1To = 1, Tk(Z) = II /r7 ,\ for k = 1,2, ...

m=l

when the interpolation points are on R.

(4.8)

(4.9)

The functions {4>n(Z)} obtained from the functions {CPn(z)} through the Cayley transform

(4.4) are orthogonal functions corresponding to the base {En}.

The situation on R (equivalent to a situation on T through the Cayley transform) when a

finite number of interpolation points are cyclically repeated has been studied in detail in

[25, 29, 46,47,48,49, 50, 53]. General moment problems related to these have also been

studied in [15], where other methods than the theory of orthogonal functions have beenused.

Let in particular all the points an on T coalesce to one point, for simplicity the point 1.

The functions in £n are in this situation of the form

n bk

R( z) = L -(- ...k=O Z - 1

This is essentially a polynomial situation. By the Cayley transform it is carried over to a

situation where the orthogonal functions are orthogonal polynomials on R, corresponding

to all the interpolation points on R coalescing at infinity. The associated moment problem

(wi th certain restrictions on the interpolating function, see [30]) is the classical Power

Moment Problem (PMP). See [1, 27, 56,58, 61].

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o. Njastad

Department of Mathematics

University of Trondheim-NTH

N-7034 Trondheim

Norway

25