Orthogonal Polynomials in Two Variables [3] 9056991671, 9789056991678

Presenting a comprehensive theory of orthogonal polynomials in two real variables and properties of Fourier series in th

425 115 15MB

English Pages 368 [376] Year 1999

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Orthogonal Polynomials in Two Variables [3]
 9056991671,  9789056991678

Table of contents :
General properties of polynomials orthogonal over ..............1
Some typical examples and special cases ..............37
Classical Appells orthogonal polynomials ..............63
Admissible differential equation for polynomials ..............87
Potentially selfadjoint equation and Rodrigues ..............131
Harmonic polynomials orthogonal over a domain ..............163
Polynomials in two variables orthogonal on ..............187
Generalized orthogonal polynomials in ..............223

Citation preview

Analytical Methods and Special Functions An International Series of Monographs in Mathematics

Orthogonal Polynomials in Two Variables P.K. Suetin

Gordon and Breach Science Publishers

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/orthogonalpolynoOOOOsuet

Orthogonal Polynomials in Two Variables

ANALYTICAL METHODS AND SPECIAL FUNCTIONS

An International Series of Monographs in Mathematics EDITOR IN CHIEF: A.P. Prudnikov (Russia) ASSOCIATE EDITORS: C.F. Dunkl (USA), H.-J. Glaeske (Germany) and M. Saigo (Japan)

Volume 1 Series of Faber Polynomials P.K. Suetin Volume 2 Inverse Problems for Differential Operators V. A. Yurko Volume 3 Orthogonal Polynomials in Two Variables P.K. Suetin Additional Volumes in Preparation Bessel Functions and Their Applications B.G. Korenev Hypersingular Integrals and Their Applications S.G. Samko Fourier Transforms and Approximations AM. Sedletskii

This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

ebomus j. bate Library

fRENI UNIVERSITY PETERBOROUGH, ONTARIO

Orthogonal Polynomials in Two Variables P.K. Suetin

Technical University of Communication and Informatics Moscow, Russia

Translated from the Russian by E.V. Pankratiev

GORDON AND BREACH SCIENCE PUBLISHERS Australia • Canada • China • France • Germany • India • Japan • Luxembourg Malaysia • The Netherlands • Russia • Singapore • Switzerland

Qf\

I

Copyright © 1999 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. All rights reserved. Originally published in Russian in 1988 by Nauka, Moscow © 1988 No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Singapore.

Amsteldijk 166 1st Floor 1079 LH Amsterdam The Netherlands

The camera-ready copy for this book was typeset using the AMS-TEX macro software package.

British Library Cataloguing in Publication Data Suetin, P. K. Orthogonal polynomials in two variables. - (Analytical methods and special functions; v. 3) 1. Orthogonal polynomials I. Title 515.5’5

ISBN: 90-5699-167-1 ISSN: 1027-0264

CONTENTS

Preface

ix

Preface to the English Edition

xv

Notation

xix

Chapter I. General properties of polynomials orthogonal over a domain § 1. Polynomials in two variables orthogonal over a domain § 2. The existence theorem and criteria of orthogonality

1 1

§ 3. Algebraic properties

6 10

§ 4. Monic orthogonal polynomials

18

§ 5. Normal biorthogonal systems

24

§ 6. Fourier series of orthogonal polynomials in two variables

28

§ 7. Fourier series for differentiable functions

31

Chapter II. Some typical examples and special cases of orthogonality over a domain § 1. Different products of classical orthogonal polynomials

37

37

§ 2. Various cases of connection between orthogonality over a domain and orthogonality on an interval

42

§ 3. Some theorems in the case of a weight function with separating variables

48

§ 4. Conditions of interconnection between the weight function and the domain of orthogonality § 5. Examples of computations of weight function moments

Chapter III. Classical Appell’s orthogonal polynomials § 1. Rodrigues formula for Appell’s polynomials

52 57 63

63

§ 2. Representation of the Appell polynomials via the hypergeometric 69

function of two variables § 3. Differential equation for the Appell polynomials

72

§ 4. Orthogonality of eigenfunctions of the Appell equation

75

§ 5. Normal biorthogonal Appell system

79

§ 6. Series in the Appell polynomials

83

Chapter IV. Admissible differential equation for polynomials orthogonal over a domain

87

§ 1. The main differential operator and a theorem on orthogonality

87

§ 2. Admissibility conditions for the main differential equation

92

v

CONTENTS

VI

§ 3. Some examples and properties of admissible differential equations

97

§ 4. Affine transformations of the arguments of the main differential

101

equation § 5. Transformation of the coefficients of the characteristic polynomial

105

§ 6. Normal forms of the admissible differential equation

115

§ 7. Normal forms when reducing the degree of the characteristic polynomial

123

Chapter V. Potentially self-adjoint equation and Rodrigues formula

131

§ 1. Potentially self-adjoint operators

131

§ 2. Admissible and potentially self-adjoint equations

135

§ 3. Rodrigues formula for polynomials orthogonal over a domain

146

§ 4. Weight functions and the Rodrigues formula in the most typical 153

cases Chapter VI. Harmonic polynomials orthogonal over a domain

163

§ 1. Homogeneous harmonic polynomials

163

§ 2. An analogue of the Christoffel-Darboux formula

169

§ 3. Harmonic polynomials orthogonal in the unit disk

173

§ 4. Harmonic polynomials orthogonal over a domain in the general case

176

§ 5. Harmonic polynomials superorthogonal over a domain

179

Chapter VII. Polynomials in two variables orthogonal on a contour

187

§ 1. Alain definitions and the simplest properties

187

§ 2. Polynomials in two variables orthogonal on an algebraic curve

191

§ 3. Harmonic polynomials orthogonal on a contour

196

§ 4. Fourier series in harmonic polynomials orthogonal on a contour

200

§ 5. Harmonic polynomials superorthogonal on a contour

206

§ 6. Examples of superorthogonal systems of harmonic polynomials

213

Chapter VIII. Generalized orthogonal polynomials in two variables

223

§ 1. Alain definitions and the simplest properties

223

§ 2. The existence theorem in the most general form

228

§ 3. Fourier series in generalized orthogonal polynomials in two variables § 4. Atonic orthogonal polynomials under minimal conditions

233 241

§ 5. Generalized generating functions for monic orthogonal polynomials

247

Chapter IX. Other results concerning the connection between orthogonal polynomials and differential equations

253

§ 1. Canonical admissible differential equation and monic orthogonal polynomials

253

§ 2. Necessary consistency conditions of the canonical operator and the functional

258

CONTENTS

vii

§ 3. Sufficient conditions of consistency of the canonical operator and the functional

262

§ 4. The deduction of the differential equation from the Pearson equation system § 5. An admissible partial differential equation of an arbitrary order Chapter X. Original results of T. Koornwinder

268 276

285

§ 1. Examples of the representation of polynomials orthogonal over a domain via the Jacobi polynomials § 2. Orthogonal polynomials in two conjugate complex variables

285 291

§ 3. The Chebyshev polynomials in two conjugate complex variables for the Steiner domain

296

§ 4. Another generalization of the Jacobi polynomials onto the case of two variables Chapter XI. Some recent results

308

313

§ 1. A new generalization of the Appell polynomials

313

§ 2. Some properties of the Koornwinder-Steiner polynomials

318

§ 3. A two-dimensional analogue of the Chebyshev-Laguerre polynomials

319

Comments and Supplements

323

References

329

Author Index

343

Subject Index

345

PREFACE

The theory of orthogonal polynomials has been developed considerably and many results of this theory are of a complete nature. The properties of different types of orthogonal polynomials under general enough conditions have been investigated. The most important and specific classes of orthogonal polynomials have been fully studied. There exists a large number of works on the theory of orthogonal polyno¬ mials distinguished by an original approach and an in-depth study of the problem. Also, the number of theoretical and applied problems solved by using orthogonal polynomials constantly grows. The first examples of classical orthogonal polynomials, as is well-known, were considered by A.M. Legendre, P.S. Laplace, J.L. Lagrange and N.H. Abel. Later P.L. Chebyshev, an outstanding Russian mathematician, developed the general theory of orthogonal polynomials and investigated some important special cases of classical orthogonal polynomials. Most further important results on the theory of orthogonal polynomials were obtained by T. Stieltjes, C. Jacobi, C. Hermite, E.

Laguerre, C. Posse and Yu.V. Sokhotskii. An effective method of investigating the asymptotic properties of classical or¬

thogonal polynomials was developed by V.A. Steklov. This method was applied in detailed study of the asymptotic properties of Jacobi, Chebyshev-Hermite and Chebyshev-Lagerre polynomials. Classical results in the theory of orthogonal polynomials were found by G. Szego. He discovered a new method of studying the asymptotic properties of polynomials orthogonal over a circumference with arbitrary weight. With the help of this method Szego derived asymptotic formulae for orthogonal polynomials outside the circum¬ ference and on the circumference itself. Further, Szego found a formula for rep¬ resenting polynomials orthogonal over a segment through polynomials orthogonal over a circumference. This formula was used for a detailed investigation of asymp¬ totic properties of polynomials orthogonal over a segment. S.N. Bernstein has developed another method for investigating the asymptotic properties of polynomials orthogonal over a segment. This method is based on the results of the theory of function approximation. The methods of Steklov, Szego and Bernstein are set forth in the monograph by Szego [A48], first published in 1939. The Russian translation of this book printed in 1962 contains some additions made by Ya.L. Geronimus giving a detailed survey of all the results on the theory of orthogonal polynomials obtained during the previous 20 years. The monograph by Geronimus [A20] develops further the methods of Szego and Bernstein. Szego introduced polynomials orthogonal over a contour in the complex domain and with the help of his method considered their asymptotic properties under the simplest conditions. Concurrently T. Carleman and S. Bergmann introduced com¬ plex polynomials orthogonal over a domain. IX

PREFACE

X

Very important results on the theory of orthogonal polynomials were obtained by E.A. Rakhmanov [C118-C120]. In his works he gives complex counter-examples connected to V.A. Steklov’s well-known problem on the theory of orthogonal poly¬ nomials [B13] and also gives different estimates for orthogonal polynomials under the most general conditions for the weight functions. V.M. Badkov [C9-C11] gives an in-depth study of the asymptotic properties of orthogonal polynomials with power and logarithm singularities of the weight function. Multiple usage of classical orthogonal polynomials in numerical analysis, mathe¬ matical physics, quantum mechanics and many other branches of science is wellknown. As an example, we can indicate operational calculus [A2, A3] and different fA'oblems of numerical analysis [A38]. However, classical orthogonal polynomials find most important use in the spectral methods of calculation and designing au¬ tomatic control systems [A44]. In a bibliographic reference book on the theory of orthogonal polynomials [A42] published in 1940 approximately 2,000 papers are mentioned. Further lists of papers and some historical background are given in monographs [A16, A19, A20, A32, A45, A46, A40]. During the last few years the number of papers on the theory of orthogonal polynomials and their applications has been growing. The text above refers only to the case of orthogonality in one variable. However, in the cases of two and more variables orthogonal polynomials have been studied much less, although their main definitions and simplest properties were considered more than 100 years ago. When generalizing classical orthogonal polynomials for two and more variables, the necessity of considering the so-called biorthogonal systems of polynomials appears. In 1865 C. Hermite considered two pairs of biorthogonal systems of polynomials in two variables when the domain of orthogonality was the whole plane or a unit disk. Later on, Hermite polynomials were generalized to the case of many variables. In 1881 P. Appell introduced polynomials in two variables biorthogonal over a triangle. Hermite and Appell polynomials are analogues and generalizations of classical orthogonal polynomials in case of two and more variables because these polynomials are the eigen-functions of some linear partial differential operators of the second order. In 1881 G. Orlov [A37] considered some orthogonal polynomials in two variables which are determined by an analogue of the Rodrigues formula. In 1926 P. Appell and J. Kampe de Feriet published a big monograph [A4], where the properties of Appell polynomials in two variables and Hermite polynomials in many variables were described in detail. The main instrument for investigations were the generalized hypergeometric functions of two and more variables. In 1938 D. Jackson [C67] considered the simplest properties of polynomials in two variables orthogonal over a domain with arbitrary weight. It turned out that the weight function and orthogonality domain for each natural n determine a linear

PREFACE

xi

space of dimension n + 1 of orthogonal polynomials of degree n. For such polyno¬ mials algebraic and extremal properties analogous to the one-dimensional case have been established and Fourier series have also been analysed for them. A detailed survey of all the above-mentioned results on the theory of orthogonal polynomials in two and more variables can be found in the monograph-reference by H. Bateman and A. Erdelyi [A12]. All the papers on the theory of orthogo¬ nal polynomials in two and more variables published before 1940 are listed in the bibliographical reference [A42]. A paper rich in content by H. Krall and I. Sheffer was published in 1967 [C94], It substantiated and generalized D. Jackson’s results concerning polynomials in two variables orthogonal over a domain. They also considered some linear par¬ tial differential operators of the second order whose eigenfunctions are polynomials orthogonal over a domain. Krall and Sheffer considered some two-dimensional ana¬ logues of classical orthogonal polynomials which are solutions of linear partial differential equations of the second order. Similar results were obtained by G.K. Engelis [C43] although he used another method of investigation and gave a more detailed list of linear partial differential equations of the second order for which orthogonal polynomials in two variables are solutions. Also, Engelis derived the Rodrigues formula for some classes of orthogonal polynomials in two variables. A considerable number of results on the theory of orthogonal polynomials in two variables were obtained by T. Koornwinder in the papers listed in the references. In these papers new systems of orthogonal polynomials were considered, new properties of the already known orthogonal polynomials were determined and much attention was given to the relationship between orthogonal polynomials and differential equations. Some of T. Koornwinder’s papers are of a review nature, where the results obtained by other authors are also analysed, and the applica¬ tions of orthogonal polynomials in two variables are described and a detailed list of reference literature is given. As is well-known [A26], orthogonal polynomials in one variable are irreplaceable tools for constructing quadrature formulae of different types. A similar situation exists in the case of orthogonality in two and more variables. And since at present the need in cubic formulae increases, the necessary properties of orthogonal polyno¬ mials in two and more variables are also studied while constructing such formulae. Many papers on this subject were published by I.P. Mysovskikh. All the results of this area of investigations are presented in a very comprehensive monograph by Mysovskikh [A32]. Most important, from the computational point of view, cubic formulae are considered there and some necessary properties of orthogonal poly¬ nomials in two and more variables are described. An analysis of the content of the monograph by Mysovskikh shows that the use of orthogonal polynomials for constructing cubic formulae outstrips and stimulates development of the theory of orthogonal polynomials in two and more variables. However, it should be noted that while orthogonal polynomials in one variable already have numerous and varied applications in many fields of science, the theory of orthogonal polynomials in two and more variables is applied insufficiently widely. It seems that some new results can be expected here.

PREFACE

xii

It follows from the above that by now many important results on the theory of orthogonal polynomials in two variables have been found. Most of them have been published in separate articles. A systematic exposition of some problems is con¬ tained only in the monographs by Appell and Kampe de Feriet [A4] and Mysovskikh [A32]. That is why it seems expedient to set forth some of the most important properties of orthogonal polynomials in two variables in a separate monograph. The main content of the first two chapters of this monograph are the results of D. Jackson [C67], S. Agakhanov [C2], S. Ogawa, S. Arioka, and S. Kida [C111]. Chapter III deals mainly with the monograph by Appell and Kampe de Feriet [A4], In this monograph much attention is paid to the relationship between orthogonal polynomials in two variables and differential equations. Chapters IV, V and IX are devoted to this subject. As has already been mentioned, in some cases orthogonal polynomials in two variables are eigenfunctions of linear partial differential equa¬ tions of the second order. Such equations are called admissible. Conditions of admissibility and a classification of admissible equations were first considered by Krall and Sheffer [C94], Another classification of admissible equations was given by Engelis [C43]. Chapter IV is based on the results obtained by Engelis although many formula¬ tions were changed and a new classification of admissible equations containing all the previously known and some new is given. Chapter V is based on the paper by Engelis [C43]. Some results in Chapters VI and VII are new and published for the first time. At the end of these chapters the results obtained by A.A. Tsygankov [C160-C162] on superorthogonal harmonic polynomials are given, all the formulations having been changed. The whole of Chapter VIII and the first three sections in Chapter IX are based on the paper by Krall and Sheffer [C94], all the formulations having been changed and strengthened. In Section 4 of Chapter IX a very important result obtained by Agakhanov [C2] is described and in Section 5 of the same chapter a result of Krall and Sheffer [C94] is given. Some results from T. Koornwinder’s papers [B9, C82-C84] are described in Chapter X. As long as in the description of the results of other authors many formulations and proofs have been changed, all the mentioned authors, whose priority is undisputable, do not bear any responsibility for the presentation of their results adopted in this monograph. On the other hand, the author hopes that he has not missed too much either in the way of choosing, presenting and reviewing the results or in making up the list of references. To read this monograph knowledge of the main properties of orthogonal poly¬ nomials in one variable is necessary, e.g., based at least on one of the monographs [A16, A48, A46], Also, in presenting the material all the main concepts and methods from the theory of functions of a real variable [A25, A39], the theory of functions of the complex variable [A27, A30] and also the theory of differential equations [A51, A31, A49] are used.

PREFACE

xiii

The author expresses his deepest gratitude to Professor B. Golubov and post¬ graduate A. Shishkin for their careful review of the manuscript, detailed discussion of many results and a number of valuable comments to the text.

Scheme of chapter dependence

PREFACE TO THE ENGLISH EDITION

The Russian edition of this book was published in 1988. Since then many works have been published concerning the theory of orthogonal polynomials in two variables. The works have been written by famous authors as well as by young mathematicians only beginning their investigations in the theory of orthogonal poly¬ nomials. However, as before, the main results among the ones obtained during the last two dozens years remain those of T. Koornwinder, some of which are is given in Chapter X of this book. Koornwinder’s results are continued and extended in different areas. Among them the following can be distinguished. (1) In the papers of Koornwinder some interesting examples of orthogonal poly¬ nomials in two variables are constructed which can be called model. Among them we should mention the polynomials in two conjugate complex variables orthogonal over the Steiner domain (Section 3 of Chapter X). Actually, this is a new class of orthogonal polynomials in two variables. In this class one can consider analogues of Chebyshev polynomials of the first and the second kind as well as analogues of Jacobi polynomials. In all these cases, one can consider asymptotic properties of these polynomials inside the domain, in the closed domain, as well as outside the Steiner curve. Similarly to the case of the Jacobi polynomials, the weight estimates play a very essential role here. And most importantly, one can consider the Fourier series in these orthogonal polynomials and carry over the known properties of the trigonometric Fourier series and the Taylor series, of course, with appropriate changes and coun¬ terexamples, onto these series. (2) Furthermore, the main achievement of Koornwinder is the introduction of new classes of orthogonal polynomials in two variables which are two-dimensional analogues of the classical orthogonal polynomials, in the sense, for instance, that they are the eigenfunctions of partial differential operators of the second order defined in the corresponding orthogonality domains. No doubt, the method due to Koornwinder can be applied for the construction of other new classes of orthogonal polynomials in two variables which are connected with partial differential equations. (3) T. Koornwinder also considered examples of domains and weight functions such that the corresponding orthogonal polynomials in two variables can be represented by complicated formulae via the orthogonal polynomials in one variable. These examples may also be considered as model; they should be taken into consideration. In this connection, new problems arise concerning asymptotic properties of the Jacobi and Laguerre polynomials when their parameters increase [Cl46].

xv

PREFACE TO THE ENGLISH EDITION

XVI

In all these areas problems arise concerning the convergence of the Fourier series in orthogonal polynomials in two variables and the characteristic of the classes of functions in two variables which can be represented by these series. The author of this monograph considers as very important and constructively essential the new results obtained by Ch.F. Dunkl [C27-C36]. On the basis of these results, one can emphasize a series of interesting areas in the theory of orthogonal polynomials in two variables. In the additional list of references (Section VIII), the works concerning the theory of orthogonal polynomials in two and several variables are given which are known to the author and have been published in recent years. For general information, the author has included in the additional list some of the most important works concerning the theory of orthogonal polynomials in one variable (real or complex) as well as new studies containing new applications of orthogonal polynomials in one variable. It is known that A.A. Gonchar [C52] first established a closed connection between the asymptotics of a ratio of orthogonal polynomials in a complex domain and the convergence of diagonal sequences of the Pade approximations. In the theory of the Pade approximations, this connection turned out to be very fruitful. Later on, the most important results in this area were obtained by E.A. Rakhmanov [Cl 18, C121] and G. Lopes [C97-C99]. In the work of A.I. Aptekarev [C4-C6], the asymptotic properties of different classes of orthogonal polynomials in a complex domain are investigated in detail. I.I. Sharapudinov [C126-C128] has considered the asymptotic properties of or¬ thogonal polynomials of a discrete variable as a function of the increasing number of the nodes of discrete orthogonality. In the calculus the closed connection between the tridiagonal Jacobi matrices, the theory of orthogonal polynomials of a real variable and the theory of Pade approximations is well-known. A large amount of work is dedicated to this problem, because here the problems of spectral theory of self-adjoint operators, the properties of orthogonal polynomials and convergence conditions of the Pade approximations intertwine. In the work of V.A. Kalyagin [C73-C75] the connection of the finitelydiagonal matrices defining non-self-adjoint operators with the corresponding sys¬ tem's of quasi-orthogonal polynomials and consistent Pade approximations is investigated in detail. Furthermore,

numerous

applications

of

orthogonal

polynomials

in

radio

engineering are explained in scientific publications, in monographs and even in textbooks for students specializing in radio engineering. And in recent years in the work of V.I. Gusevskii [C54, C55] the orthogonal polynomials of one and two vari¬ ables are applied to the solution of different problems of synthesis from the antenna theory. The tendency of steady extension of the domains of application of orthogonal polynomials in applied problems is also being maintained. Thus, for example, in the recent works [C143-C145] the orthogonal polynomials of one and two variables are applied to the solution of the problems of the filtration theory of one- and two-dimensional discrete signals.

PREFACE TO THE ENGLISH EDITION

xvn

During the translation the text of the Russian edition remains almost without changes. A new Chapter XI is added in which some results of A.D. Shishkin [C130C138] are presented. The publishing house, Gordon and Breach Publishing Group shows me great honour for the second time; although this monograph is far from perfection and the author’s merits in this area are rather modest. The editor in chief of the journal Integral Transforms and Special Functions and the series of monographs ‘Analytical Methods and Special Functions’, Professor A.P. Prudnikov again gave essential help to the author in the preparation of the manuscript to the translation into English. In this connection the author expresses his heartfelt gratitude to Professor A.P. Prudnikov. The author wishes successes to young mathematicians who begin the study of the great theory of orthogonal polynomials — one of the main parts of modern mathematical analysis.

NOTATION

{Tn(x)}

the Chebyshev polynomials of the first kind,

{Un(x)}

the Chebyshev polynomials of the second kind,

{Pn{x;a, f3)}

the Jacobi orthogonal polynomials,

{Hn(x)}

the Chebyshev-Hermite orthogonal polynomials,

{Ln(x;a)}

the Chebyshev-Laguerre orthogonal polynomials,

r

a closed rectifiable Jordan curve,

G

the interior of the curve F.

D

the exterior of the curve T.

h(x,y)

a weight function in the domain G (differential weight),

dF{x, y)

an integral weight function in the domain G.

{Fnk{x,y)}

the basic orthonormal polynomials.

{Fnk(x,y)}

the basic orthogonal polynomials with unit principal coeffi¬ cient.

{§nk{x,y)}

the monic orthogonal polynomials.

{$nk(x,y)}

the monic orthogonal polynomials with unit principal coeffi¬ cient.

{Ankix, y)}

the classical Appell polynomials.

{ Ank(x, y)}

the basic orthogonal Appell polynomials.

(n, k)

the order of an algebraic polynomial Pnk(x,y) = Pn,k(x,y).

hnk

the power moments of the weight function h(x,y)

over the

domain G. ^nk

the

Grammians

of

the

monomial system

{xpyq}

with the

weight h(x,y) over the domain G. {Ckn}

the binomial coefficients, i.e. Ck = i(n—l)..■(„ — »+!)

wn

k\ the space of polynomials of degree

Wnk

the set of polynomials of order (n, k).

wnk

the set of polynomials of order

n orthogonal with the

weight h(x,y) over the domain G. (n,k)

with unit principal

coefficient. {ank{f)}

the Fourier coefficients of the functions f(x,y) with respect to orthogonal polynomials in two variables.

{Snk (x, y)j

the partial sums of the Fourier series in orthogonal polyno¬ mials in two variables,

r (p)

the Eiler gamma function,

B{p,q) Lip a

the beta function. the set of functions satisfying the Lipschitz condition. xix

NOTATION

XX

the Pochhammer symbol denoting the product a (a + l)(a+

(a )n

2)...

(a + n — 1) of n numbers,

C(p,a)

a class of mulitply smooth curves (Ch. VI, § 4).

w =

0.

the space of functions square integrable over the domain G

L2{h,G)

with the weight h(x,y). II/IIg

the norm of the function f(x,y) in the space L2(h,G).

(p,Q)

the inner product in the space L2(h,G).

L2(h,T)

the space of functions square integrable on the contour T with the weight h(x,y).

ll/lir

the norm of the function f(x,y) in the space L2(h,T).

H2(h,G)

the space of harmonic functions square integrable over the domain G with the weight h(x,y).

En(f,G)

the best uniform approximation of a function f(x,y) in the closed domain G by polynomials of degree no higher than n.

E(n\f,G)

the best approximation of a function f(x,y) by polynomi¬ als of degree no higher than n in the metric of the space L2(h,G).

D i — Dx

the partial derivation with respect to x.

D2 = Dy

the partial derivation with respect to y.

D[n}

the main linear partial differential operator of second or-

{ An}

the eigenvalues of the operator D[u).

a(x:y)

the characteristic polynomial of the operator D[u\.

der(Ch. IV, § 1).

F(x; a, b, a)

the hypergeometric function in one variable.

F2(x,y;a,b,c ,a,/3)

the hypergeometric Appell function in two variables.

$(x,y;z,w)

the generalized generating function of monic orthogonal polynomials.

{Pn{z,z)}

the orthogonal polynomials in two conjugate complex vari¬

{Tn{z,z)}

the Chebyshev polynomials of the first kind in two con¬

ables. jugate complex variables for the Steiner domain,

{un{z, z)}

the Chebyshev polynomials of the second kind in two con¬ jugate complex variables for the Stenier domain,

Pnk(z, Z\ O')

the Koornwinder-Steiner polynomials

Fnk(u, v; q, (3 ,7)

the orthogonal polynomials over a domain bounded by two straight lines and a parabola (the Koornwinder polynomi¬ als).

Enm(x, y;

ti, v)

the partial sum of a bilinear series in orthogonal polynomi¬ als in two variables. the inverse image of the circumference |io| = R > 1 under the mapping w = $(z).

CHAPTER I GENERAL PROPERTIES OF POLYNOMIALS ORTHOGONAL

OVER A DOMAIN

§ 1. Polynomials in two variables orthogonal over a domain

When considering algebraic polynomials in two variables x and y, one needs first of all to order the set of monomials constructed of products of powers of these independent variables, i.e., the set of monomials of the form {x”yfe}. Usually, the ordering is accepted, when the monomials considered are listed in the form of the sequence 1 ,x,y,x2,xy,y2,.. .,xn,xn~1y,.. .,yn,... . (1) Using this ordering, we divide the set of monomials into the blocks of the form ,xn 1y,xn V,

n— 1

■ ,xy

(2)

Each block (2) contains n + 1 monomials of one and the same total degree n with respect to the variables x and y. In this case, we begin with the maximal power of the variable x, then the degree of monomials in x decreases and simultaneously its degree in y increases. Of course, one can accept another ordering, for example, to interchange x and y in the systems (1) and (2). However, ordering (1) and (2) is preferable, because introducing the complex variable we obtain the formula z — x + iy. For the sake of the better visibility, sequence (1) can be represented in the form of an infinite triangular table 1,

x,y, x2,xy,y2,

(3) ,xn y ..., xyn 1 ,yn,

The rows of this table consist of the blocks of monomials of the form (2). Of course, if it is necessary, one can assume that the monomials in table (3) are ordered in the same way as in sequence (1). For the sake of brevity, table (3) can be rewritten in the form {xn-y},

n = 0,l,2,..., 1

* = 0,1,2,

n.

(4)

I. POLYNOMIALS ORTHOGONAL OVER A DOMAIN

2

Let now a table of real numbers

coo, Cio, Cll, C20, C21, C22,

(5) CnO> cn 1, • • • , Cn,n —1,

be given.

cnn,

The first index of any number in this table denotes the index of the

block, and the second index denotes the location of the number in this block, and the indices of the blocks as well as the indices of the numbers in the blocks begin from 0. Table (5) can, of course, be rewritten in form (4). After multiplying cm, by the monomial xm~’y3 and summing the products ob¬ tained from the indices 0 and 0 to the indices n and k, we obtain as a result the algebraic polynomial in two variables n— 1

Pnk(x,y)=

m

k

££cmszm-V + I>njx"-V. 771 = 0 5=0

(6)

5=0

The index n in this formula denotes the number of the last block in table (3), from which the homogeneous polynomials of the highest degree are taken, and the index k denotes the highest power of the variable y in the last block. Thus, n in formula (6) denotes the total degree of the polynomial with respect to the variables x and y. It is clear that the power of y in formula (6) may be greater than k. The second sum in (6) contains the homogeneous monomials of total degree n with respect to the variables x and y.

Thus, the polynomial (6) has the leading

coefficients CnO, Cnl, ■ • • , Cn,k — 1, Cnk-

(7)

The last of these coefficients, i.e. the number cnk will be called the principal coef¬ ficient of polynomial (6). It is natural to assume that the principal coefficient of the polynomial is non-zero. In this case, for the sake of brevity, we shall say that polynomial (6) has the order (n, k). Clearly, some of the coefficients (7), except the principal one, may be equal to zero. Let us count the number of terms in the polynomial (6). First of all, note that the first (double) sum corresponds to the first n blocks of table (3), counting the block with the index 0. Whence, the double sum contains at most 0.5n(n-|-l) terms. Polynomial (6) also contains k + 1 terms of the highest degree from the block with the index n. Thus, the number of terms in polynomial (6) does not exceed N{n,k) = n(n2+ 1} +k + l

(8)

If the polynomial (6) contains all monomials from the block with index n, i.e. k = n, then formula (8) implies N(n, n) =

n(n + 1)

2

+ n + 1 =

(n + 1) (n + 2)

2

§ 1. POLYNOMIALS IN TWO VARIABLES ORTHOGONAL OVER A DOMAIN

3

Lemma 1. Let every polynomial in the system of order (n, k) Poo{x,y),

pio{x, y),Pn{x,y), P2o{x,y),P2i{x,y),P22{x,y),

Pno(x, y), Pni(x, y),, Pn k_

(9)

i(x, y),

Pnk[x,

y)

have a non-zero principal coefficient. Then any polynomial Qnk(x, y) of order (n, k) can be uniquely represented in the form n— 1 m

k

Qnk{x,y)

O'ms Pms ix,y) + j:an,Pns(x,y). m=0 j=0

(10)

j=0

Proof. Similarly to formula (6), for the given polynomial Qnk{x,y) we intro¬ duce the expansion n— 1 m Qnk{l,

y)

>zm~V + ^6njxn~y

(11)

s=0

m=0 5—0

Then we represent all polynomials (9) in the form (6), and introduce the upper indices of the coefficients. Comparing the coefficients of the equal monomials in formulae (10) and (11), we obtain the equalities 1

_

“nk —

(r» 1 fc)

i (n,fc) Vn,k — 1 — anfcCn i

+ an,k-lCnk_1 ,

f>n,k-2 — ankcnk-2

T dn k-lCn k_2

,

(n,k- 1)

(n,k-2) + Un,A:-2Cn,/c-2

>

(12)

i _ (n,fc) i (n,/s-l) . (0,0) Oq0 — ankcoo d" an,k —1^00 T ^OOCqo Thus, for the determination of the unknown coefficients {am3} we obtain a trian¬ gular linear system of N equations, where N is determined by formula (8). The de¬ terminant of system (12) is equal to the product of all principal coefficients of polynomials (9). Since all these principal coefficients are non-zero, system (12) has a unique solution. The lemma is proved. Sometimes it is necessary to consider polynomials, whose principal coefficients are equal to 1. In this case we use the notation Qnk{x,y), i.e. we have n—1 m

k—1

Qnk(x,y)= £ J>m,X™-V + £cn,x"-V+x"-y. m=0 5=0

(13)

5=0

We are now going to the definition of polynomials in two variables orthogonal over a domain.

4

I. POLYNOMIALS ORTHOGONAL OVER A DOMAIN Let a finite simply connected domain G in the plane xOy be given which is

bounded by a rectifiable Jordan curve T. A non-negative function h(x,y) is called a weight function in the domain G, or simply a weight, if it is integrable over the domain G and is not equivalent to zero, i.e. the condition holds

0
s a weight function, we can introduce the function space L2{h, G), the inner product in which is defined by the formula

(/;V?) =

JJ

Hx,y)f{x,y) 0. If the domain G is finite and the condition

0
1, which is non-negative in the domain G the condition holds jI h(x,y)Pnk(x,y) dx dy > 0.

(1)

G PROOF.

Indeed, suppose that contrary to our claim this integral is equal to

zero. Then by virtue of (1.14) and the properties of the Lebesgue integral, we find that Pnk{x,y) = 0 almost everywhere in the domain G. However the polynomial Pnk{x,y) is of order (n,k) and can vanish only on a set of zero planar measure. The lemma is proved. Theorem 1.

For any weight function h(x, y) defined in a domain G there exists

a unique system of polynomials {Fnk(x, y)} orthonormal over the domain G with the weight h(x, y). Proof.

For a polynomial of order zero we have Foo(x, y) = c00 > 0,

JJ h(x, y)co0 dx dy = 1, G

and by virtue of (1.14), the value coo is determined. Now we are going to use the method of induction and suppose that the orthonor¬ mal polynomials ^oo(x.y),

F\o{x, y), Fn(x,y),..., Fn,kc_i(ar, y)

(2)

are determined. Consider the process of determinating the polynomial Pnk(x,y) of order (n,k). By Lemma 1, this unknown polynomial can be represented in the form n—1

m

k —

1

Fnk{x,y)= Y] y^cmsFms(x,y) + y^cni,Fna(x,y) + c^k\xn~kyk. m=0 s=0

(3)

s=0

Note that if k — 1 = n in (2), then we should pass from (n, n) to (n + 1, 0). The coefficients of expansion (3) should be taken in such a way that the polynomial (3) is orthogonal to all polynomials (2). By virtue of (1.18), this means that the equalities are valid {Fnk]Fms) = 0,

(m,s)

k = 0,l,...,n,

(8)

i>n,(x,y) = ^a,q^riq(ar,y),

s = 0,1,..n.

(9)

y) == ^ ^

y) i

Q — 0, 1, . . . , fl.

m—0

Then, substituting these polynomials into (9), we obtain the formulae n

ipn,{x,y) = ^2 d,mtpnm(x,y),

s = 0,

(10)

m=0

Thus, the polynomials (9) are represented via the polynomials (8) by formulae (10). Let us prove that the polynomials (8) and (9) are biorthogonal. Actually, by virtue of the biorthogonality of the system (2) and the orthogonality of the matrix (4.20), we have n

n

n

{fnk > V’ni) — ^ \ y ^ QkpQsqi^np > ^ nq) = ^ ^ &kpasp — &k j • p=0 q=0

p—0

Choosing for any n an orthogonal matrix of form (4.20), we obtain a normal biorthogonal system {lPnk{x,y)-,rpms(x,y)}.

(11)

The condition of orthogonality of the functions ||/-S„„||,

ll/-5„„|| > ||/-5;n||,

I. POLYNOMIALS ORTHOGONAL OVER A DOMAIN

30

which imply two equalities

ii/-5„„n = iiz-sui, q

= q*

'

This result is also obvious because for any function /(x, y) from the space Z/2(fi, G) there exists a unique polynomial of order (n, n) approximating this function in the best way in the metric of the space L2(^,G)-

Note that in contrast to equality

(11) the partial sums Snk{x,y) and S^k(x,y) of the series (3) and (10), for k ± n, do not, in general, coincide, because by virtue of formula (4.22) any powers of the variable y may be present in the polynomial Qm,(x,y), but not only the powers less than or equal to s, as occurs for the polynomial Fm,(x,y). Furthermore, subtracting from identity (11) the analogous identity corresponding to the number n — 1, we obtain the very important identity n

n

(12)

Y^Ar>sFn,{x,y) = '^2bn,Qn3(x,y). 3=0

3=0

Note once again that in the left-hand side of this identity the basic orthonormal polynomials are present which can be uniquely determined by the weight function h(x,y), and the polynomials

!/)} in the right-hand side are determined

by formula (4.22) with the help of an arbitrary orthogonal matrix of form (4.20). Squaring both sides of the identity (12) and integrating over the domain G with the weight h(x,y), we obtain

3=0

3=0

Hence, Bessel’s inequality (6) and the condition (7) hold for any general orthonor¬ mal polynomials as well as for the basic ones. Thus, orthogonal series (3) and (10) have equal internal sums. Denoting these sums by n

Fn

1

y)

— ^

'

^ns Qns(x, y),

(13)

s-0

we can write both of these series in the form oo 52Bn(x,y)• n=0

The sum (13) can be simplified, if we mean a fixed function f(x, y). Indeed, for the coefficients in sum (13), by virtue of formula (4.22), we have

bn, = jj h(x,y)f(x,y)Qn,(x,y)

dx dy

G

= Yh aip) JJ Hx> y)f{x, y)Fnp(x, y) p=o

~ Z_^ a»p AnP ’ P—0

s = 0,1,..., n.

dx dy

(14)

§ 7. FOURIER SERIES FOR DIFFERENTIABLE FUNCTIONS

31

Since the function f(x, y) is fixed, the Fourier coefficients {A„p} are defined. These coefficients can be considered as coordinates of a vector in a n+ 1-dimensional space. Furthermore, since the numbers {a^} form an orthogonal matrix of the (4.20), system (14) can be considered as an orthogonal transformation of the vector {Anp } into the vector {6nj}. It is known from linear algebra that for any fixed vector {A„p} there exists an orthogonal transformation with the matrix ||aiP^|| which transforms a given vector into a vector {bns} with only one non-zero component. Without loss of generality, we can assume that after such an orthogonal transformation we have bn0 = bn 1 — ‘ ‘

— ^n,n —1 — 0,

bnn ^ 0.

In this case, formula (13) is reduced to the form Bn(®i!/) — bnnQnn(x, y).

Assuming that these transformations are fulfilled for all numbers n, we obtain instead of the series (10) the series Yl^=o bnnQnn{x,y)- Note once again that this series is obtained for a special choice of orthogonal polynomials depending on the function /(x, y). For another function the orthogonal polynomials in an analogous series will be different. Let us give one more result concerning the connection between basic and general orthogonal polynomials corresponding to a given weight function h(x,y) in a domain G. The equality (12) can be represented in the form

du dv

y ^ Tnj(u, u)Fns(x, y) .5=0

du dv.

Since the function f(u, v) in this equality is arbitrary, we have n y

n

] Fns(n> v)Fns{xt y) =

5=0

y ] Qns(u, v)Qns(x>

J/)-

5=0

This identity implies that for general orthonormal polynomials the ChristoffelDarboux formula obtained at the end of Section 3 for basic orthonormal polynomials also holds. § 7. FourieT series for differentiable functions

In this section we consider some sufficient conditions under which a twice contin¬ uously differentiable function of two variables can be expanded into Fourier series with respect to orthogonal polynomials in two variables. Let a finite simply connected domain G be bounded by a smooth Jordan curve r. For the distance between points M(x,y) and P(£,t?) we introduce the notation r

=

x/(£

- *)2

+ (v- y)2-

(1)

32

I. POLYNOMIALS ORTHOGONAL OVER A DOMAIN

Then we introduce the function v(Z, ry, x, y) = In r = In

— x)2 + (t? - j/)2.

(2)

This function is called the fundamental solution of the Laplace equation on the plane [A51, A31, A49]. Let a function f(x,y) of two variables be given in the domain G. Suppose that f{x,y) is continuous in the closed domain G and its partial derivatives up to the second order are also continuous in the closed domain G. Then we have for f(x,y) the integral representation

r + 7^ ff {InrjAfi^rjjdfdy.

(3)

G

In this formula M(x,y) is an internal point of the domain G, A is the Laplace operator, u is the direction of the- external normal to the curve T at the point P(f,T]) and ds is the differential of the arc. Formula (3) is well-known [A31] and can be obtained from the second integral Green formula on the plane. Formula (3) can be used when investigating the sufficient conditions for ex¬ panding a function of two variables with respect to orthogonal polynomials in two variables. Let a weight function h(x, y) be defined in the domain G and let the correspond¬ ing system of the basic orthonormal polynomials be (4)

{Fnk{x,y)}.

Suppose that the function (2) can be expanded into the Fourier series in poly¬ nomials (4), i.e. we have oo

n

ffir = ^^anfe(£,77)Fnfc(x,y),

(5)

n=0 k—0

where the coefficients can be determined by the formula

dnk

h(x, y) In rFnk(x, y) dx dy.

(6)

Suppose that the following conditions hold: (1) if a point M(x, y) is fixed inside the domain G, then the series (3) converges uniformly with respect to the point P{f,r]) on T; (2) series (5) can be differentiated term by term in the direction of the external normal v at the point P(f, rf) on T, i.e. we have the expansion

d In: dv

— (J -YlYl'Qf, n=0 fc=0

Fnk{x, y),

(7)

§ 7. FOURIER SERIES FOR DIFFERENTIABLE FUNCTIONS

33

and the series obtained converges uniformly with respect to the point P(£, y) on the contour T when the point M(x,y) is fixed inside the domain G; (3) when the point M(x,y) is fixed inside the domain G and the point P{£,rj) runs over the domain G, series (5) converges in such a way that it can be integrated term by term over the area of the domain G, i.e. the equality holds

[ f {lnr)Af{Z,T))d£dTi = a

// an/=(£, v)&f(S, v) d£ dy Fnk{x, y).

(8)

"=°*=o g

If all these conditions hold, then, substituting expansions (5), (7), (8) into formula (3), we obtain

OO

71

1

f(x,y) = ^2^2Fnk(x,y) n=0k=0 oo

r

2^

/

f)

f{Z,V)-Q^[°-nk{Z,V)]ds

p n

If

f) f

ank(ty)-£ds n=0k-0 OO

TX

n=0

k=0

p -»

PC

(9)

Now we introduce the Fourier coefficients by the formula

anfc(/) = ^ J ^f{Cv)^nk^,r)) - ank{^,y)^

ds

r

+^

JJ

ank{Z,y)Af{£,y) d£ dy.

(10)

The equality (9) implies the expansion oo

n

f(x,y) = ^^ant(/)Fnk(i,y)-

(11)

n-0k=0

Thus, if the three conditions formulated above hold, then any function f(x,y) which has continuous derivatives up to the second order in the closed domain G can be expanded into the Fourier series with respect to orthogonal polynomials in two variables. Let us note that the three conditions mentioned depend only on the weight function h(x,y). Let us consider formula (10) in detail.

Taking into account formula (6), we

I. POLYNOMIALS ORTHOGONAL OVER A DOMAIN

34

obtain

ank(f) =

2~

J

JJ

LG

r

-

JJ

^ r

h(x,y)\nrFnk(x,y)dxdy

Lg

+ 2?r // A-^,T?) G

JJ /i(a;.y)lnrirnfc(x-y)dxdy L G

5 In r =

d£ dy

JJh{x,y)Fnk{x,y)lj^J f{£,

5/

,)-sT-|n,'a;

ds

G

+

JJ In

rAf(£, y) d£ dy

1

dx dy.

(12)

By virtue of formula (3) the whole expression in the curly brackets is equal to /(x,y). Hence equality (12) is reduced to the form

ank(f) =

JJ

h{x,y)Fnk{x,y)f{x,y) dxdy.

G

Thus, series (11) is the usual Fourier series of the function f{x,y) with respect to the polynomial system (4). Further, the right-hand side of (3) contains three integrals

My) = hj%hiris' r h(x,y)-2J m,n)

gv

(13)

ds,

(14)

r f3(x,y)

= ^ JJ

Af(£,y) In rd£dy.

(15)

G

These integrals are called logarithmic potentials of the simple layer, double layer, and over the domain, respectively. If all these logarithmic potentials can be ex¬ panded into Fourier series with respect to the polynomials (4), then the equality (11) holds. However the conditions of representation of the three functions (13)—(15) by Fourier series in polynomials (4) can be different in the sense of restrictions on the function f{x,y) and on the weight function h(x,y). Therefore, instead of three functions (13)—(15) depending on one function f(x,y) it is convenient to consider

§ 7. FOURIER SERIES FOR DIFFERENTIABLE FUNCTIONS

35

logarithmic potentials with three different distribution functions

J

f\(xj y) —

¥>i (£,»?) lords,

(16)

r

...

1

[

Mx>y) = ^ J



,31nr ,

,

.

(17)

r

fa{x,y)=~

jj

y)

=

x

&nm-

6

(^

)

§ 2. ORTHOGONALITY OVER A DOMAIN AND ON AN INTERVAL

45

Now, similarly to (10) we consider the polynomials Fn+m,m{x,y) = Qn{x-,m)Rm(x,y)}

n,m = 0,l,...

(19)

Let us prove that these polynomials are orthonormal over the domain G = {(*, y) -y2 < x < a}

(20)

h(x,y) = h1(y/y/x)h2(x).

(21)

with the weight function

Indeed, by virtue of formulae (18), (19), and (21), we have

JJ

h(x, y)Fn+mrn(x, y)Fk+SiS(x, y) dx dy

= JJ

=J

h{x, y)Qn{x] m)Rm(x, y)Qk(x] s)Rs(x, y) dx dy

h2{x)Qn(x-,m)Qk(x]s)

J

Rm(x,y)Rs(x,y) dy

dx

— y/x

a

= Sn

J

h2(x)Qn(x-,m)Qk(x-,s)xN dx,

(22)

where we use the notation N — 0.5(tti + s + 1). If m ^ s, then the right-hand side of this equality is equal to zero, because Sms = 0. And if m — s and n ^ k, then the right-hand side of (22) is equal to zero by virtue of (14). Thus, the polynomials (19) are orthonormal over domain (20) with weight (21).

4. Consider the next case. Let us make the change of variable in the integral (1) t = y/{ 1 - x),

(23)

where x is fixed in the interval (0, 1). Then equality (1) is reduced to the form

(24)

Now we introduce the polynomials {Qn{x,rn)} orthonormal on the interval (0, 1) with the weight h2{x){ 1 — ar)2m+1. Then, as usual, from formula (4), taking into account the change of variable (23), we obtain

(25) = (i-x)-"STakyk('-x)n-k k=0

= (1 - x)~nRn{x,y).

II. EXAMPLES OF ORTHOGONALITY OVER A DOMAIN

46

Here the function Rn(x,y) is an algebraic polynomial, even in the case when the weight function hi(t) is not even. Consider the polynomials

Fn+m,m{z) y) — Qn(x, m) Rm{x

> J/)i

Tl,W.

0, 1, ■ ■ ■

(^6)

Let us prove that these polynomials are orthonormal with the weight function

h(x,y) = hi

M*)

(27)

over the domain G = {{x,y) : |y| < 1 - x).

(28)

Indeed, similarly to (22), using formulae (24) and (25), we find

IL

h(x, y)Fn

{x, y)Fk+s,s{x, y) dx dy

1

"

/ h2(x)Qn(x\m)Qk(x]s)

0

1—x

/ *i ( ,

j Rm(x,y)Rs(x,y)dy

dx

x-l

-

j

Mx)(l ~ x)m+s+1Qn(x; m)Qk(x\ s) dx.

0

Thus, polynomials (26) are orthonormal with weight (27) over domain (28). 5. Let a domain G - {(i, y) : xy > 0, a < x + y < /?}

(29)

be given. We assume that a > 0. If a > 0, then domain (29) is a trapezium, and if a — 0, then we have a triangle. The case a < (3 < 0 can be considered similarly. The mapping

(30)

transforms the domain (29) into the rectangle D — {(f,r) : a < t < /?, a < r < 6).

(31)

The Jacobian of this transformation has the form

J(f,r)

D{x,y) _ D(t, r)

t b — a

(32)

§ 2. ORTHOGONALITY OVER A DOMAIN AND ON AN INTERVAL

47

Furthermore, formulae (30) imply x + y = t, ax + by

(33) = T.

x + y Further, let {.Pm(r)} denote the polynomials orthonormal on the segment [a, 6] with the weight function /ii(r), i.e. O

j

hi(T)Pm(r)P,(T) dr-

(34)

Similarly, let {Qn(t] m)} denote the polynomials orthonormal on the segment [a, /?] with the weight function

h3(t) = h2(t)t2m+\

(35)

i.e. we have p

^ ^3 (f)^rc(^> 7u) Q k (^ j

dt — dnk ■

77l)

(36)

Now, let us introduce in the domain (29) the weight function

h(x, y) = hx

ax + by' x + y

Mx + y)-

(37)

We consider the polynomials ax + by

fn-\-m ,m {x 1 y)



Qn{x +

y?

ITl) Pn

x + y

{x + y)r

(38)

Let us prove that these polynomials are orthonormal over the domain (29) with weight function (37). Passing from the domain (29) to domain (31) and taking into account formulae (30) and (32), we obtain

J^'S)

=

JJ h(x,y)Fn+m]m(x,y)Fk+St,(x1y)dxdy

-sl G

h\ (ax + by) h2(x + y)Qn{x + y,rn)Pm fax + by j [x + y)7 V x + y J \ x + y ax + by

x Qk{x + y, s)P,

— t ~

=

(x + y)s dx dy x+ y

[[ hi(T)h2{t)Qn{t-,m)PTn(T)tmQk(t]s)P,(r)t3 + 1dtdT

a JJd y

b

J

J

h2{t)Qn(t] m)Qk(t] s)fm+5+1 dt

/ii(r)Pm(r)PJ(r) dr.

(39)

48

II. EXAMPLES OF ORTHOGONALITY OVER A DOMAIN

If m / s, then by virtue of (34), the second integral in the product (39) is equal to zero. And if m = s and n ^ k, then by virtue of (35) and (36), the first integral in product (39) is equal to zero. Thus, the polynomials (38) are orthogonal over the domain (29) with weight function (37). Various special cases of the results described have been considered by F. Didon, G. Orlov, L. Koschmieder, H. Larher and others. The general cases are described in the papers by S.A. Agakhanov [C2] and G.K. Engelis [C42], § 3. Some theorems in the case of a weight function with separating variables Let two continuously differentiable functions 0, x + y < 1}

(1)

and a weight function

be given in this domain. We shall assume that the parameters of the weight function (2) satisfy the conditions

a > 0,

0> 0,

7>a + /?—1.

(3)

For the sake of brevity, we denote p =

7

— a — /?> —1.

(4)

We introduce the standard notations (a)0 = 1,

(a)„ = a(a + 1) . .. (a + n — 1).

(5)

Here a denotes the first factor and n denotes the number of factors. Let us consider the function i y)



, y, OL, /?, 7)

[x

x1

dn+r

ay1'

(i

(6)

2>l—n — ocyl — m — ft

y)P dxndy”

(Ct)n(/?)m(l

-x-yy+n +m

Let us prove that for any non-negative integers n and m this function is a polyno¬ mial in x and y.This can be done with the help of the Leibniz formula for higher derivatives of the product of two functions



(««)(") = k-0 63

(7)

III. CLASSICAL APPELL’S ORTHOGONAL POLYNOMIALS

64

First, we apply this formula to the partial derivative with respect to x in the equality (6). We obtain as a result, using notation (5), an

n

— [xa+n~H\ -X- y)p+n+m] = y C£(xa+n-1)[(l - X - J,)P+»+”'](*) dxn

t'o n

= ^ Ckn{a + n - l)(a + n - 2)...(a + k)xa+k~l k=0

x (p+n + m)(p + n + m— 1)... (p + n + m — A: + 1)(1 — x — yy+n+m~k (—1)*

n =

Y,

Cn(« + *)(n-*)Z“+*-1(-l)*(p + n + rn - * + 1)*(1 - X -

(g)

yy^rn-k

k —0

After substituting the sum (8) into (6), we differentiate the result m times with respect to y, using the formula (7), and as a result, we obtain am

m

ii_^/3+m-i(i_x_y)P+n+m-fcj _ ^c^(y/3+m'1)(m-s)[(l-x-i/)p+n+m-/c]^)

5=0 m

= Y,Csm(P + m - 1)(/? + m - 2)... (/? + s)y

I/) =

% 7~o\ 77

\77

(a)n(/?)m( 1 - x-y)P n

x

Ck(a + k)(n_k)Xa+k l(p + n + m — A-f-l)^( — l)fc

k=0 m

xJ]C^(/3 + s)(m_,)/+s-1(-l)5(p + n + m-A:-s + l)J(l-x-y)p+n+m-/!-A 5=0 Finally, after obvious cancellations, we have the formula

Anm(x, y)



,

,

n

m

y~! r. ^n(Q +

k)(n_k)X

{a)n(p)m k=Qs=0

x (p+n + m—Ar + l)^ (p+n + m — A; — s + l)s X

(-l)fe+*C^(/3 + s)(m_5)t/*(l

- X -

(IQ)

Thus, the function (6) is a polynomial in x and y. This polynomial is called the Appell polynomial, and formula (6) is called the Rodrigues formula for the Appell

§ 1. RODRIGUES FORMULA FOR APPELL’S POLYNOMIALS

65

polynomials. Formula (10) implies that the degrees of the Appell polynomial in x as well as in y are equal to n + m. The same is valid for the total degree of this polynomial. Let us transform formula (10). First of all, we have the equalities

Ct

n(n-l)...(n-* + l)

1

,

lWc,

,

--= pP1) (-")*.

m(m — 1)... (m — s -f 1) 1 , lN.,, x Cm - -- = ^j(-l) (-m)fI (p + n + m — k — s + l)s(p + n + m - k + 1)* = (p + n + m — A: — s + l)(p + n + m — A; — s + 2) .. . . . . (p + n + m - k)(p + n + m - k + l)(p + n + m - k + 2) ... (p + n + = {p+n + m-k-s + l)(fe+J) = (~l)fc+s(-n - m-p)k+s. With the help of these equalities, the coefficient in formula (10) can be reduced to the form (-l)fc+*(-n)fc(-m),(-n - m - p){k+s)(a + k){n_k)(/3 + s)(m_5)

(11)

(Ad)(s!)(a)n(/?)m Then, using the notation (5), we obtain (q + Ar)(n_fc) _ (a + k)(a + A: + 1) . . . (a + A: + n - A: - 1) _ (a)„

(a)(a+1).. .(a + n - 1)



1 (a)*

Hence, for (11) we have the representation (-l)fc+,(-n)fe(-m)s(-n - m-p)[k+s) . ------= Bks(n, m, p, a, p).

(12)

(k\)(s\){a)k(p)s

Substituting this expression into (10), we obtain as a result the formula

Anm{x,y) = Y]y^Bks(n,m,p,a,P)xkya(l - x - y)n

-j-m

—k —s

(13)

k=0s—0

Taking into account (12), we obtain one more equality Anm(x, y) = (1 - x - y)n+m

EE

(-n)k(-m)3(-n - m - p)(k+s) x + y — \)

(k\)(s\)(a)k(/3)a

\x + y -

k=03-0

1

(14)

Let us compute several of the first Appell polynomials. First of all, formula (6) for n = m =

0

implies xl A0o{x,y)

-

(1

“y1 P

(1 - x - y)P =

— x — y)p [ xl

ayl P

1.

III. CLASSICAL APPELL’S ORTHOGONAL POLYNOMIALS

66

From the same formula (6) for n — 1 and m = 0 we obtain

Aio (x,y) =

x1~ay1~P d [xayp *(1 - x - y)p+1] a(l — x — y)P dx „1 — a [axa-1(l — x — y)p+1 - xa{p + 1)(1 - x - y)p] a (1 - X - y)P

= (1 - x - y)-(p + l)x. a Similarly, for n = 0 and m = 1 we have

1 An(x,y) = (1 - x-y) - -(p + l)y. Now let n — m = 1. From the Rodrigues formula (6) we obtain

An(x, y) =

y_ .„a.a.,[rV( 1 af3( 1 — x — y)p dxdy

x

- y)p+2]

= (1 - x - y)[a/3( 1 - x - y) - P(p + 2)x - a(p + 2)y] + xy(p+ l)(p+ 2). Similarly, the polynomials A2o(x,y), A02{x,y) and the following ones can be com¬ puted.

Of course, one can use formulae (13) and (14) instead of the Rodrigues

formula (6). As a whole, the set of Appell’s polynomials can be represented in the form of the table A00(x,y), Aio(x,y),A0i(x,y), A2q(x, y),An(x, y):A02{x, y), (15) Ano{x, y), An_i,i(x, y),..., A0n(x, y),

Thus, all Appell’s polynomials can be divided into blocks, with each one containing the polynomials of the same total degree.

However, contrary to the triangular

tables considered in the first chapter, all polynomials in an arbitrary row of table (15) have one and the same total degree as well as the degree with respect to every variable. Of course, the Appell polynomials are not monic. Let us now establish the orthogonality property of the Appell polynomials over the domain (1) with weight function (2) under the condition (3).

We prove that

the Appell polynomial Anm(x, y) is orthogonal to any polynomial Pn{x, y) of total degree N < n + m.

By virtue of the Rodrigues formula (6), taking into account

notation (4), we have

{Pn, Anm) —

JJ h[x, y)P (x, y)A jy

nm

(*E) y) dx dy (1 - X - y)p+n

+m‘

(16)

§ 1. RODRIGUES FORMULA FOR APPELL’S POLYNOMIALS

67

This integral can be computed with the help of a formula of multiple integration by parts for double integrals. Let G be a simply connected domain bounded by a piecewise smooth curve T on which two functions /(x, y) and F(x, y) be defined continuously differentiable in the closed domain G. By virtue of the Green formula, we obtain

JJfl*.

y)F'x(I,y)dIdy =

G

jf

j

f(x,y)Fx(x,y) dx\ dx dy

Jo

G

x

= =

JJ Jx f(x'y)F(x’y) ~ J fx(x>y)F{x,y)

dx

dx dy

J f{x,y)F{x,y) dy- JJ fx{x,y)F(x,y) dx dy. (17)

If the condition F(x,y) = 0

(18)

holds on the contour T, then the line integral is equal to zero and (17) can be reduced to the form f(x,y)F^(x,y)dxdy=- ff f,x{x,y)F{x,y)dxdy.

(19)

Similarly to (17), we have

f

y

JJ f{x,y)F^x,y)dxdy = J J J- J f(x,y)F'y{x,y)dy

dx dy

y

d_

f(x,y)F(x,y) -

dy

J f'y(x,y)F(x,y) dy

dx dy

yo

= -

J f{x,y)F(x,y) dx- JJ fy{x, y)F(x, y) dx dy. (20)

Hence, under the same condition (18) we have

JJ /(x, y)Fy(x, y) dx dy =

-

JJ /' (a, y)F(x, y) dx dy.

(21)

Of course, formulae (19) and (21) hold if condition (18) on the contour V is replaced by the condition f(x,y) = 0. Using formulae (19) and (21), one can easily obtain the following equality

SS

SS F(i'y)

dn+mf{x,y) dxndym

dx dy.

(22)

III. CLASSICAL APPELL’S ORTHOGONAL POLYNOMIALS

68

This equality holds if the function F(x,y) and all its partial derivatives except the last mixed derivative of order n + m vanish on the contour T, or if the function f(x,y) satisfies the same condition. Now consider the function (23)

F{x, y) = xn+a-1ym+p-1{ 1 - x - yy+n+m.

By virtue of (3) and (4), all partial derivatives of this function except the last two mixed derivatives of order (n — 1) + m and n + (m — 1) vanish on the contour T which is the boundary of domain (1). Nevertheless, it can easily be proved that the function (23) satisfies (22) if the condition n + m > 1 holds, i.e. at least one of the numbers n and m is non-zero. Let n > 1. By virtue of (17), we have dn+mF{x,y)

!Lf(x-y)

- J f(x,y)

dx dy dxndym

dn-1+mF{x,y) dy dxn~ldym (24)

[!L fU *,y)d-^£Adxdy. dxn~ldyn

Consider the partial derivative dn-1+mF{x,y) (25) dxn~ldym Formula (23) implies that this partial derivative vanishes on the vertical leg and on the hypotenuse of the contour T. On the horizontal leg we have y = 0. Hence, the line integral in formula (24) is equal to zero. If m > 1, then formula (20) implies

SLnx'y)

dn+mF(x,y)

dn+m-lF(xty)

= - / f{x,y)

dx dy dxndym

dxndy,m — 1

dx (26)

dn+m-lF{xy)

-!L

fy^’V)

dxdy■

dxndy'

Similarly to the previous case, considering the partial derivative

dn+m-1F{X,y)

dxn dym~1 instead of (25), we obtain that the line integral in (26) is equal to zero. Thus, function (23) satisfies equation (22) although in some cases the integral in it may be improper. Applying formula (22) to the integral in equality (16), we obtain

(7?TV)24nm) —

(-1)

n+m

(“)n(/?)r

dxndyn

(27)

Since N < n + m, the polynomial PN(x,y) contains only monomials of the form xkys, where k + s < n + m. Therefore, we have dn+mPN(x,y) _Q dxndym

(28)

§ 2. REPRESENTATION VIA HYPERGEOMETRIC FUNCTION

69

Hence, the integral (27) is equal to zero, i.e. we have {Pn] 2lnm) = 0,

N