Series of Faber Polynomials

321 Pages·1998·48.241 MB·English

by  P. K. Suetin, E.V. Pankratiev

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Series of Faber Polynomials
P.K. Suetin
Technical University of Conununication and lnjormatics
Moscow, Russia
Translated from the Russian by E.V Pankratiev
GORDON AND BREACH SCIENCE PUBLISHERS
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ANALYTICAL METHODS AND SPECIAL FUNCTIONS
An International Series of Monographs in Mathe1natics
EDITOR IN CHIEF: A.P. Prudnikov (Russia)
ASSOCIATE EDITORS: C.P. Dunk! (USA), H.-J. Glaeske (Germany) and M Saigo Qapan)
®
-.
Volume 1
Series of Faber Polynomials
P.K. S11etin
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
'
Orthogonal Polynomials in Two Variables
P.K. Suetin
Inverse Problems for Differential Operators
V.A. Yurko
This book is part o~ a series. The publisher will accept continuation orders which may be cancelled at
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cation. Please write for details.
Copyright © I 998 OPA (Overseas Publishers Association) Amsterdam B. V. Published under
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All rights reserved.
No part of this book may be reproduced or utilized in any fonn or by any means, electronic or
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system, without permission in writing from the publisher. Printed in Singapore.
Originally published in Russian in 1984 as Riad Mnogochlenov Faber by Nauka, Moscow
Amsteldijk 166
lst Floor
1079 LH Amsterdam ..
The Netherlands
British Library Cataloguing In Publication Data
Suetin, P.K.
Series of Faber Polynomials -(Analytical methods and special functions; v. I)
I. Polynomials
I. Title
512.9'42
JSBN 90-5699-058-6
CONTENTS
Pre~e ~
Preface to the English Edition xv
Notation xix
Chapter I. Some results of approximation theory 1
§ 1. The best approximations of real functions 1
§ 2. Lebesgue inequalities for Fourier and Taylor series 7
§ 3. Jackson approximating sums 15
§ 4. Estimates of the Fourier and Taylor coefficients 22
§ 5. Analogues of the Weierstrass theorem on approximation
in a complex domain 28
Chapter II. The elementary properties of Faber polynomials 33
§ 1. Fundamental definitions and examples 33
§ 2. Algebraic properties of Faber polynomials 39
§ 3. The simplest asymptotic properties 42
§ 4. Generalized Faber polynomials 44
Chapter III. Faber series with the simplest conditions 49
§ 1. Faber polynomial series 49
§ 2. Faber series of analytic functions 51
§ 3. The direct theorem of Bernstein & Walsh 54
§ 4. The inverse theorem of Bernstein & Walsh 56
Chapter IV. As;Ymptotic properties of Faber polynomials 60
§ 1. Estimation of Faber polynomials inside a domain 60
§ 2. Asymptotic representations outside a domain 62
§ 3. Faber polynomials with singularities of the weight function 70
§ 4. Faber polynomials with singularities of the contour 73
v.
Chapter Convergence of Faber series inside a domain 77
§ 1. Some conditions' for convergence inside a domain 77
§ 2. The main theorem of Faber series 83
§ 3. Expansion of Cauchy-type integrals with respect to area 86
§ 4. The table of conditions for Faber series convergence 90
§ 5. Generalized Faber series 93
v
vi CONTENTS
Chapter VI. Series of Faber polynomials 100
§ 1. Conditions for convergence inside a domain 100
§ 2. Boundary properties of series of Faber polynomials 103
§ 3. Uniqueness of Faber polynomial series 108
§ 4. Series of generalized Faber polynomials 113
Chapter Vll. Some propei-ties of Faber operators 117
§ 1. The Faber operator and its simplest properties 117
§ 2. Boundary properties of Faber operators 120
§ 3. Estimat~s of the norms of Faber operators 123
§ 4. Transformation of rational and meromorphic functions 128
§ 5. Generalized Faber operators 131
Chapter VIII. Faber series in a closed domain 1?6
§ 1. Domains with a strongly smooth boundary . 136
§ 2. Theorem concerning the inverse Faber operator 142
§ 3. Al'per's principal results 148
§ 4. Another summation formula 156
§ 5. Generalized Faber series in a closed domain 160
Chapter IX. Faber polynomials and the theory of univalent
functions 168
§ 1. The method of areas in the theory of univalent functions . 168
§ 2. Lebedev's results 172
· § 3. Pommerenke's results for Faber polynomials 177
§ 4. The result~ of Kova.ri & Pommerenke 184
§ 5. The results of Lesley, Vinge & Warschawski 191
Chapter X. Fabe.r series in canonical domains 196
§ 1. The awdliary propertie.5 of Taylor series 196
§ 2. Bounded functions in a canonital domain 199
§ 3. The general case of an arbitrary continuum 205
§ 4. The case of poles on a level line 208
Chapter XI. Faber series and the Riemann boundary problem 214
§ 1. The approximate solution of Riemann problem 214
§ 2. Some properties of Faber series of the second kind 218
§ 3. Faber series and generalized functions 223
§ 4. Riemann problem in the class of generalized functions 230
· Chapter XII. The summation formula of Dzyadyk 233
§ 1. Dzyadyk's formula for Faber series summation 2·33
§ 2. Summation of the generalized Faber series 237
§ 3. Summation of Faber series in a closed domain 241
§ 4. Faber approximation of the Cauchy kernel 245
§ 5. Nikol'skii's problem in a complex domain 248
Chapter XIII. Generalization of Faber polynomials and series 252
§ 1. Faber-Walsh polynomials 252
§ 2. Faber-Laurent series 255
CONTENTS vii
§ 3. Faber-Dz11Fbashyan rational functions 257
§ 4. Basic systems of Fabcr-Erokhin 262
Chapter XIV. Some recent results 270
§ l. Faber series of matrices and operators 270
§ 2. Faber operators with singularities of the weight function and
m
~oom
§ 3. Faber series with singularities of weight function and contom 276
§ 4. The Faber-Dirichlet operaoor 278
§ 5. The Faber series in the Dirichlet problem 280
Comments and Supplements 283
References 287
Authors Index 297
Subject Index 299
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PREFACE
The classic Faber polynomials play an important role in modern approximation
theory for the functions of a complex variable. Series of Faber polynomials are
used for the representation of analytic functions in simply connected domains, apd
many theorems on the approximation of analytic functions are proved with the qelp
of these series. By their construction, the Faber series are a natural generalization
of the Taylor series from a disk to a simply connected domain.
In tills book the main properties of Faber polynomials and series are presented.
Attention is mainly given to the asymptotic properties of the Faber polynomials
and to the approximative properties of Faber series. In many cases, known results
about Taylor series are carried over to Faber series.
Any function f(z), analytic in the disk lz - .zol < R, can be expanded in this
disk into the Taylor series
=I00:
1cz) en(z - zor, (1)
n=O
and this series converges uniformly inside the disk lz - .zol < R, i.e. uniformly on
every compact subset of thls disk.
In 1885 C. Runge proved that for any function /(z), analytic in a bounded, simply
connected domain G, there exists a polynomial sequence {Q n(z)}, converging to the
function f(z) uniformly inside the domain G. This theorem can be considered as one
of the simplest analogues of K .. Weierstrass' theorem in the case of approximation
of analytic functions by polynomials. This theorem implies the representation
00
f(z) = Qo(z) + L IQn(z) -Qn-1(z)], z E G, (2)
n=l
and the series converges uniformly inside the domain G.
In 1903 G. Faber ID7], emphasizing the ambiguity of definition of the terms of
series (2), had formulated the following problem on constructing a generalization
of the Taylor series (1) ,to the case of an arbitrary simply connected domain G.
As with the process which associates with any disk, lz - .zol < R, the system of
polynomials {(z-zor} having the prope.rty that any function f(z) analytic in this
disk can be expanded in a series of these polynomials, it is required, for an arbitrary
bounded, simply connected domain G, to determine a system of polynomials { ~(z)}
such that any function f(z) analytic in the domain G and possibly satisfying some
additional conditions, can be expanded into a series of the form
E00
=
f(z) an4?n(z), z E G, (3)
n=O
ix
x PREFACE
where the coefficients {an} are defined by the function /(z) and depend on the do
main G. lo solving this problem, Faber considered only the case when the boundary
r
of the simply coM.ected domain G is a regular analytic curve; however, the polyno
mials introduced by Faber, and later called the Faber polynomials, are determined
uniquely for any bounded continuum with a connected complement.
Let the complement of a bounded nondegenerate continuum K be a simply con
nected domain D. Denote by w = 4}(z) a function mapping the domain D confor
mally and univalently onto the domain lwl > 1 under the conditions <I>(oo) = oo
and <I>'(oo) > 0, and let z = 1/J(w) be the inverse function. Then the sequence of
Faber polynomials {wn(z)} for the continuum K is defined with the help of the
generating function by the formula.
Vi'(w) Lo:> <I>n(z)
- (- ) - = +i , z E K, iwl > 1. {4)
'I/; w - z n=O w~
r
Let G denote the set of internal points of the continuum K and let be its boundary.
In Faber's first paper [D7], the case when the continuum K is the closure of a
r
simply connected domain G with a regular analytic boundary is considered. In
this case any fun~tion f(z) analytic in the domain G can be expanded into the
Faber series (3), converging absolutely and uniformly inside the domain G, and the
coefficients of this expansion are determined by the formula
J
1 /(()<!>'(()
an = 211"i <]?n+ 1 { () d('
r,.
r lwl
where the curve is the inverse image of the circumference = p under the
p
mapping w = <P(z), and the number p < 1 is chosen such that the function 'l/;(w) is
lwl
analytic in the domain ~ p.
In his other paper [D8] Faber proved that expansion (3) holds when the function
f(z) is analytic on a bounded continuum with a connected complement. In this
case series {3} converges absolutely and uniformly on the whole K. These two cases
considered by Faber can be easily reduced from one to the other. The second of
them is described in A.I. Markushevich's book [AlO, II].
Interesting applications of Faber polynomials were given in Faber's third paper
[D9], published in 1920. Here Faber considered the properties of the Chebyshev
polynomials deviating least from zero in a closed domain G, bounded by a regular
r.
analytic curve In this paper, for the first time, the problem concerning the
best uniform approximation of analytic functions in a closed domain was stated,
and with the help of series of Faber polynomials the simplest estimates of the best
approximation had been found.
In 1942 W. Sewell published a monograph [A21] in which the questions of ap
proximation of analytic functions by polynomials were considered.
In this monograph, with the help of Faber series, the direct and inverse theorems
on the order of the best uniform approximation are obtained, in the case when
the function f(z) is analytic in a domain G and continuous in the closed domain
G, and the boundary r of the domain G is a. regular analytic curve. Under these