Convergence problems of orthogonal series

358 Pages·2014·11.198 MB·English

by  Alexits, György

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II
CONVERGENCE PROBLEMS
OF
ORTHOGONAL SERIES
by
Prof. G. ALEXITS
TECHNICAL UNIVERSITY, BUDAPEST
MEMBER OF THE HUNGARIAN ACADEMY
OF SCIENCES
PERGAMON PRESS
NEW YORK · OXFORD · LONDON · PARIS
1961
PERGAMON PRESS INC.
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PERGAMON PRESS LTD.
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PERGAMON PRESS S. A. R. L.
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PERGAMON PRESS G.m.b.H.
Kaiserstrasse 75, Frankfurt am Main
Translated
by
I. FÖLDES
Copyright © 1961
AKADÉMIAI KIADO, BUDAPEST
Library of Congress Card No. 61-14566
PRINTED IN HUNGARY
IV
PREFACE TO THE ENGLISH EDITION
The text of this book is an improved and extended ver-
sion of the German original, for, since the issue of the latter
many interesting results were published which I have thought
necessary to include in the text. At the same time, I have
corrected some errors and misprints of the original text. In
this task I have derived much assistance from the valuable
remarks of ÀKOS CSÂSZÂR and GÉZA FREUD. I am particularly
indebted to KAROLY TANDORI for having revised the complete
English text and read its proof-sheets. Finally, I express my
gratitude to ISTVAN FÖLDES for the careful translation.
PREFACE
The questions of convergence and summation of the gen-
eral orthogonal series forms perhaps the most impressive
domain of application of the Lebesgue or of the Stieltjes—
Lebesgue concept of integral, respectively. Many methods of
inquiry owe their discovery to the investigation of this sphere
of problems. In spite of their great generality, some of the
results obtained provide wider knowledge of the convergence
features than the remaining theorems, shaped specially to the
expansion in question, even in case of applications to clas-
sical orthogonal expansions. Thus, for instance, the MEN-
CHOFF—RADEMACHER convergence theorem for general ortho-
gonal series ensures the convergence almost everywhere of cer-
tain Fourier series with irregularly distributed lacunarities, while
the special theorems achieved up to the present time on the
VII
PREFACE TO THE ENGLISH EDITION
The text of this book is an improved and extended ver-
sion of the German original, for, since the issue of the latter
many interesting results were published which I have thought
necessary to include in the text. At the same time, I have
corrected some errors and misprints of the original text. In
this task I have derived much assistance from the valuable
remarks of ÀKOS CSÂSZÂR and GÉZA FREUD. I am particularly
indebted to KAROLY TANDORI for having revised the complete
English text and read its proof-sheets. Finally, I express my
gratitude to ISTVAN FÖLDES for the careful translation.
PREFACE
The questions of convergence and summation of the gen-
eral orthogonal series forms perhaps the most impressive
domain of application of the Lebesgue or of the Stieltjes—
Lebesgue concept of integral, respectively. Many methods of
inquiry owe their discovery to the investigation of this sphere
of problems. In spite of their great generality, some of the
results obtained provide wider knowledge of the convergence
features than the remaining theorems, shaped specially to the
expansion in question, even in case of applications to clas-
sical orthogonal expansions. Thus, for instance, the MEN-
CHOFF—RADEMACHER convergence theorem for general ortho-
gonal series ensures the convergence almost everywhere of cer-
tain Fourier series with irregularly distributed lacunarities, while
the special theorems achieved up to the present time on the
VII
VIII PREFACE
convergence of the Fourier series are unable to answer this
question in such cases. Furthermore, the convergence prob-
lems of the orthogonal series are very tightly bound up with
some other branches of Analysis, especially with probability
theory. It may even be stated that a set of theorems from
the theory of orthogonal series and from the theory of proba-
bility are basically only bilingual terms for the same mathe-
matical fact.
The large range and the depth of the convergence theory
of the orthogonal series justifies a systematic treatment of
this theory. Although a programme of such a kind was
excellently carried out in the well-known book of KACZMARZ
and STEINHAUS, the zeal of the mathematicians has opened
a way to new, beautiful and important discoveries during the
25 years which have elapsed since its publication. In view of
this circumstance, it seems to me reasonable to hope that this
book will not call forth the sentiment that it is superfluous.
I have attempted to represent the actual state of the theory
of convergence and summation of the general orthogonal
series with hints as to the connexion of the general theory with
the corresponding questions of the classical expansions. On
the other hand, however, I have not dealt with other impor-
tant ranges of ideas, unrelated to questions of convergence,
as for instance those connected with the theorem of YOUNG—
HAUSDORFF or the theorem of PALEY.
I endeavoured to formulate the text in such a way that
any reader, acquainted with the most important facts from
the theory of functions of a real variable and from the theory
of the Fourier series, will find all the rest completely proved
in this book, excepting only the parts printed in smaller
type, containing various topics: theorems with complete or
with only sketched proof or even without proof, references
to unsolved problems, remarks for the better classification of
the main text, etc. I have also striven to give a hint of the
origin of the several theorems, not only by indicating the
place where the theorem in question has been formulated
for the first time in its most general form, but frequently also
PREFACE IX
by referring to the older literature in which the fundamental
idea of the proof has first appeared.
I am very much indebted, relative to the form as well as
to the content, to the monograph of KACZMARZ and STEIN-
HAUS, to the book of SZEGÖ on orthogonal polynomials and
to the well-known work of ZYGMUND on trigonometrical
series whose new, greatly extended edition, however, could
unfortunately not actually be utilized for this book. The
appendix of GUTER and ULIANOFF provided for the Russian
translation of the book of KACZMARZ and STEINHAUS (MOSCOW,
1958) has likewise been very useful for our purposes.
I avail myself of the opportunity to express my deepest
gratitude to my colleagues B. SZ.-NAGY and K. TANDORI for
their generous help and very valuable comments during the
writing of this book. Their remarks, advice and assistance
have contributed appreciably to the improvement of the text. I
have also to thank to the publishing house of the Hungarian
Academy of Sciences, as well as to the printing office of
Szeged, for their careful production of the book.
Budapest
G. ALEXITS.
CHAPTER I
FUNDAMENTAL IDEAS. EXAMPLES OF SERIES
OF ORTHOGONAL FUNCTIONS
§ 1. Orthogonality, orthogonalization,
series of orthogonal functions
We take an arbitrary closed interval [a, ft], the so-called
interval of orthogonality, as a basis for our investigations
relative to orthogonal series of functions. We shall assume
throughout that this interval is finite; in what follows we do
not intend to concern ourselves with questions relating to
infinite intervals of orthogonality.1 Following usual notations,
we shall denote by (a, b) the open interval with the end-
points a, b; moreover, we shall occasionally denote by [a, b)
and (a, b] the corresponding half-open intervals, open on the
right or on the left side, respectively.
We introduce the notion of orthogonality by means of the
Stieltjes—Lebesgue integral. Let μ(χ) be a positive, bounded,
monotone-increasing function in the interval of orthogonality
[α, 6], whose derivative μ'(χ) ^ 0 vanishes at most in a set
of measure zero (in the sense of Lebesgue). The (real) function
f(x) is called Ζ,μ-integrable, if it is μ-measurable and more-
over the condition
b
JV(x)|</Kx)<°o
a
is fulfilled.
The condition that μ'(χ) may vanish at most in a set
of measure zero would not be necessary for the definition
1 Among the series of orthogonal functions with infinite interval of
orthogonality there are only two rather special systems, which are gene-
rally considered, i. e. those of the Laguerre and Hermite polynomials.
As to their theory, see G. SZEGO'S book "Orthogonal Polynomials" (New
York, 1939).
1
2 CONVERGENCE PROBLEMS OF ORTHOGONAL SERIES
of the Stieltjes—Lebesgue integral; nevertheless, we still
prefer to retain it in order that the sets of μ-measure zero
should at the same time be sets of measure zero in the sense
of Lebesgue also. The state of things is, viz., as follows:
Without our condition a set of ^-measure zero is not neces-
sarily of measure zero in the sense of Lebesgue. But if μ'(χ)
vanishes at most in a set of measure zero (in the sense of
Lebesgue), we may denote by a(x) the absolutely continuous
part of μ(χ) and by TV a set of μ-measure: ιη (Ν) = 0. Then
μ
we have
0 = πι (Ν) = \άμ(χ) i= \da(x) = \a\x)dx.
μ
N N N
Since on TV a(x) = μ\χ) > 0 holds almost everywhere, the
last integral can not vanish unless TV is a set of measure
zero also in the sense of Lebesgue. Therefore the sets of μ-
measure zero are at the same time sets of measure zero in
the sense of Lebesgue also. For this reason in what follows
we may by the term "almost everywhere" mean simply "in
the interval of orthogonality everywhere with the exception at
most of a set of measure zero in the sense of Lebesgue".
If μ(χ) is absolutely continuous and 9(χ) = μ'(χ), then for
any Z^-integrable function f(x) the relation
$/(χ)άμ(χ) = $/(χ)ρ(χ)αχ
a a
is valid. In this case we shall say that f(x) is an L^-inte-
grable function and we shall call Q(X) a covering function
or weight function. If, in particular, ρ(χ)Ξΐ, then we shall
say — in accordance with the usual terminology — that f(x)
is L-integrable.
A function f(x) is called L^9 - or L9^ -integrable, if it is
Ζ, - or L ()-integrable, respectively, and if furthermore
μ 0 r
b b
\ρ(χ)άμ(χ)<°ο or \p(x)Q(x)dx<°o
a a
holds, respectively. We shall talk about an L2-integrable
function, if ρ(χ) = \.
FUNDAMENTAL IDEAS 3
A finite or denumerably infinite system {<p (x)} of LJi-in-
n
tegrable functions is said to be orthogonal with respect to
the distribution άμ(χ), or, if it is not likely to be misunder-
stood, simply orthogonal, if
b
(1) J ψτη(χ)ψη{χ)άμ{χ) = 0 (ηιφη)
a
holds and none of the functions cp {x) vanishes almost
n
everywhere. The system {<p (x)} is called orthonormal, if
n
besides the condition of orthogonality the conditions
b
(2) \ 1(χ)αμ(χ)=\ («=0,1,...)
Ψ
a
are also fulfilled. Every orthogonal system {ψη(χ)} can be
converted into an orthonormal system by means of multiply-
ing every one of its members by a suitably chosen constant
factor. For, since none of the functions ip {x) can vanish
n
almost everywhere, the functions
*.(*)= MX) x
T
if l
' a i
exist and it is immediately evident that they constitute an
orthonormal system with respect to αμ(χ). If, in particular,
μ(χ) = χ, i.e. μ'(χ) = ρ(χ)=1, then {<p (x)} is simply an
n
orthonormal system in the ordinary sense.
It is not necessary to assume the orthogonal system to be denumer-
able. For according to F. RIESZ [1] every set of Ζ,^-integrable functions,
whose any two different elements <p and ψ satisfy the condition
b
§<ρ(χ)ψ(χ)αμ(χ) = 0,
a
is denumerable. As this fact has no importance from the point of view
of convergence, it seems to us preferable, for the sake of simplicity,
to embody denumerability in the definition of orthogonal systems.
It is easy to give a general procedure for constructing
entirely different orthogonal systems. For that purpose we