Fourier Analysis and Approximation

573 Pages·1971·25.27 MB·English

by  Paul Leo Butzer

FOURIER ANALYSIS AND APPROXIMATION
Volume I
This is Volume 40 in
PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Editors: PAULA . SMITHA ND SAMUEELI LENBERG
A complete list of titles in this series appears at the end of this volume
FOURIER ANAL YSIS AND APPROXIMATION
Volume 7 One-Dimensional Theory
Paul L. Butzer Rolf J. Nessel
Technological University of Aachen
Academic Press
New York and London
1971
COPYRIGHT 0 1971, BY BIRKH~USVEERRL AG BASEL.
(LEHRB~CHUENRD MONOQRAPHJEN AUS DEM GEBIETDEER
EXAKTEWNI SSENSCHAFTEMNA, THEMATJSCRHEEJ HE,
BAND4 0)
ISBN 3 7643 0520 7 (BirkhiIuser Verlag)
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(PUREAN D APPLIED MATHEMATICAS, Series of
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AMS 1970 Subject Classification 4141,4241, 26-01
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At the international conference on ‘Harmonic Analysis and Integral Transforms’,
conducted by one of the authors at the Mathematical Research Institute in Oberwolfach
(Black Forest) in August 1965, it was felt that there was a real need for a book on
Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans-
forms from a transform point of view, (ii) treatment of Fourier transforms in
LP(Rn)-
space not only for p = 1 and p = 2, (iii) classical solution of partial differential
equations with completely rigorous proofs, (iv) theory of singular integrals of convolu-
tion type, (v) applications to approximation theory including saturation theory,
(vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel
potentials, (viii) Fourier transform methods on locally compact groups. This study
aims to consider these aspects, presenting a systematic treatment of Fourier analysis
on the circle as well as on the infinite line, and of those areas of approximation theory
which are in some way or other related thereto. A second volume is in preparation
which goes beyond the one-dimensional theory presented here to cover the subject for
functions of several variables.
Approximately a half of this first volume deals with the theories of Fourier series
and of Fourier integrals from a transform point of view. This parallel treatment easily
lends itself to an understanding of abstract harmonic analysis; the underlying classical
theory is therefore presented in a form that is directed towards the case of arbitrary
locally compact abelian groups, which are to be discussed in the second volume. The
second half is concerned with the concepts making up the fundamental operation of
Fourier analysis, namely convolution. Thus the leitmotiv of the approximation
theoretic part is the theory of convolution integrals, the ‘smoothing’ of functions by
such, and the study of the corresponding degree of approximation. Special as this
approach may seem, it not only embraces many of the topics of the classical theory but
also leads to significant new results, e.g., on summation processes of Fourier series,
conjugate functions, fractional integration and differentiation, limiting behaviour of
solutions of partial differential equations, and saturation theory.
On the other hand, no attempt is made to present an account of the theory of Fourier
series or integrals per se, nor to prepare a book on classical approximation theory as
such. Indeed, the theory of Fourier series is the central theme of the monumental
treatises by A. ZYGMUND(1 935, 1959) and N. K. BARI( 1961). With respect to the
theory of Fourier integrals we have aimed to bring portions of the fine treatises by
E. C. TITCHMAR(1S9H3 7) and S. BOCHNJB-K. CHANDRASEKHA(R19A49N) up to date,
yet complementingt hem with Fourier transforms on the circle. Furthermore, a number
of excellent books giving a broad coverage of approximation theory has appeared in
viii PREFACE
the past years; for the classical ones we can single out N. I. ACHIESER(1 947) and I. P.
NATANSO(1N9 49). In contrast, the present volume is meant to serve as a detailed intro-
duction to each of these three major fields, emphasizing the underlying, unifying prin-
ciples and culminating in saturation theory for convolution integrals.
Whereas many texts on approximation treat the matter only for continuous func-
tions (and in LP-space, if at all, separately), the present text handles it in the spaces
C,, and LS,, 1 I p < m, simultaneously. The parallel treatment of periodic questions
and those on the line, already mentioned in connection with Fourier transforms, is a
characteristic feature of the entire material as presented in this volume. This exhibits
the structures common to both theories (compare the treatment of Chapters 6 and 11,
for example), usually discussed separately and independently. Whenever the material
would be too analogous in statement or proof, emphasis is laid upon different methods
of proof. However, the reader mainly interested in the periodic theory can proceed
directly from Chapters 1 and 2 to Chapter 4 and from there to the relevant parts indi-
cated in Chapter 6 and Sec. 7.1. He may then turn to Chapter 9, Sec. 10.1-10.4, Sec.
11.4-1 1.5, Sec. 12.1-12.2. On the other hand, Chapters 4 and 5, together with the basic
material in Chapters 1 and 3 (Sec. 1.1-1.2, 1.4,3.1-3.2) may serve as a short course on
(classical) Fourier analysis; for selected applications one may then consult Chapters 6
and 7. As a matter of fact, Chapter 7 gives the first and best-known application of
Fourier transform methods, namely to the solution of partial differential equations.
In Chapters 10-13 these integral transform methods are developed and refined so as to
handle profounder and more theoretical problems in approximation theory. A brief
introductory course on classical approximation theory for periodic functions may be
based on Chapters 1 and 2.
The present treatment is essentially self-contained; starting at an elementary level,
the book progresses gradually but thoroughly to the advanced topics and to the
frontiers of research in the field. Many of the results, especially those of Chapters
10-13, are presented here for the first time, at least in book-form. Although the
presentation is completely rigorous from the mathematical point of view, the lowest
possible level of abstraction has been selected without compromising accuracy. In
many of the proofs intricate analysis is required. This we have carried out in detail not
only since we believe it is more important to save the reader’s time than to save paper,
but because we believe firmly that the student reader should be able to follow each
step of a proof. Despite the virtues of elegant brevity in the presentation of proofs,
many recent texts have gone to the extreme of sacrificing understanding to the cost of
all but the more expert in their fields. Although we have attempted to range both in
depth and breadth, it remains inevitable that several themes have been slighted in a
subject of rapidly increasing diversity and development. Presumably no apology is
necessary for the fact that we have been guided in our selection by pursuing those topics
which have caught our imagination; however, in the process we have attempted to
illustrate a variety of analytical techniques.
With this step-by-step development the volume can be readily utilized by senior
undergraduate students in mathematics, applied mathematics, and related fields such
as mathematical physics. It is also hoped that the book will be useful as a reference for
workers in the physical sciences. Indeed, the central theme is Fourier analysis and
PREFACE ix
approximation, subjects of wide importance in many of the sciences. The princi-
pal prerequisites would be a solid course on advanced calculus as well as some
working knowledge of elementary Lebesgue integration and functional analysis. To
make the presentation self-contained these foundations are collected in a preliminary
Chapter 0.
Following each section there is a total of approximately 550 problems, many con-
sisting of several parts, ranging from fairly routine applications of the text material to
others that extend the coverage of the book. Many of the more difficult ones are
supplied with hints or with references to the pertinent literature. The results of prob-
lems are often used in subsequent sections. Each chapter ends with a section on ‘Notes
and Remarks’. These contain historical references and credits as well as detailed
references to some 650 papers or books treating or supplementing specific results of a
chapter. Important topics related to those treated but not included in the text are out-
lined here. Although we have tried our best to give everyone his full measure of credit,
we apologize in advance for any oversights or inaccuracies in this regard. Here,
as well as in the subject matter, we will appreciate any corrections suggested by the
reader.
The second volume will deal with more abstract aspects of the material. Special
emphasis is placed upon the theory in Euclidean n-space. Fourier transforms will be
discussed in the setting of distribution theory, and a systematic account given of those
parts of approximation theory concerned with functions of several variables. Included
will be characterizations of saturation classes of singular integrals with radial or
product kernels by Lipschitz conditions, Riesz transforms and fractional integrals,
Bessel potentials, etc. by means of embedding theorems.
The material presented here first took form during several one-semester courses on
Fourier series, on Fourier transforms, and on approximation theory given at the
Technological University of Aachen during the past decade by one of the authors and
assisted by the other. The third and final typewritten version was begun in the summer
of 1966, as a joint effort of both authors. We have been especially fortunate with the
assistance of several members of our team of collaborators. Dr. EBERHARLD. STARK
read and checked the whole manuscript, gave helpful suggestions, edited every chapter,
assisted in reading the proofs, and prepared the index. It is certain that without his
patient and unstinting work nothing on the scale of the volume could have been com-
pleted. Drs. ERNSTG ORLICHan d WALTETRR EBELgSa ve valuable advice and criticism,
read parts of the manuscript and set the authors straight on many a vital point;
several portions of the manuscript were written in collaboration with Dr. TREBELS,
including Chapter 11. Mr. FRIEDRICHES SERa ssisted in reading the proofs. We are
particularly indebted to our secretary Miss URSULAC OMBACwHh o typed the final
version cheerfully and with painstaking care; the earlier version had been capably
typed by Mrs. KARINK OCHa nd Mrs. DORISE WERS.
We also wish to thank Mr. C. EINSELoEf Birkhauser Verlug for his patience, and the
staff of William Clowes and Sons Ltd. for their skill and meticulous care in the produc-
tion of this book.
Aachen, February, 1970 P. L. BUTZERa nd R. J. NESSEL