Table Of ContentGraduate Texts in Mathematics 12
Managing Editors: P. R. Halmos
C. C. Moore
Richard Beals
Advanced
Mathematical
Analysis
Periodic Functions and Distributions,
Complex Analysis, Laplace Transform
and Applications
Springer Science+Business Media, LLC
Richard Beals
Professor of Mathematics
University of Chicago
Department of Mathematics
5734 University Avenue
Chicago, Illinois 60637
Managing Editors
P. R. Halmos C. C. Moore
Indiana University University of California
Department of Mathematics at Berkeley
Swain Hali East Department of Mathematics
Bloomington, Indiana 47401 Berkeley, California 94720
AMS Subject Classification
46-01, 46S05, 46C05, 30-01,43-01
34-01,3501
Library of Congress Cataloging in Publication Data
Beals, Richard, 1938-
Advanced mathematical ana!ysis.
(Graduate texts in mathematics, v. 12)
1. Mathematica! analysis. 1. TitIe. II. Series.
QA300.B4 515 73-6884
AII rights reserved.
No part of this book may be trans!ated or reproduced in any form
without written permission from Springer-Verlag.
© 1973 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc in 1973
ISBN 978-0-387-90066-7 ISBN 978-1-4684-9886-8 (eBook)
DOI 10.1007/978-1-4684-9886-8
to Nancy
PREFACE
Once upon a time students of mathematics and students of science or
engineering took the same courses in mathematical analysis beyond calculus.
Now it is common to separate" advanced mathematics for science and engi-
neering" from what might be called "advanced mathematical analysis for
mathematicians." It seems to me both useful and timely to attempt a
reconciliation.
The separation between kinds of courses has unhealthy effects. Mathe-
matics students reverse the historical development of analysis, learning the
unifying abstractions first and the examples later (if ever). Science students
learn the examples as taught generations ago, missing modern insights. A
choice between encountering Fourier series as a minor instance of the repre-
sentation theory of Banach algebras, and encountering Fourier series in
isolation and developed in an ad hoc manner, is no choice at all.
It is easy to recognize these problems, but less easy to counter the legiti-
mate pressures which have led to a separation. Modern mathematics has
broadened our perspectives by abstraction and bold generalization, while
developing techniques which can treat classical theories in a definitive way.
On the other hand, the applier of mathematics has continued to need a variety
of definite tools and has not had the time to acquire the broadest and most
definitive grasp-to learn necessary and sufficient conditions when simple
sufficient conditions will serve, or to learn the general framework encompass-
ing different examples.
This book is based on two premises. First, the ideas and methods of the
theory of distributions lead to formulations of classical theories which are
satisfying and complete mathematically, and which at the same time provide
the most useful viewpoint for applications. Second, mathematics and science
students alike can profit from an approach which treats the particular in a
careful, complete, and modern way, and which treats the general as obtained
by abstraction for the purpose of illuminating the basic structure exemplified
in the particular. As an example, the basic L2 theory of Fourier series can be
established quickly and with no mention of measure theory once L 2(O, 21T) is
known to be complete. Here L2(O, 21T) is viewed as a subspace of the space of
periodic distributions and is shown to be a Hilbert space. This leads to a dis-
cussion of abstract Hilbert space and orthogonal expansions. It is easy to
derive necessary and sufficient conditions that a formal trigonometric series
be the Fourier series of a distribution, an L2 distribution, or a smooth
function. This in turn facilitates a discussion of smooth solutions and distri-
bution solutions of the wave and heat equations.
The book is organized as follows. The first two chapters provide back-
ground material which many readers may profitably skim or skip. Chapters
3, 4, and 5 treat periodic functions and distributions, Fourier series, and
applications. Included are convolution and approximation (including the
vii
viii Preface
Weierstrass theorems), characterization of periodic distributions, elements of
Hilbert space theory, and the classical problems of mathematical physics. The
basic theory of functions of a complex variable is taken up in Chapter 6.
Chapter 7 treats the Laplace transform from a distribution-theoretic point of
view and includes applications to ordinary differential equations. Chapters 6
and 7 are virtually independent of the preceding three chapters; a quick
reading of sections 2, 3, and 5 of Chapter 3 may help motivate the procedure
of Chapter 7.
I am indebted to Max 10deit and Paul Sally for lively discussions of what
and how analysts should learn, to Nancy for her support throughout, and
particularly to Fred Flowers for his excellent handling of the manuscript.
Richard Beals
TABLE OF CONTENTS
Chapter One Basis concepts
§1. Sets and functions .
§2. Real and complex numbers 5
§3. Sequences of real and complex numbers 10
§4. Series 14
§5. Metric spaces 19
§6. Compact sets 23
§7. Vector spaces 27
Chapter Two Continuous functions
§1. Continuity, uniform continuity, and compactness 34
§2. Integration of complex-valued functions 38
§3. Differentiation of complex-valued functions . 42
§4. Sequences and series of functions . 47
§5. Differential equations and the exponential function 51
§6. Trigonometric functions and the logarithm 57
§7. Functions of two variables 62
§8. Some infinitely differentiable functions 67
Chapter Three Periodic functions and periodic distributions
§1. Continuous periodic functions 69
§2. Smooth periodic functions . 72
§3. Translation, convolution, and approximation 77
§4. The Weierstrass approximation theorems 81
§5. Periodic distributions 84
§6. Determining the periodic distributions 89
§7. Convolution of distributions 94
§8. Summary of operations on periodic distributions 99
Chapter Four Hilbert spaces and Fourier series
§1. An inner product in <'C, and the space 22 103
§2. Hilbert space 109
§3. Hilbert spaces of sequences 113
§4. Orthonormal bases . 116
§5. Orthogonal expansions . 121
§6. Fourier series 125
ix
x Table of Contents
Chapter Five Applications of Fourier series
§1. Fourier series of smooth periodic functions and periodic dis-
tributions 131
§2. Fourier series, convolutions, and approximation 134
§3. The heat equation: distribution solutions 137
§4. The heat equation: classical solutions; derivation 142
§5. The wave equation . 145
§6. Laplace's equation and the Dirichlet problem 150
Chapter Six Complex analysis
§1. Complex differentiation 155
§2. Complex integration 159
§3. The Cauchy integral formula 166
§4. The local behavior of a holomorphic function 171
§5. Isolated singularities 175
§6. Rational functions; Laurent expansions; residues 179
§7. Holomorphic functions in the unit disc . 184
Chapter Seven The Laplace transform
§1. Introduction 190
§2. The space 2 . 193
§3. The space 2' 197
§4. Characterization of distributions of type 2' 201
§5. Laplace transforms of functions . 205
§6. Laplace transforms of distributions 210
§7. Differential equations . 213
Notes and bibliography 223
Notation index 225
Subject index 227
Advanced Mathematical Analysis
