Selected Problems in Real Analysis

382 Pages·1992·3.61 MB·English

by  M. G. Goluzina, A. A. Lodkin, A. N. Podkorytov

TRANSLATIONS OF
MATHEMATICAL
MONOGRAPHS
VOLUME 107
TRANSLATIONS
MATHEMATICAL
MONOGRAPHS
.107
B. M. Makarov, M. G. Goluzina,
A.A. Lodkin, and A. N. Podkorytov
Selected
Problems in
Real Analysis
Mathematical Society
Translated from the Russian by H. H. McFaden
1991 Mathematics Subject Classification. Primary 26-01, 28-01.
Library of Congress Cataloging-in-Publication Data
lzbrannye zadachi p0 matematicheskomu analizu. English.
Selected problems in real analysis/B. M. Makarov...[et all.
p. cm.—(Translations of mathematical monographs, ISSN 0065-9282; V. 107)
Includes bibliographical references.
ISBN 0-8218-4559-4 (alk. paper)
1. Functions of real variables. 2. Mathematical analysis. I. Makarov, B. M. II. Title.
III. Series.
QA331.5 19313 1992 92-15594
515'.8—dc2O CIP
Copying and reprinting. Individual readers of this publir.ation, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy a chapter for use
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reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also
be made by e-mail to
Contents
Foreword
Notation ix
Problems Solutions
CHAPTER I. Introduction 3 143
§1. Sets 3 143
§2. Inequalities 8 147
§3. Irrationality 13 152
CHAPTER II. Sequences 17 157
§1. Computation of limits 17 157
§2. Averaging of sequences 19 162
§3. Recursive sequences 21 164
CHAPTER III. Functions 25 167
§1. Continuity and discontinuities of functions 25 167
§2. Semicontinuous functions 28
§3. Continuous and differentiable functions 28 171
§4. Continuous mappings 31 173
§5. Functional equations 33 175
CHAPTER IV. Series 35 179
§1. Convergence 35 179
§2. Properties of numerical series connected with
monotonicity 36 180
§3. Various assertions about series 38 185
§4. Computation of sums of series 40 190
§5. Function series 41 192
§6. Trigonometric series 43 194
CHAPTER V. Integrals 47 201
§1. Improper integrals of functions of a single variable 47 201
§2. Computation of multiple integrals 49 204
CONTENTS
Problems Solutions
CHAPTER VI. Asymptotics 53 213
§1. Asymptotics of integrals 53 213
§2. The Laplace method 55 217
§3. Asymptotics of sums 59 221
§4. Asymptotics of implicit functions and recursive
sequences 63 227
CHAPTER VII. Functions (continuation) 67 231
§1. Convexity 67 231
§2. Smooth functions 73 237
§3. Bernstein polynomials 77 244
§4. Almost periodic functions and sequences 80 253
CHAPTER VIII. Lebesgue Measure and the Lebesgue
Integral 87 263
§1. Lebesgue measure 87 263
§2. Measurable functions 89 267
§3. Integrable functions 91 268
§4. The Stieltjes integral 98 280
§5. The e-entropy and Hausdorif measures 99 282
§6. Asymptotics of integrals of higher multiplicity 104 290
CHAPTER IX. Sequences of Measurable Functions 109 301
§1. Convergence in measure and almost everywhere 109 301
§2. Convergence in the mean. The law of large numbers 111 302
§3. The Rademacher functions. Khintchine's inequality 114 305
§4. Fourier series and the Fourier transform 120 311
CHAPTER X. Iterates of Transformations of an Interval 125 317
§1. Topological dynamics 125 317
§2. Transformations with an invariant measure 132 333
Answers 347
Appendix I 357
Appendix II 361
Appendix III 363
Bibliography 367
Subject Index 369
Foreword
This problem book is intended first and foremost for students wishing to
deepen their knowledge of mathematical analysis, and for lecturers conduct-
ing seminars in university mathematics departments. It is somewhat different
from the usual problem books in the greater difficulty of the problems, which
include a number of well-known theorems in analysis. Despite this, no special
preparation is required to solve the problems in Chapters 1—Vu and in §1 of
Chapter X, and many of them are accessible even to first-year students in the
second semester. All the facts needed to solve these problems are contained
in the standard university analysis texts, in particular, in the books of Zorich
[7], Kudryavtsev [16], Rudin [23], and Fikhtengol'ts [29]. The problems in
Chapters VIII and IX and §2 of Chapter X require a somewhat higher level
of preparation of the reader and presuppose that he is familiar with the fun-
damental concepts of measure theory. The corresponding facts can be found
in the last chapter of the cited text of Rudin and, in more complete form, in
the books [14] of Kolmogorov and Fomin and [4] of Vulikh.
The contents of the first seven chapters, which include about two thirds
of all the problems, do not go outside the framework of the classical topics
of analysis (functions, derivatives, integrals, asymptotics). Both here and
in the subsequent chapters we make no attempt at maximal generality and,
when we have to choose between a more general and less general formula-
tion of a problem, often show preference to the latter. Chapters VIII—X are
less traditional for a problem book in analysis. Besides the tastes of the au-
thors, the program of the mathematical analysis course adopted at Leningrad
University served as a criterion in choosing the material for these chapters.
Problems going outside its framework and relating to the "theory of func-
tions of a real variable" (in spite of the arbitrariness of this division) are not
included in the book. For example, we do not use many attractive problems
whose solution is based on the Lebesgue theorem on differentiation of an
integral with respect to a variable upper limit. Also;almost no reflections of
problems connected with complex analysis are found. We refer the readers
interested in this circle of questions to the widely known collection of Pãlya
and Szegô [21] and to the book of Titchmarsh [27].
FOREWORD
We tried to combine problems dealing with separate topics or methods
into cycles within whose confines it would be possible to exhaust, step by
step, various circles of questions with sufficient thoroughness. Partly for this
reason we were not able to avoid a certain lack of homogeneity in the degree
of difficulty of neighboring problems, which can grow markedly in the course
of a single cycle. Therefore, it is not rare that more difficult problems give
way to relatively simple problems, and the reader who has not solved some
problem must not feel disheartened and can hope in full for success in the
solution of subsequent problems.
Brief but often detailed solutions of most of the problems are given in the
second part of the book. However, we advise the reader not to be in a hurry
to use this part of the book and thus miss the chance of devising a better
solution than the one presented there.
The literature in analysis and, in particular, textbooks and problem col-
lections, contains an enormous amount of material, and we think that only
a few of the problems presented can pretend to be original. We saw it as
our goal first and foremost to try to systematize and introduce into everyday
practice problems contained (sometimes in implicit form) in almost inac-
cessible sources (especially for students) and in the mathematical folklore.
The experienced reader will notice that along with the traditional material
are borrowings from "Matematicheskoe Prosveshchenie," Mathe-
matical Monthly," and the collections [19], [22], [24]—[26], [38], and others.
Problems are accompanied by references to the literature only in exceptional
cases.
The general editing of this problem book was done by B. M. Makarov.
We express sincere gratitude to our friends and colleagues A. B. Alek-
sandrov, D. A. Vladimirov, E. D. Gluskin, Yu. G. Dutkevich, V. V. Zhuk,
K. P. Kokhas', M. Yu. Lyubich, G. I. Natanson, A. V. Osipov, A. I. Plotkin,
0. I. Reinov, B. A. Samokish, S. V. Khrushchev, and D. V. Yakubovich,
whose frequent advice and numerous critical remarks were of great help to
us. We are also obligated to them for a number of elegant problems.
The following system was used for numbering problems and for references.
Within a single chapter the problems are labelled by two numbers, the first
denoting the section and the second the problem in that section. In refer-
ring to a problem in another chapter we first indicate the chapter number (a
Roman numeral). For example, problem Vll.2.5 is problem 2.5 in Chapter
VII.
We will be grateful to all readers for their opinions and comments.
The authors
Notation
N is the set of natural numbers;
Z is the set of integers;
is the set of rational numbers;
R is the set of real numbers;
is the extended set of real numbers;
C is the set of complex numbers;
R" is the additive n-dimensional space;
0 is the empty set;
A x B is the direct (Cartesian) product of sets A and B;
card(A) is the cardinality of a set A;
fog is the composition of mappings (functions) f and g:(fo g)
(x) = f(g(x)).
f(A) is the image of a set A under a mapping f;
r'(A) is the complete inverse image of a set A under a mapping
1;
is the closure of a subset A of the space
B(x, r) is the open ball of radius r about a point x;
is the ball B(O, r) in
is the ball
is the unit sphere about zero in
(a, b) stands for any one of the four kinds of intervals ([a, b], (a, b),
[a,b),or (a,b]);
11, (11) denote the character of monotonicity of a function f
(nonincreasing, nondecreasing);
11 A, (fT A) denote the equalities limx..,a f(x) = A(A E for a
nonincreasing (nondecreasing) function f;
f(x) = O(g(x)) for x E A (or on the set A) means that If(x)I �
C is some positive number;
1(x) = O(g(x)) as x a (or f(x) xa O(g(x))) means that
f(x) = O(g(x)) on some neighborhood of the point a;
f(x) g(x) as x —, a (or 1(x) g(x)) means that f(x) =
9,(x)g(x), where —, 1 as x —, a;