Approximation Problems in Analysis and Probability

259 Pages·1989·9.5 MB·English

by  M.P. Heble

APPROXI MATI0 N PROBLEMS IN
ANALYSIS AND PROBABILITY
NORTH-HOLLAND MATHEMATICS STUDIES 159
(Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
and
University of Rochester
New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM ' NEW YORK OXFORD TOKYO
APPROXIMATION PROBLEMS IN
ANALYSIS AND PROBABILITY
M.P. HEBLE
Department of Mathematics
University of Toronto
Toronto, Canada
1989
NORTH-HOLLAND - AMSTERDAM ' NEW YORK OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress Cataloging-in-Publication Data
Heble. M. P.
Approximation problems in analysis and probability / M.P. Heble.
p. cm. -- (North-Holland mathematics studies ; 159)
Includes bibliographical references.
ISBN 0-444-88021-6
1. Approximation theory. 2. Mathematical analysis.
3. Probabilities. I. Title. 11. Series.
.
OA221 H375 1989
511'.4--dc20 89-16147
CIP
.ISBN: 0 444 88021 6
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989
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Printed in the Netherlands
To
My mother Girijabai
My uncle Rama Rao
and
Sushila, Ajay and Sucheta
This Page Intentionally Left Blank
Vii
Table of Contents
Introduction ix
Chapter I. Weierstrass-Stone theorem and generalisations - a brief survey 1
8 1. Weierstrass-Stone theorem 2
2. Closure of a module - the weighted approximation problem 6
3. Criteria of localisability 19
4. A differentiable variant of the Stone-Weierstrass theorem 31
5. Further differentiable variants of the Stone-Weierstrass theorem 34
Chapter II. Strong approximation in finitedimensional spaces 41
1. H. Whitney’s theorem on analytic approximation 41
2. C” -approximation in a finitedimensional space 61
Chapter III. Strong approximation in infinitedimensional spaces 77
§ 1. Kurzweil’s theorems on analytic approximation 77
5 2. Smoothness properties of norms in LP-spaces 95
§3. C”-partitions of unity in filbert space 99
§4. Theorem of Bonic and Frampton 101
5 5. Smale’s Theorem 103
§6. Theorem of Eells and McAlpin 107
§ 7. Contribution of J. Wells and K. Sundaresan 111
§8. Theorems of Desolneux-Moulis 121
§9. Ck-approximation of Ck by Cw-a theorem of Heble 127
0 10. Connection between strong approximation and earlier ideas
of Bernstein-Nachbin 153
fill. Strong approximation - other directions 154
Chapter IV. Approximation problems in probability 169
1. Bernstein’s proof of Weierstrass theorem 170
5 2. Some recent Bernstein-type approximation results 172
53. A theorem of H. Steinhaus 178
$4. The Wiener process or Brownian motion 183
5. Jump processes - a theorem of Skorokhod 189
Appendix 1 : Topological vector spaces 20 1
Appendix 2: Differential Calculus in Banach spaces 215
Appendix 3: Differentiable Banach manifolds 223
Appendix 4: Probability theory 229
Bibliography 237
Index 243
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ix
Introduction
The classical Weierstrass-Stone theorem and the Bernstein-type weighted approx-
imation theorems were greatly extended by L. Nachbin. Another aspect of approxima-
tion theory, now called strong approximation and initiated by H. Whitney, had simul-
taneously developed, with contributions in finite-dimensional spaces as also in infinite-
dimensional spaces, by various individuals. At the same time, several approximation re-
sults in a probabilistic setting - from the elegant probabilistic proof of Weierstrass’ the-
orem by S. Bernstein to the later results on convergence of stochastic processes estab-
lished by A.V. Skorokhod and other later authors - were being added to the literature.
The material in this book covers some special aspects of the approximation theory
of functions, viz. strong approximation in function spaces, as also certain approximation
results concerning stochastic processes. The choice of topics reflects only the author’s
taste. Within the narrow range of topics chosen, I have tried to do as thorough justice
as I could, to the subject as also to the contribution of various individuals active in their
respective areas; any possible omission of names is unintentional.
This book is meant to be a monograph, of interest to research workers in the fields
of analysis, probability, and stochastic processes. Graduate students, hopefully, will find
it useful not merely as a source of information but also as an incentive to spur then on
to do further work. The author has noted other monographs recently published, cover-
ing related topics. However, the contents of these books show that the overlap between
these and my present monograph is negligible (e.g., cf. K. Sundaresan and S. Swami-