Approximation of Continuously Differentiable Functions

257 Pages·1986·9.7 MB·English

by  José G. Llavona (Eds.)

APPROXIMATION OF
CONTINUOUSLY DIFFERENTIABLE FUNCTIONS
NORTH-HOLLAND MATHEMATICS STUDIES 130
Notas de Matematica (11 2)
Editor: Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas,
Rio de Janeiro
and University of Rochester
NORTH-HOLLAND -AMSTERDAM NEW YORK 'O XFORD 'TOKYO
APPROXIMATION
OF CONTINUOUSLY
DIF FEREN TlABLE FU NCTIONS
Jose G. LMVONA
Facultad de Matematicas
UniversidadC omplutense de Madrid
Madrid, Spain
YH c
1986
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD *TOKYO
(cl Elsevier Science Publishers B. V., 1986
All rights reserved. No part of this publication may be reproduced, storedin a retrievalsystem,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 70128 1
Publishedby:
ELSEVIER SCIENCE PUBLISHERS B.V.
P.O. BOX 1991
1000 BZ AMSTERDAM
THE NETHERLANDS
Sole distributors for the U.S.A. and Canada:
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Library of Congress Catalogingin-F'ubliertion Data
Llavona, Joe6 G.
Approximation of continuously differentiable
functions.
(Notas de matem6tica ; 112) (North-Holland mathematics
studies ; 130)
Includes index.
1. Differentiable functions. 2. Approximation theory.
3. Banach spaces. I. Title. 11. Series: Notas de
matedtica (Rio de Janeiro, Brazil) ; no. 112.
111. Series: North-Holland mathematics studies ; v. 130.
QU.N86 no.ll2 CQ4331.53 510 s C515.83 86-19924
ISBN 0-444-70128-1
PRINTED IN THE NETHERLANDS
To Ana, A ida and Bea
This Page Intentionally Left Blank
vii
The purpose of this book is to expose the basic results about
approximating continuously differentiable real functions. The first chap-
ter refers to functions defined on manifolds locally of finite dimension,
and includes, among other things, Nachbin's theorem about description of
dense subalgebras i n the algebra of continuously differentiable functions
i n the spirit of Weierstrass-Stone theorem for continuous functions ,
published i n 1949; and also density theorems for topological and poly-
nomial algebras of continuously differentiable functions. The rest of the
book is devoted to the approximation of continuously differentiable func-
tions on a Banach space.
There has been considerable interest during the last few years
in function theory in infinite dimensional spaces, and in particular to
approximation of "complicated" functions defined on a Banach space by
"simpler" o "nicer" functions. For example, in the complex case, there
has been work done on polynomial approximation of analytic functions ,
defined on Runge or polynomially convex sets i n infinite dimensional
spaces. In the real case, there has been interest in the general problem
of approximating i n one of several topologies, certain classes of differ-
entiable functions by smoother ones, such as polynomials or real analytic
functions. In this book we make a systematic study of this problem with
respect to five topologies of normal use, and also of the classes of
continuously differentiable functions associated with them. We present
the versions of Whitney and Nachbin theorems for infinite dimensional
spaces. Finally we show important results about homomorphisms i n algebras
of continuously differentiable functions and a version of the Paley-Wiener
-Schwartz theorem i n infinite dimensions.
To summarize, we can say that the main objetive of this book is
to present, taking the classic results of the theory as a starting point,
viii Foreword
the different contributions in the last few years of mathematicians such
as Abuabara, Aron, Bombal, Ferrera, Gomez, Guerreiro, Lesmes, Nachbin,
Prolla, Restrepo, Sundaresan, Valdivia, Wells, Wulbert, Zapata and myself
among others.
The main features of this book are:
1.- For the first time the work knits together some important
and very recent results in approximation of continuously differentiable
functions such as: extension of Wells' theorem and Aron's theorem for the
fine topology of order m ; extension of Bernstein's and Weierstrass'
theorems for infinite dimensional Banach spaces ; extension of Nachbin's
and Whitney's theorem for infinite dimensional Banach spaces ; automatic
continuity of homomorphisms in algebras of continuously differentiable
...
functions etc.
2.- The book describes some of the most important moderin fea-
tures of a very rapidly expanding area, which abounds in quite interesting
and challenging oper: problems.
.-
3 Very accessible. Sel f-cont.ained.
A more detailed description of the book:
Chapter I shows the most important general results about appro-
ximation of continuously differentiable functions on real manifolds local-
ly of finite dimension. It starts with Weierstrass' theorem about polyng
mial approximation of continuously differentiable functions and shows
Nachbin's theorem about dense subalgebras in the algebra of Cm functions
endowed with the compact open topology. In order to study the problem of
describing dense subalgebras in topological algebras of continuously dif-
ferentiable functions , we introduce m-admissible algebras and characterize
m-admissible algebras among their closed subalgebras. Finally we study
modules on strongly separating algebras, obtaining a description of dense
polynomial algebras related to Stone and Nachbin conditions.
The rest of the book is devoted to the approximation of continu
ously differentiable functions on a Banach space E.
Chapter I1 is dedicated to approximation for the fine topology
of order m. Wells' and Aron's theorems are extended and we present a non-
linear characterization of superreflexive Banach spaces.
Foreword ix
Chapter I11 brings out several results on approximation for the
compact-compact topology of order m, and furthermore a characterization
of finite type continuous polynomials space completion for this topology.
Chapter I V is an exhaustive study concerning the principal spaces
of weakly continuous functions on Banach spaces. The bw-topology and the
completion of these spaces are studied. Specifically treated i s the poly
nomial case.
Chapter V shows the uniformly weakly differentiable functions
class and presents an extension of Bernstein's theorem.
Chapter V I deals with approximation for the compact open topo-
logy of order m. An extension of Weierstrass' theorem for infinite
dimensional Banach spaces is given.
Chapter V I I goes into the weakly differentiable functions class.
We introduce the bounded weak approximation property and offer some results
on polynomial approximation of weakly differentiable functions.
Chapter V I I I is dedicated to the approximation property in con-
tinuously differentiable function spaces. Many of the density results
obtained in the previous chapters and €-products of continuously differ-
entiable function spaces are used.
Chapter I X deals with polynomial algebras of continuously differ
entiable functions. An extension of Nachbin's theorem is found.
Chapter X delves into the closure of continuously differentiable
function modules. An extension of Whitney's theorem i s given.
Chapter X I develops a study of homomorphisms between algebras of
uniformly weakly differentiable functions. The automatic continuity problem
of these homomorphisms is treated and the functions inducing these homo-
morphisms are characterized.
Chapter XI1 finally shows a version of the Paley-Wiener-Schwartz
theorem in infinite dimensions.
The book is finished up with an appendix dedicated to Whitney's
Spectral Theorem.
This book can be used by graduate students that have taken cour-
ses in Differential Calculus , Topology and Functional Analysis and are
interested in the Approximation Theory and Infinite Dimensional Analysis.