Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory

459 Pages·1979·10.27 MB·English

35
NORTH-HOLLAND
MATHEMATICS STUDIES
Notas de Matematica
editor: Leopoldo Nochbin
Approximation Theory and
Functional Analysis
J. B. PROLLA Editor
'"lRTH-HOLLAND
APPROXIMATION THEORY
AND FUNCTIONAL ANALYSIS
This Page Intentionally Left Blank
NORTH-HOLLAND
MATHEMATICS STUDIES 35
Notas de Matematica (66)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Approximation Theory
and Functional Analysis
Proceedings of the International Symposium on Approximation Theory,
Universidade Estadual de Campinas (UNICAMP)
Brazil, August 1-5, 1977
Edited by
Joio B. PROLLA
Universidade Estadual de Campinas, Brazil
1979
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM. NEW YORK. OXFORD
© North-Holland Publishing Company, 1979
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
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 7204 1964 6
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM. NEW YORK. OXFORD
Sole distributors for the U.S.A. and Canada:
ELSEVIER NORTH-HOLLAND, INC.
52 V ANDERBILT A VENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
International Symposium on Approximation Theory,
Universidade Estadual de Campinas, 1977.
Approximation theory and functional analysis.
(Notas de matematica ; 66) (North-Holland
mathematics studies ; 35)
Papers in English or French.
Includes index.
1. Functional analysis--Congresses. 2. Ap
proximation theo~y--C?ngresses. I. Prolla,
Jo~o B. . II. Un~vers~dade Estadual de Campinas.
III. T~tle. IV. Series.
QAl.N86 no. 66 [QA320] 510' .8s [515' .7] 78-26264
ISBN 0-444-85264-6
PRINTED IN THE NETHERLANDS
FOREWORD
This book contains the Proceedings of the International Sympo
sium on Approximation Theory held at the Universidade Estadual de
Campinas (UNICAMP), Brazil, during August 1 -5, 1977. Besides the
texts of lectures delivered at the Symposium, it contains some papers
by invited lecturers who were unable to attend the meeting.
The Symposium was supported by the International Mathematical
Union, by the Funda9ao de Amparo a Pesquisa do Estado de Sao Paulo
(FAPESP), by German and Spanish government agencies, and by UNICAMP
itself.
The organizing committee was constituted by Professors Silvio
Machado, Leopoldo Nachbin, Joao B. Prolla (chairman), and Guido
Zapata.
We would like to thank Professor Ubiratan D'Ambrosio, director
of the Institute of Mathematics of UNICAMP, whose support made the
meeting possible. Our special thanks are extended to Miss Elda Morta.ri
who typed this volume.
Joao B. Prolla
v
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~ABLE OF CONTENTS
R. ARON, Po.tynomia.t appJtoximation and a que.6tion 06 G.E. SlUlov.
J. BARROS NETO, Ana.tytic. hypoe.t.tiptic.ity 06 opeJtatoJt.6 06 pJt.£n-
c.ipa.e type. . . . . . . . .. .......... 13
H. BAUER, KoJtovkin appJtoximation in 6unc.tion .6pac.e.6 . .. 19
K. D. BIERSTEDT, A JtemaJtk on vec.toJt-va.tued appJtoximation on
c.ompac.t .6et.6, appJtoximation on pJtoduc.t .6et.6, and the
appJtoximation PJtopeJtty .... 37
B. BROSOWSKI, The c.omp.tetion 06 paJttia.t.ty oJtdeJted ve.c..tOJt .6pac.e..6
and KoJtovkin'.6 theoJtem . . . . . . . . . . . . . . 63
P. L. BUTZER, R. L. STENS and M. WEHRENS, AppJtoxima.tion bya.t-
gebJtaic. c.onvo.tu.t.£on integJta.t.6 . . . . . . . . 71
J. P. Q. CARNEIRO, Non-aJtc.himedean weighted appJtoximation ... 121
J. P. FERRIER, TheoJtie .6pec.tJta.te en une in6inite de vaJtiabte..6 . . 133
P. M. GAUTHIER, MeJtomoJtphic. uni60Jtm appJtoximation on
.6ub.6et.6 06 open Riemann .6uJt6ac.e.6 .... • 139
C. S. GUERREIRO, Whitney'.6 .6pec.tJta.t .6ynthe.6i.6 theoJtem in in-
• • 159
G. G. LORENTZ and S. D. RIEr~NSCHNEIDER, Rec.ent PJtoge.6.6 in
BiJtkh066 inteJtpo.tation .... • 187
P. MALLIAVIN, AppJtoximation po.tynomia.te. pondeJtee et pJtoduit.6
c.anonique..6 ........•....• • • 237
vi i
vi i i TABLE OF CONTENTS
R. MEISE, Spac.e-& 06 di 66 elr. entiable nun c.tion-& and the app1toxima-
tio n pita p etr.ty. . . . . . • • 263
L. NACHBIN, A look at appltoximation theolty ........... 309
L. NARICI and E. BECKENSTEIN, Banac.h alg e bltCL6 ovelt va..fued 6ie.fM. . 333
on
Ph. NOVERRAZ, Appltoximation plulti-&ubhaltmonic. 6unc.tion-& . ... 343
O. T. W. PAQUES, The appltoximation pltopeltty 601t c.elttain -&pac.M
06 holomo~phic. mapping-&. · . 351
J. B. PROLLA, The appltoximation pltopeltty nolt Nac.hbin -6pac.e-6 . .. 371
I. J. SCHOENBERG, On c.altdinal -&pline -&moothing ... • . 383
M. VALDIVIA, A c.haltac.teltization 06 ec.helon Kothe-Sc.hwMtz -6pac.M . . 409
D. WULBERT, The Itational appltoximation 06 Iteat nunc.tion-6 .... 421
G. ZAPATA, Fu ndamentat -6 eminoltm-6 . . . . • . . . . . . . . . . . 429
Index . .................•.....•..... 445
Apppo~imation TheopY and FunationaZ Analysis
J.B. FPoZZa (ed.)
© No#h-HoZZand Publishing Company, 1979
POLYNOMIAL APPROXIMATION AND A QUESTION OF G. E. SHILOV
RICHARD M. ARON
Instituto de Matematica
Universidade Federal do Rio de Janeiro
Caixa Postal 1835, zc-OO
20.000 Rio de Janeiro, Brazil
and
School of Mathematics
Uni vers i ty of Dub 1 in
39 Trinity College
Dublin 2, Ireland
ABSTRACT Let E be an inf ini te dimens ional real or complex Banach
space. For n = 0 ,1,2, ... , 00, let an (E) be the algebra generated by
all continuous polynomials on E which are homogeneous of degree < n.
We discuss the completion of antE) with respect to several natural
topologies, in the real and complex case. In particular, we prove that
when E is a complex Banach space whose dual has the approximation
1
property, then the T w - completion of a (E) can be iden ti f ied wi th
those holomorphic functions f:E + ~ whose derivative df: E +E'is
compact.
Let E be a Banach space over JK = lR or <I:, with closed unit
ball B . For each n E IN, let P (nE ,F) be the space of n-hcr.ogeneous
l
continuous polynomials P: E + F, normed by
=
sup {lIp{x)1I : x E B } (P{oE,F) F). ~(E,F) is the space of m-con-
I
tinuously Frechet differentiable functions from E to F and H(E,F)
is the space of holomorphic mappings from E to F, where E and F
are complex Banach spaces. Throughout, if the range space F is