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 This Page Intentionally Left Blank ~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