Functional Analysis, Holomorphy and Approximation Theory: 1st: Seminar Proceedings: 1st

495 Pages·1982·12.86 MB·English

by  Jorge Alberto Barroso

FUNCTIONAL ANALYSIS, HOLOMORPHY
AND APPROXIMATION THEORY
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NORTH-HOLLAND
MATHEMATICS STUDIES 71
Notas de Matematica (88)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Functional Analysis, Holomorphy
and Approximation Theory
Proceedings of the Seminario de Analise Functional, Holomorfia
e Teoria da AproximaGGo, Universidade Federal do Rio de Janeiro,
August 4-8,1980
Edited by
Jorge Alberto BARROSO
lnstituto de MatemBtica
Universidade Federal do Rio de Janeiro
1982
NORT-HOLLAND PUBLISHING COMPANY-AMSTERDAM NEW YORK OXFORD
North-Holland Publishing Company, 1982
All rights reserved. No part of this publication may be reproduced, stored in 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 86527 6
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM NEW YORK OXFORD
Sole distributorsf or the U.S.A.a nd Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE
NEW YORK,N.Y. 10017
Lihrary of Congress Cataloging in Publication Data
Semingrio de Anslise Funcional, Holomorfia e Teoria
da Aproximaf % o (1980 : Universidade Federal do Rio
de Janeiro)
Functional analysis, holomorphy, and approximation
theory.
(North-Holland mathematics studies ; 71) (Notas de
matematica ; 88)
1. Functional analysis--Congresses. 2. Holomorphic
function--Congresses. 3. Domains of holomorphy--
Congresses. 4. Approximation theory--Congresses.
I. Barroso, Jorge Alberto. 11. Title. 111. Series.
IV. Series: Notas de matematica (North-Holland Publishing
Company) ; 88.
QAl.N86 no. 88 [QA320] 510s L515.71 82-18908
ISBN 0-444-86527-6
PRINTED IN THE NETHERLANDS
FOREWORD
This volume is the Proceedings of the Semindrio de Andlise
Funcional, Holomorfia e Teoria da Aproximapzo, held at the Insti-
tuto de MatemBtica, Universidade Federal do Rio de Janeiro (UFRJ)
in August 4-8, 1980. It includes papers of a research or of an
advanced expository nature. Some of them were presented at the
Seminar. Others are contributions of prospective participants,
that, for one or another reason, could not attend the Seminar.
The participant mathematicians are from Brazil, Chile, England,
France, Spain, United States, Uruguay, West Germany and Yugoslavia.
The members of the organizing committee were J.A. Barroso
(Coordinator), S. Machado, M.C. Matos, L. Nachbin, D. Pisanelli,
J.B. Prolla and G. Zapata.
Our warmest thanks are due to the support of the Conselho
de Ensino para Graduados e Pesquisa (CEPG) of UFRJ, Mainly through
its President, Profeseor SQrgio Neves Monteiro, and to the I.B.M.
do Brasil.
We are happy to thank Professor Paulo Emidio de Freitas Bar-
bosa, Dean of the Centro de CiGncias Matemgticas e da Natureza
(CCMN) of UFRJ, in whose facilities the Seminar was very comforta-
bly held.
Our gratitude and admiration to our friend Professor Leopoldo
Nachbin, whose experience and support made the task of preparing
this volume easier.
We also tbank Wilson Goes for a competent typing job.
Jorge Albert0 Barroso
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vii
TABLE OF CONTENTS
Rodrigo Arocena and On a lifting theorem and its rela-
Mischa Cotlar tion to some approximation problems 1
Klaus D. Bierstedt,
R.G. Meise and Kt)the sets and Kbthe sequence spaces 27
W.H. Summers
Bruno Brosowski Parametric approximation and
optimization 93
M.T. Carrillo and Maximal convolution operators and
M. de Guzm6n approximations 117
J.F. Colombeau and Convolution equations in infinite
Mdrio C. Matos dimensions: Brief survey, new
results and proofs 13 1
J.F. Colombeau and Holomorphic and differentiable
Jorge Mujica mappings of uniform bounded type 179
J.F. Colombeau and Finite-difference partial differ-
O.W. Paques ential equations in normed and
locally convex spaces 201
Ed Dubinsky Approximation properties in
nuclear Fr6chet spaces 215
R6mi Langevin Geometry of the neighbourhood of
a singularity 23 5
Pierre Lelong A class of Fr6chet complex spaces
in which the bounded sets are
C-polar sets 255
R.G. Meise and An interpretation of T,,, and
Dietmar Vogt T6 as normal topologies of
sequence spaces 273
Reinhard Mennicken and Well located subspaces of
Manfred MOller LF-spaces 287
viii TABLE OF CONTENTS
P.S. Milojevi6 Continuation theory for A-proper
and strongly A-closed mappings and
their uniform limits and nonlinear
perturbations of Fredholm mappings 299
V.B. Moscatelli New examples of nuclear Fr6chet
spaces without bases 373
Michael 0’ Carroll A survey of some recent results
on the inverse spectral and
scattering problems for differ-
ential operators 379
Peter Pflug Various applications of the
existence of well growing holo-
morphic functions 391
On the Stone-Weierstrass theorem
for modules over non-Archimedean
valued fields 4 13
Laurent Schwartz Semi-Martingales and measure theory 433
Manuel Valdivia On semi-Suslin spaces and dual
metric spaces 445
G. Zapata On the approximation of functions
in inductive limits 461
Functional Analysis, Holomotphy and
Approximation Theory, JA. Barroso led.)
0N orth-Hollond hblishing Company. 1982
In memory of A. MONTEIRO, an extraordinary man and teacher
ON A LIFTING THEOREM AND ITS RELATION
TO SOME APPROXIMATION PROBmMS
Rodrigo Arocena and Mischa Cotlar
SUMMARY
We point out that there is a close relation between some
approximation problems and a lifting theorem studied in previous
papers. A new simplified proof and an improved version of the
theorem, more adequated to our aim, are given.
1. INTRODUCTION AND NOTATIONS
In this self-contained paper we continue the study of some
questions considered in [2] and related to a lifting theorem. We
show that this lifting theorem allows to approach some classical
approximation problems. Conversely, these approximation problems
yield a natural motivation of the lifting and suggest the corres-
ponding theorem. Thus, we give a new simplified proof and an im-
proved version of that lifting theorem, adapted to and motivated by
approximation questions.
We shall work in the unit circle TN [0,2rr] and use the
following notations: en(t) = exp(int), P+ = {analytic polynomials,
‘kn=O ckek(t), n ;r 03, P- = Ex,, -1 ck e k(t ), n=1,2 ,...3 , eJP+ =
= {x;+n ckek( t), n t 0) , p = P+ + P- , dt = the Lebesgue measure
-n
in T, P - =~ { c ckek(t), n > 03, HP = {f 6 LP(T) : 2.(n)= 0,
k=-1