Topics in Approximation Theory
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Harold S. Shapiro Department of Mathematics The University of Michigan, Ann Arbor, MI/USA
Topics in Approximation Theory
Springer-Verlag Berlin· Heidelberg· New York 1971
AMS Subject Classifications (1970): 41-XX, 42A04, 42A08, 42A64 - 42A72, 42A88, 42A96, 44A35, 46B99,46ClO, 46E15, 46E20, 26A 72, 30A31, 30A38, 30A 76, 30A80
ISBN 3-540-05376-X Springer-Verlag Berlin' Heidelberg' New York ISBN Q-387-05376-X Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasring, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number73-151323. Printed in Germany. Offsetdruck: Julius Beltz, Weinheim/Bergstr.
PREFACE
These notes are based on a course given by me at the Royal Institute of Technology in Stockholm in the Fall Term, 1969, and on some seminar talks I gave in the Stockholm area. In addition, two chapters (presented as appendices) were kindly contributed to the present volume, at my request, by Jan Boman and Torbjorn Hedberg respectively. The audience at my lectures consisted largely of research scholars from the Mittag-Leffler Institute with strong backgrounds in analysis, and therefore I took for granted some familiarity with measure theory, distribution theory, Fourier analysis, functional analysis and even more specialized topics like HP spaces and vectorvalued integration. On the other hand, no prior knowledge of approximation theory was assumed. In writing up the notes, I tried to smooth the path of the beginner by giving references for those notions not likely to be familiar; I did not always do this, h01l'eYer,for fairly standard topics of real, complex, or functional analysis (e.g. the Hahn-Banach theorem) since good text-books on these subjects are legion. It should be stressed that these notes are not meant to serve as a text-book for a course in approximation theory, although portions of i t might advantageously be used in connection with such a course or seminar. The choice of topics is somewhat arbitrary, the coverage of these is mostly not systematic, and there is considerable disparity in level and style between different chapters. (Thus, for example, there is no discussion of such fundamental topics as splines, polynomial and rational approximation in the complex domain, or algorithms for constructing best approximations.) It was my intention that this book should be used in conjunction with existing works, and I have frequently referred to the literature rather than repeat demonstrations in readily available sources. The exercises also vary Widely in character; a serious
reader should not just skip over
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