Approximation Theory in Tensor Product Spaces
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1169
WA. Light E.W Cheney
Approximation Theory in Tensor Product Spaces
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1169
WA. Light E.W Cheney
Approximation Theory in Tensor Product Spaces
Springer-Verlag Berlin Heidelberg New York Tokyo
Authors
William Allan Light Mathematics Department, University of Lancaster Bailrigg, Lancaster LA 1 4YL, England Elliott Ward Cheney Mathematics Department, University of Texas Austin, Texas 78712,USA
Mathematics Subject Classification (1980): Primary: 41 A63, 41 A65 Secondary: 41-02, 41 A30, 41 A45, 41 A50
ISBN 3-540-16057-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16057-4 Springer-Verlag New York Heidelberg Berlin Tokyo 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, broadcasting, 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
DEDICATION
This work is dedicated to the memory of Robert Schatten (1911 - 1977)
who did much of the pioneering work in the theory of tensor products of Banach spaces.
PREFACE
In the past two decades, a new branch of approximation theory has emerged; it concerns the approximation of multivariate functions by combinations of univariate ones. The setting for these approximation problems is often a Banach space which is the tensor product of two or more simpler spaces. Approximations are usually sought in subspaces which are themselves tensor products. While these are infinite dimensional, they may share some of the characteristics of finite-dimensionalsubspaces. The usual questions from classical approximation theory can be posed for these approximating subspaces, such as (i) Do best approximations exist? (ii) Are best approximations unique? (iii) How are best approximations characterized? (iv) What algorithms can be devised for computing best approximations? (v) Do there exist simple procedures which provide "good" approximations, in contrast to "best" approximations? (vi) What are the projections of least norm on these subspaces? and (vii) what are the projection constants of these subspaces? This volume surveys only a part of this growing field of research. Its purpose is twofold: first, to provide a coherent account of some recent results; and second, to give an exposition of the subject for those not already familiar with it. We cater for the needs of this latter category of reader by adopting a deliberately slow pace and by including virtually all details in the proofs. We hope that the book will be useful to students of approximation theory in courses and seminars. Expert readers may wish to
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