Nonlinear Approximation Theory
The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approxima tion is strongly connected with his name. By making use of his ideas, the theories of best uniform ap
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Editorial Board
R. L. Graham, Murray Hill J. Stoer, Wurzburg R. Varga, Cleveland
Dietrich Braess
Nonlinear Approximation Theory With 38 Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Profmor DL DiBtrich Braess Fakultat fUr Mathematik. Ruhr-Universitat Bochum Postfach 102148.0-4630 Bochum 1
Mathematics Subject Classification (1980): 41-02. 41A15. 41A20. 41A21. 41A25. 41A30. 41ASO. 41A52. 41A55.
41A65. 65010. 65015. 65032
ISBN-131:978-3-642-64883-0 e-ISBN-13: 978-3-642-61609-9 001: 10.1007/978-3-642-61609-9
Library of Congress Cataloging-In-Publication Data Braess, Dietrich. 1938Nonlinear approximation theory. (Springer series in computational mathematics; 7) Bibliography: p. Includes index. 1. Approximation theory. I. TItle. II. Series. OA221.B67 1986 511'.4 86-10101 ISBN-131:978-3-642-;648!l3-0 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, afee is payable to "Verwertungsgesellschaft Wort". Munich. © Springer-Verlag Berlin Heidelberg 1986 Typesetting: Asco Trade Typesetting Ltd., Hong Kong Printing and bookbinding: Graphlscher Betrieb Konrad Triltsch, WUrzburg 2141 / 31~543210
To Anneliese
Preface
The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approximation is strongly connected with his name. By making use of his ideas, the theories of best uniform approximation by rational functions and by polynomials were developed over the years in an almost unified framework. The difference between linear and rational approximation and its implications first became apparent in the 1960's. At roughly the same time other approaches to nonlinear approximation were also developed. The use of new tools, such as nonlinear functional analysis and topological methods, showed that linearization is not sufficient for a complete treatment of nonlinear families. In particular, the application of global analysis and the consideration of flows on the family of approximating functions introduced ideas which were previously unknown in approximation theory. These were and still are important in many branches of analysis. On the other hand, methods developed for nonlinear approximation problems can often be successfully applied to problems which belong to or arise from linear approximation. An important example is the solution of moment problems via rational approximation. Best quadrature formulae or the search for best linear spaces often leads to the consideration of spline functions with free nodes. The most famous problem of this kind, namely best interpolation by polynomials, is treated in the appendix of this book. The monograph grew out of lectures which the author gave on numerous
