Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . Band 7 A Series of Modern Surveys in Mathematics Editorial Board E. Bombieri, Princeton S. Feferman, Stanford N. H. Kuiper, Bures-sur-Yvette P. Lax, New York R. Remmert (Managing Editor), Munster W Schmid, Cambridge, Mass. J-P. Serre, Paris J. Tits, Paris Allan Pinkus n-Widths in Approxilllation Theory Springer-Verlag Berlin Heidelberg New York Tokyo 1985 Allan Pinkus Technion Israel Institute of Technology Department of Mathematics Haifa 32000, Israel AMS Subject Classification (1980): 41-02, 41A46, 41A65 ISBN-13 :978-3-642-69896-5 e-ISBN-13: 978-3-642-69894-1 DOl: 10.1007/978-3-642-69894-1 Library of Congress Cataloging in Publication Data Pinkus, Allan, 1946- N-widths in approximation theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 7) Bibliography: p. Includes index. 1. Approximation theory. I. Title. II Series. QA221.P56 1985 511'.4 84-13902 ISBN-13 :978-3-642-69896-5 (U.S.) 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 Softcover reprint of the hardcover 1st edition 1985 Typesetting: Daten-und Lichtsatz-Service, Wiirzburg 2141/3140-543210 To Rachel Preface My original introduction to this subject was through conservations, and ultimate ly joint work with C. A. Micchelli. I am grateful to him and to Profs. C. de Boor, E. W. Cheney, S. D. Fisher and A. A. Melkman who read various portions of the manuscript and whose suggestions were most helpful. Errors in accuracy and omissions are totally my responsibility. I would like to express my appreciation to the SERC of Great Britain and to the Department of Mathematics of the University of Lancaster for the year spent there during which large portions of the manuscript were written, and also to the European Research Office of the U.S. Army for its financial support of my research endeavors. Thanks are also due to Marion Marks who typed portions of the manuscript. Haifa, 1984 Allan Pinkus Table of Contents Chapter I. Introduction . . . . . . . . 1 Chapter II. Basic Properties of n-Widths . 9 1. Properties of dn • • • • • • • • • • 9 2. Existence of Optimal Subspaces for dn • 15 3. Properties of dn • • • • • • 17 4. Properties of b 20 n • • • • • • 5. Inequalities Between n-Widths 22 6. Duality Between dn and dn • • 27 7. n-Widths of Mappings of the Unit Ball 29 8. Some Relationships Between dn(T), dn(T) and bn(T) . 32 Notes and References . . . . . . . . . . . . . . 37 Chapter III. Tchebycheff Systems and Total Positivity 39 1. Tchebycheff Systems 39 2. Matrices . . . . 45 3. Kernels. . . . . 52 4. More on Kernels . 56 Chapter IV. n-Widths in Hilbert Spaces 63 1. Introduction. . . . . . . . . . . 63 2. n-Widths of Compact Linear Operators 63 3. n-Widths, with Constraints . . . . . 73 3.1 Restricted Approximating Subspaces 73 3.2 Restricting the Unit Ball and Optimal Recovery. 77 3.3 n-Widths Under a Pair of Constraints . . . . . 80 3.4 A Theorem of Ismagilov . . . . . . . . . . . 93 4. n-Widths of Compact Periodic Convolution Operators 95 4.1 n-Widths as Fourier Coefficients . . 95 4.2 A Return to Ismagilov's Theorem . . . 98 4.3 Bounded mth Modulus of Continuity . 101 5. n-Widths of Totally Positive Operators in L2 107 5.1 The Main Theorem . . . . . . . . 108 5.2 Restricted Approximating Subspaces 117 X Table of Contents 6. Certain Classes of Periodic Functions . . . . . . . . . 125 6.1 n-Widths of Cyclic Variation Diminishing Operators. 126 6.2 n-Widths for Kernels Satisfying Property B . 128 Notes and References . . . . . . . . . . . . 136 Chapter V. Exact n-Widths of Integral Operators 138 1. Introduction. . . . . . . . . . . . . 138 2. Exact n-Widths of .Yt'oo in Lq and ~ in L1 139 3. Exact n-Widths of .Yt':, in Hand .Yt'; in V 154 4. Exact n-Widths for Periodic Functions 169 5. n-Widths of Rank n + 1 Kernels 188 Notes and References . . . . . . 195 Chapter VI. Matrices and n-Widths 198 1. Introduction and General Remarks 198 2. n-Widths of Diagonal Matrices. . 202 2.1 The Exact Solution for q ~ p and p = 1, q = 2 202 2.2 Various Estimates for p = 1, q = 00 . • . 212 3. n-Widths of Strictly Totally Positive Matrices 223 Notes and References . . . . . . . . . . . . 231 Chapter VII. Asymptotic Estimates for n-Widths of Sobolev Spaces 232 1. Introduction. . . . . 232 2. Optimal Lower Bounds . . . 234 3. Optimal Upper Bounds. . . 236 4. Another Look at <>n(Br); Loo) . 241 Notes and References . . . . . 246 Chapter VIII. n-Widths of Analytic Functions 248 1. Introduction. . . . . . . . . . . . . . 248 2. n-Widths of Analytic Functions with Bounded mth Derivative 249 3. n-Widths of Analytic Functions in H2 . 258 4. n-Widths of Analytic Functions in HOO 263 5. n-Widths of a Class of Entire Functions 270 Notes and References 275 Bibliography. . . . 277 Glossary of Selected Symbols. 288 Author Index 289 Subject Index. 291 Chapter 1. Introduction In this short chapter we introduce the subject matter. We hope that this introduc tion whets the reader's appetite for the more systematic treatment which will follow. Let X be a normed linear space and Xn any n-dimensional subspace of X. For each x E X, E (x; Xn) shall denote the distance of the n-dimensional subspace Xn from x, defined by If there exists a y* E Xn for which E (x; X n) = II x - y* II , then y* is a best approxi mation to x from Xn. Problems of existence, uniqueness, and characterization of the best approximation are of central importance in approximation theory. (Since Xn is here a finite dimensional subspace of X, a best approximation always exist.) For example, consider X = C [a, b], the space of real-valued continuous functions on the finite interval [a, b], endowed with the uniform norm. The classical Haar Theorem delineates those n-dimensional subspaces of C [a, b], called Chebyshev systems, for which there is a unique best approximation to every fEe [a, b]. This unique best approximant is characterized by the fact that the error function equioscillates on at least n + 1 points in [a, b]. Let us now suppose that instead of a single element x, we are given a subset A of X. How well does the n-dimensional subspace Xn of X approximate the subset A? A commonly used definition is to set E(A;Xn) = sup {E(x;Xn): x E A} = sup inf II x - y II. xeA yeXn E(A;Xn) is the deviation of A from Xn. Thus E(A;Xn) measures the extent to which the "worst element" of A can be approximated from Xn. Many results in approximation theory are concerned with this particular quantity for specific choices of A and X n • Given a subset A of X, one might also ask how well one can approximate A by n-dimensional subspaces of X. Thus, we shall consider the possibility of allow ing the n-dimensional subspaces Xn to vary within X. This idea was first pro pounded by Kolmogorov [1936]. Definition 1. Let X be a normed linear space and A a subset of X. The n-width, in the sense of Kolmogorov, of A in X (or the Kolmogorov n-width of A in X) is 2 I. Introduction given by dn(A;X) = inf{E(A;Xn): Xn an n-dimensional subspace of X} = inf sup inf II x - y Ilx, x" xeA yeXn the left-most infimum being taken over all n-dimensional subspaces Xn of X. (Some authors prefer the expression "n-diameter" rather than "n-width". We shall always use the latter term.) A subspace Xn of X of dimension at most n for which is called an optimal subspace for d (A; X). n Since d (A; X) measures the extent to which A may be approximated by n n-dimensional subspaces of X, it is, in a certain sense, a measure of the "thickness" or "massivity" of A. It is, of course, generally impossible to obtain d (A; X) and determine optimal n subspaces Xn for dn(A;X) (if they exist) for all A and X. Nonetheless, much effort has been devoted to this task for specific choices of A and X. In this work we survey numerous choices of A and X for which both dn(A;X) and Xn are in fact obtained, or at least characterized. However in some surprisingly simple cases (see for example Chapter VI) these quantities have not as yet been calculated, though hardly through lack of effort. It is also of considerable interest to determine the asymptotic behavior of dn(A; X) as n i 00, since typically an optimal Xn cannot be explicitly obtained or if it can, too much computational effort is involved in its determination. In many cases very simple n-dimensional subspaces may approximate A in an asymp totically optimal manner. Thus the n-width, in providing a lower threshold on the degree of approximation of A by n-dimensional subspaces, tells us how well a given n-dimensional subspace (e.g., algebraic or trigonometric polynomials, splines with fixed knots, etc ...) approximates A relative to the theoretical lower bound. On this basis it is then possible to judge whether the additional time, effort, and perhaps money, involved in using better but more complicated subspaces is, in fact, justified. Kolmogorov, aside from simply defining d (A; X), also computed this quan n tity in two particular instances. The first of these we give here; the second is deferred to Chapter IV. Let L2 = L2 [0,211:] denote the usual space of square integrable functions on [0,211:] with norm )1/2 21< 11111 = ( 1/211:! II(xWdx . For any given positive integer r, let Wj') denote the (Sobolev) space of2 11:-periodic, real-valued, (r - I)-times differentiable functions whose (r - l)st derivative is ab- I. Introduction 3 solutely continuous and whose rth derivative is in L2. Thus wt) = {f:j<,-1) abs. cont., j<')e L2, j

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