Table Of Content

Grundlehren der mathematischen Wissenschaften 304 A Series of Comprehensive Studies in Mathematics- George G..Lorentz. Manfred v. Golitschek Yuly Makovoz Constructive Approximation Advanced Problems -Springer Grundlehren der mathematischen Wissenschaften 304 A Series of Comprehensive Studies in Mathematics Editors M. Artin S. S. Chern J. Coates J. M. Fro- hlich H. Hironaka F. Hirzebruch L. HOrmander C. C. Moore J. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt S. Watanabe Managing Editors M. Berger B. Eckmann S. R. S. Varadhan Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo George G. Lorentz Manfred v. Golitschek Yuly Makovoz Constructive Approximation 10 10 Advanced Problems With 1.0 Figures Springer George G. Lorentz Yuly Makovoz Department of Mathematics Department of Mathematics The University of Texas University of Massachusetts at Austin at Lowell Austin, TX 78712-1082, USA Lowell, MA 01854, USA combs @math.utexas.edu makovozy @ woods.uml.edu Manfred v. Golitschek Universitat Wurzburg Institut fur Angewandte Mathematik and Statistik D-97074 Wurzburg, Germany goli @mathematik.uni-wuerzburg.de Library of Congress Cataloging-in-Publication Data Lorentz, G. G. Constructive approximation advanced problems / George G. : Lorentz, Manfred von Golitschek, Yuly Makovoz. p. cm. -- (Grundlehren der mathematischen Wissenschaften 304) Includes bibliographical references (p. - ) and index. ISBN 3-540-57028-4 (Berlin hardcover alk. paper). -- ISBN : : 0-387-57028-4 (New York hardcover alk. paper) : : 1. Approximation theory. I. Golitschek, Manfred von, 1943- II. Makovoz, Yuly, 1937- III. Title. IV. Series. OA221.L63 1996 96-298 515'.83--dc20 CIP Mathematics Subject Classification (1991): Primary: 41 xx; Secondary: 30xx, 31 xx, 26Bxx, 26Cxx, 26Dxx, 42xx, 46xx ISBN 3-540-57028-4 Springer-Veriag Berlin Heidelberg New York ISBN 0-387-57028-4 Springer-Veriag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Editing and reformatting of the authors' input files by Springer-Verlag in using a Springer TEX macro package SPIN: 10124042 41/3143-5 4 3 2 1 0 Printed on acid-free paper Preface In the last 30 years, Approximation Theory has undergone wonderful development, with many new theories appearing in this short interval. This book has its origin in the wish to adequately describe this development, in particular, to rewrite the short 1966 book of G. G. Lorentz, "Approximation of Functions." Soon after 1980, R. A. DeVore and Lorentz joined forces for this purpose. The outcome has been their "Constructive Approximation" (1993), volume 303 of this series. References to this book are given as, for example [CA, p. 2011. Later, M. v. Golitschek and Y. Makovoz joined Lorentz to produce the present book, as a continuation of the first. Completeness has not been our goal. In some of the theories, our exposition offers a selection of important, representative theorems, some other cases are treated more systematically. As in the first book, we treat only approximation of functions of one real variable. Thus, functions of several variables, complex approximation or interpolation are not treated, although complex variable methods appear often. Most of the chapters of the present book can be read independently of each other. They fall into groups: Chapters 1-6 deal with polynomial and spline approximation - in some sense they continue the themes of [CA]. Chap- ters 7-10 contain a fairly complete theory of rational approximation. Chapters 12-14 treat widths and entropy of classes of functions. But even within the groups, chapters are more or less independent, except that it is advisable to read Chapter 3 before Chapter 4, while Chapter 7 is indispensable for Chap- ters 8 and 10, and Chapter 13 for Chapter 14. Most of the information about Banach function spaces needed in the two volumes of CA can be found in [CA, Chapter 21, in §7 of Chapter 1 of the present volume, and in the book of Bennett and Sharpley [B-1988]. We also provide a quick new look at some of the important approximation theorems: for polynomials in §7 of Chapter 1, for splines in § 1 of Chapter 6. Related branches of Analysis: Fourier Series, Orthogonal Polynomials, Potential Theory, Functional Analysis, even Number Theory are our allies. We use their methods; some of the needed results are collected for the reader in the four Appendices. For the development of the Approximation Theory, one cannot be suf- ficiently thankful to the Russian (Soviet) mathematicians: to Chebyshev, A. A. Markov, Bernstein, Kolmogorov and others, who built its foundations. VI Preface At present Approximation Theory is popular worldwide, with the new theories of splines, of rational approximation, of wavelets. We are very grateful to A. A. Pekarskii (Grodno, Belarus), who has pre- pared for us Chapter 10, which deals with complex methods in rational approximation. Our colleagues, Berens, R. A. Lorentz, Stahl, Erdelyi, Lubinsky, Totik have helped us with concrete problems. We are also indebted to Blatt, Buslaev, Chui, Jetter, Maiorov, Shechtman, Varga and others for useful ad- vice. Margaret Combs at the Department of Mathematics, The University of Texas, has very ably typed many chapters of the book. The book has an extensive bibliography, which can also serve as Author's Index. Each quoted journal article is followed by the number of page, where it is referred to in the text. There is also a Subject Index. The authors would be grateful for any comments or proposals of correc- tions from the readers. The Authors Contents Chapter 1. Problems of Polynomial Approximation ................ 1 § 1. Examples of Polynomials of Best Approximation .............. 1 § 2. Distribution of Alternation Points of Polynomials .................................... of Best Approximation 4 .... § 3. Distribution of Zeros of Polynomials of Best Approximation 11 § 4. Error of Approximation .................................... 20 § 5. Approximation on (-oo, oo) by Linear Combinations ..................................... of Functions (x - c)-1 23 ......... § 6. Weighted Approximation by Polynomials on (-oo, oo) 28 Spaces of Approximation Theory ............................ § 7. 33 § 8. Problems and Notes ....................................... 37 ............. Chapter 2. Approximation Problems with Constraints 39 § 1. Introduction ............................................. 39 2. Growth Restrictions for the Coefficients ...................... 39 § 3. Monotone Approximation .................................. 43 ....................... § 4. Polynomials with Integral Coefficients 49 § 5. Determination of the Characteristic Sets ..................... 59 .................................. § 6. Markov-Type Inequalities 64 § 7. The Inequality of Remez ................................... 73 § 8. One-sided Approximation by Polynomials .................... 76 ................................................ §9. Problems 81 10. Notes ................................................... 82 Chapter 3. Incomplete Polynomials .............................. 85 Incomplete Polynomials ..................................... § 1. ......................... 85 §2. Incomplete Chebyshev Polynomials 89 §3. Incomplete Trigonometric Polynomials ....................... 92 .............. § 4. Sequences o.f. P.o..ly.n.o.m..i.a.ls. .w.i.t.h. M..a.n.y. .R.e.a.l. Z..e.ro.s................ 98 § 5. Problems................................................... 104 §6. Notes 104 VIII Contents ............................... Chapter 4. Weighted Polynomials 105 ...................... §1. Essential Sets of Weighted Polynomials 105 Weighted Chebyshev Polynomials ........................... §2. 109 The Equilibrium Measure .................................. §3. 117 §4. Determination of Minimal Essential Sets ..................... 125 5. Weierstrass Theorems and Oscillations ....................... 131 §6. Weierstrass Theorem for Freud Weights ...................... 134 ................................................ §7. Problems 141 ................................................... §8. Notes 141 Chapter 5. Wavelets and Orthogonal Expansions .................. 145 Multiresolutions and Wavelets .............................. § 1. 145 § 2. Scaling Functions with a Monotone Majorant ................. 151 Periodization ............................................. § 3. ................................ 156 § 4. Polynomial Schauder Bases ............................. 160 § 5. Orthonormal Polynomial Bases 164 Problems and Notes ....................................... § 6. 172 ............................................ Chapter 6. Splines 175 I. General Facts ............................................ 175 Splines of Best Approximation .............................. § 2. 181 .......................................... § 3. Periodic Splines 189 ...................... § 4. Convergence of Some Spline Operators 196 ................................................... §5. Notes 202 ............................. Chapter 7. Rational Approximation 205 ............................................. § 1. Introduction 205 § 2. Best Rational Approximation ............................... 210 .............................. § 3. Rational Approximation of lxi 217 ............................. § 4. Approximation of ex on [-1, 1] 221 .................... § 5. Rational Approximation of e_x on [0, oo) 227 § 6. Approximation of Classes of Functions ....................... 231 ....................................... § 7. Theorems of Popov 235 § 8. Properties of the Operator of Best Rational Approximation in C and LP .............................................. 242 ... § 9. Approximation by Rational Functions with Arbitrary Powers 248 § 10. Problems ................................................ 251 §11.Notes ................................................... 252 .................................... Chapter 8. Stahl's Theorem 255 .............................. § 1. Introduction and Main Result 255 .......................... § 2. A Dirichlet Problem on [1 /2,1 / pn] 256 Contents IX § 3. The Second Approach to the Dirichlet Problem ............... 263 § 4. Proof of Theorem 1.1 ...................................... 271 ................................................... §5. Notes 276 ................................ 277 Chapter 9. Pade Approximation § 1. The Pade Table .......................................... 277 .................. § 2. Convergence of the Rows of the Pade Table 282 § 3. The Nuttall-Pommerenke Theorem .......................... 290 §4. Problems ................................................ 296 §5. Notes ................................................... 296 Chapter 10. Hardy Space Methods in Rational Approximation ...... 299 § 1. Bernstein-Type Inequalities for Rational Functions ............ 300 ............ § 2. Uniform Rational Approximation in Hardy Spaces 308 § 3. Approximation by Simple Functions ......................... 314 § 4. The Jackson-Rusak Operator; Rational Approximation of Sums of Simple Functions 320 ................. § 5. Rational Approximation on T and on [-1, 1] 322 § 6. Relations Between Spline and R.a.t.i.o.n.a.l .A..p.p.r.o.x.im..a.t.io..n.......... in the Spaces LP, 0 < p < oo 332 § 7. Problems ................................................ 341 § 8. Notes ................................................... 341 Chapter 11. Miintz Polynomials ................................. 345 § 1. Definitions and Simple Properties ........................... 345 Miintz-Jackson Theorems .................................. § 2. 347 § 3. An Inverse Miintz-Jackson Theorem ......................... 353 ............................... § 4. The Index of Approximation 360 § 5. Markov-Type Inequality for Mi.intz Polynomials ............... 362 ................................................ § 6. Problems 365 ................................................... 366 §7. Notes ........................... Chapter 12. Nonlinear Approximation 369 Definitions and Simple Properties ........................... §1. 369 Varisolvent Families ....................................... §2. 371 Exponential Sums ......................................... §3. 376 ......... §4. Lower Bounds for Errors of Nonlinear Approximation 383 § 5. Continuous Selections from Metric Projections ................ 386 § 6. Approximat.i.o.n. i.n. .B.a.n.a..c.h. S..p.a.c.e.s.: .S.u.n.s. .a.n.d. .C.h.e.b..y.s.h.e.v. S..e.ts. ....... 390 § 7. Problems................................................... 395 § 8. Notes 396