Constructive Approximation: Advanced Problems

663 Pages·2011·8.95 MB·English

by  George G. Lorentz

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
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Springer
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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
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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 develop-
ment, 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 ap-
proximation. 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