Orthogonal Polynomials for Exponential Weights

471 Pages·2001·10.439 MB·English

by  Eli Levin, Doron S. Lubinsky (auth.)

Canadian Mathematical Society
matMmatique du Canada
Soci~t~
Editors-in-Chie[
Redacteurs-en-chef
Jonathan Borwein
Peter Bonvein
Springer-Science+Business Media, LLC
CMS Books in Mathematics
Ouvrages de mathematiques de /a SMC
1 HERMAN/KuCERAlSIMSA Equations and Inequalities
2 ARNOLD Abelian Groups and Representations of Finite Partially Ordered Sets
3 BORWEIN/LEWIS Convex Analysis and Nonlinear Optimization
4 LEVIN/LuBINSKY Orthogonal Polynomials for Exponential Weights
5 KANE Reflection Groups and Invariant Theory
6 PHILLIPS Two Millennia of Mathematics
7 DEUTSCH Best Approximation in Inner Product Spaces
8 FABIAN ET AL. Functional Analysis and Infinite-Dimensional Geometry
Eli Levin Doron S. Lubinsky
Orthogonal Polynotnials
for Exponential Weights
, Springer
Eli Levin Doron S. Lubinsky
Department of Mathematics Centre for Applicable Analysis
The Open University of Israel and Number Theory
P.O. Box 39328 Department of Mathematics
Ramat Aviv Witwatersrand University
Tel Aviv 61392 Private Bag 3
Israel Wits 2050
South Africa
Editors-in-ChieJ
Redacteurs-en-cheJ
Jonathan Borwein
Peter Borwein
Centre for Experimental and Constructive Mathematics
Department of Mathematics and Statistics
Simon Fraser University
Burnaby, British Columbia V5A IS6
Canada
Mathematics Subject Classification (2000): 33C50, 42C05
Library of Congress Cataloging-in-Publication Data
Levin, A.L., 1944-
Orthogonal polynomials for exponential weights I A.L. Levin, Doron S. Lubinsky.
p. cm. - (CMS books in mathematics; 4)
Includes bibliographical references and index.
ISBN 978-1-4612-6563-4 ISBN 978-1-4613-0201-8 (eBook)
DOl 10.1007/978-1-4613-0201-8
1. Orthogonal polynomials. 2. Convergence. I. Lubinsky, D.S. (Doron Shaul), 1955-
II. Title. III. Series.
QA404.5 .U8 2001
515'.55-dc21 00-069253
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 2001
Softcover reprint of the hardcover 1s t edition 200 I
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SPIN 10745929
To AlIa and Jenny and Shira
Acknowledgements
While taking responsibility for all the errors and misprints in this book, we
acknowledge comments and corrections from Percy Deift, Arno Kuijlaars,
Hrushikesh Mhaskar, Paul Nevai, Ed Saff, Vili Totik and Walter Van Ass
che. This book is the fruit of some 18 years of collaboration between the
authors on orthogonal polynomials and related topics. During that period,
we have both been inspired by contact and collaboration with many of the
orthogonal polynomials community, and especially with Paul Nevai and Ed
Saff. Their positive and constructive leadership has contributed greatly to
the growth and advancement of the field.
In terms of research support, it is a privilege to acknowledge the aid
of the National Research Foundation of South Africa (formerly the Foun
dation for Research and Development), the Deputy Registrar Research of
Witwatersrand University, the John Knopfmacher Centre for Applicable
Analysis and Number Theory, and the Mathematics Departments of the
Open University of Israel and Witwatersrand University.
On a personal note, we would like to thank our wives AlIa and Jenny
for their patience and encouragement during the six years that this book
was written. And especially, we would like to thank Mrs. Ingrid Eitzen of
the Mathematics Department of Witwatersrand University for her friendly
help, and for her meticulous conversion of the manuscript from its original
Scientific Workplace form into the format required by Springer. During that
painstaking process, Mrs. Eitzen also helped to correct numerous misprints.
If the reader notices more, we would be most grateful to hear of them.
At Springer, we wish to thank Ms. Ina Lindemann for her speedy han
dling of our manuscript.
Contents
1 Introduction and Results 1
1.1 Background .... 1
1.2 Classes of Weights ... 6
1.3 Inequalities ....... 14
1.4 Orthogonal Polynomials: Bounds 21
1.5 Asymptotics of Extremal and Orthonormal Polynomials 23
1.6 Specific Examples. . ........... . . . . . . . . . 27
2 Weighted Potential Theory: The Basics 35
2.1 Equilibrium Measures ....... 35
2.2 Rakhmanov's Representation for Q 45
2.3 A Formula for a~ ........ 51
2.4 Further Identities Involving at . 56
3 Basic Estimates for Q, at 63
3.1 Estimates involving Q ..... 64
3.2 Estimates involving a±t, Part 1 68
3.3 Estimates involving a±t, Part 2 79
3.4 A Weight in F\F (Dini) ... 91
4 Restricted Range Inequalities 95
5 Estimates for Measure and Potential 109
5.1 Statement of Results ......... . 109
x
5.2 Lower Bounds. . 113
5.3 Upper Bounds . 114
5.4 Estimates for 'Pt 125
5.5 Estimation of Potential 129
6 Smoothness of crt 145
6.1 Statement of Results 145
6.2 Outline of the Proof and Technical Lemmas 150
6.3 The Proof of Theorem 6.3 157
6.4 Proof of Theorem 6.2. . . . . . . . . . 165
7 Weighted Polynomial Approximation 169
7.1 Statement of Results ..... . 169
7.2 Setting up the Discretisation . 174
7.3 The Proof of Theorems 7.1-7.4 176
7.4 One-Sided Approximation ... 183
7.5 Properties of the Discretisation Points 186
7.6 The Tail Terms . . 195
7.7 The Central Terms . . . . . . . . . . . 201
7.8 Global Bounds ............ . 206
7.9 Derivatives of Discretised Polynomials 212
7.10 Proof of Theorem 7.5 ...... . 227
8 Asymptotics of Extremal Errors 231
8.1 Statement of Results . 231
8.2 Proof of Theorem 8.1 . 234
8.3 Proof of Theorem 8.2 . 239
8.4 Proof of Theorem 8.3 . 248
9 Christoffel Functions 253
9.1 Statement of Results 253
9.2 Proof of Theorem 9.1 ............ . 258
9.3 Lower Bounds for Lp Christoffel Functions. 260
9.4 Upper Bounds for Christoffel Functions .. 265
9.5 Christoffel Functions for Bernstein-Szego Weights . 271
9.6 Asymptotic Lower Bounds for Christoffel Functions. 273
9.7 Asymptotic Upper Bounds for Christoffel Functions 280
10 Markov-Bernstein and Nikolskii Inequalities 293
10.1 Statement of Results ...... . 293
10.2 Bernstein Inequalities for p = 00 . 295
10.3 Bernstein Inequalities for p < 00 . 298
10.4 Proof of the Nikolskii Inequalities. 308
11 Zeros of Orthogonal Polynomials 313
11.1 Statement of Results . . . . . . . 313
xi
11.2 The Largest Zeros 315
11.3 Spacing of Zeros . 320
12 Bounds on Orthogonal Polynomials 325
12.1 The Essence of the Proof .... 326
12.2 Some Technicalities Involving Q . 333
12.3 Integrals Near and Far from x . 336
12.4 Non-negative x 341
12.5 Negative x. . . . . . . . . . . . 350
13 Further Bounds and Applications 359
13.1 Statement of Results ... . 359
13.2 Estimation of A~(x) ...... . 362
13.3 The Proof of Theorem 13.2 .. . 369
13.4 Lagrange Interpolation Polynomials 371
13.5 Proof of the Corollaries ...... . 380
14 Asymptotics of Extremal Polynomials 385
14.1 Statement of Results . 385
14.2 Technical Lemmas .. 388
14.3 Proof of Theorem 14.1 391
14.4 Proof of Theorem 14.2 394
15 Asymptotics of Orthonormal Polynomials 401
15.1 Statement of Results . 401
15.2 Proof of Theorem 15.2 404
15.3 Proof of Theorem 15.3 409
A Bernstein-Szego Lp Extremal Polynomials 419
B Bernstein-Szego Orthogonal Polynomials 435
Notes 447
References 455
List of Symbols 471
Index 475
1
Introduction and Results
1.1 Background
Let I be a finite or infinite interval and let W : I -+ [0,00) be measurable
with all power moments
1
xnw(x)dx, n = 0,1,2,3, ...
finite. Then we call W a weight and may define orthonormal polynomials
Pn(x) = Pn(w,x) = 'Yn(W)Xn +"', 'Yn(w) > 0,
satisfying
lpnPmw = omn, m,n = 0, 1,2, ....
The analysis of the orthogonal polynomials {Pn(w,· )}~=o associated
with general weights w has been a major theme in classical analysis of
the twentieth century. Probably the most elegant part of that theory is due
to Szego: if w is a weight on I = (-1,1) that satisfies Szego's condition
1 logw(X)d
x>-oo (1.1)
/ -1 J1- x2 '
then there are very precise asymptotics for Pn(z), n -+ 00, for z E C\[-l, 1).
Szego's theory had a substantial influence on the development of Hp spaces
- the notion of an outer function came from the theory.
E. Levin et al., Orthogonal Polynomials for Exponential Weights
© Springer Science+Business Media New York 2001