Asymptotics for Orthogonal Polynomials
Recently there has been a great deal of interest in the theory of orthogonal polynomials. The number of books treating the subject, however, is limited. This monograph brings together some results involving the asymptotic behaviour of orthogonal polynomia
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1265 Walter Van Assche
Asymptotics for Orthogonal Polynomials
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Walter Van Assche Senior Research Assistant of the Belgian National Fund for Scientific Research) Departement Wiskunde, Katholieke Universiteit Leuven Celestijnenlaan 200 B, 3030 Leuven, Belgium
Mathematics Subject Classification (1980): Primary: 42C05 Secondary: 33A65; 39A 10; 40A 15; 40A30; 41 A55; 41 A60; 60J80 ISBN 3-540-18023-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18023-0 Springer-Verlag 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
In recent years there has been an increased interest in the theory of orthogonal polynomials but the number of textbooks treating orthogonal polynomials is rather limited. Even at present the best reference is Szego's book [175} which was first published in 1939. Freud's book [61} is also highly recommended. The more recent introduction by Chihara [39] does not include asymptotic results and the monographs by Geronimus [75} and Nevai [138] are rather technical and treat a very specific part of the theory of orthogonal polynomials on [-l,l}. This monograph concentrates on the asymptotic theory of general orthogonal polynomials on the real line. Most of the theorems have been proved, for some of them only a sketch of the proof is given and tedious proofs out of the scope of this monograph have been omitted. I would like to express my very cordial thanks to all those who, in some way or another, have contributed to this monograph. Many thanks in particular to Jef Teugels who made me appreciate the mathematical analysis of orthogonal polynomials. Some theorems in this monograph are the result of working with other mathematicians. I am very grateful to Makoto Maejima (Chapter 3), Jeffrey Geronimo (Chapter 2, Section 4.4) and Guido Fano (Section 5.1) for a fruitful collaboration. Finally I would like to thank Daniel Bessis, Pierre Moussa, Giorgio Turchetti, Paul Nevai, Doron Lubinsky, Ed Saff, Alphonse Magnus and many others for various interesting discussions and Bea Peeters for an excellent job of typing the manuscript.
Walter Van Assche, March 1987.
TABLE OF CONTENTS
Introduction 0.1 0.2 0.3 0.4
Definitions and examples General properties Outline of this monograph A short historical note
1 7
11 12
Chapter 1 : Orthogonal Polynomials on a
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