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Discrete Orthogonal Polynomials. (AM-164):
Asymptotics and Applications (AM-164)
J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller

Paperback | 2007 | $57.50 / £39.95 | ISBN: 9780691127347
184 pp. | 8 x 10 | 14 halftones. 6 line illus.
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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case.

J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Review:

"This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications."--H.J. Glaeske, Zentralblatt MATH

"The book includes the authors' own research results developed over the last years. Their approach and proofs are straightforward constructive making this monograph accessible and valuable to undergraduate and graduate students, PhD students and researchers involved in the asymptotic analysis of systems of discrete orthogonal polynomials."--Octavian Agratini, Mathematica

"This book is very interesting. It pulls together several methods developed during the last twenty years and makes them into a general theory. Thanks to this general theory, several open problems are solved. The organization of the book is very nice."--Sylvie Corteel, Mathematical Reviews

Table of Contents:

Preface vii
Chapter 1. Introduction 1
Chapter 2. Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane 25
Chapter 3. Applications 49
Chapter 4. An Equivalent Riemann-Hilbert Problem 67
Chapter 5. Asymptotic Analysis 87
Chapter 6. Discrete Orthogonal Polynomials: Proofs of Theorems Stated in x2.3 105
Chapter 7. Universality: Proofs of Theorems Stated in x3.3 115
Appendix A. The Explicit Solution of Riemann-Hilbert Problem 5.1 135
Appendix B. Construction of the Hahn Equilibrium Measure: the Proof of Theorem 2.17 145
Appendix C. List of Important Symbols 153
Bibliography 163
Index 167

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