Baik / Kriecherbauer / McLaughlin Discrete Orthogonal Polynomials

Asymptotics and Applications
Course Book
ISBN: 978-1-4008-3713-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 184 Seiten

Reihe: Annals of Mathematics Studies

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.

Autoren/Hrsg.

Baik, J.

Kriecherbauer, T.

McLaughlin, Kenneth Dean T-R

Miller, Peter D.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik

Inhaltsverzeichnis

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

Über Autor(innen)

J. Baik is Associate Professor of Mathematics at the University of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universität Bochum in Bochum, Germany. K. T.-R. McLaughlin is Professor of Mathematics at the University of Arizona. P. D. Miller is Associate Professor of Mathematics at the University of Michigan.