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基本説明
Addresses completely general weight functions and presents a new methodology for handling the discrete weights case.
Full Description
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.
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
