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Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/4610

Title: Asymptotics of orthogonal polynomials and Riemann-Hilbert problems
Other Titles: Zheng jiao duo xiang shi de jian jin fen xi he Liman-Xierbote wen ti
正交多項式的漸進分析和黎曼-希爾伯特問題
Authors: Dai, Dan (代丹)
Department: Dept. of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2006
Publisher: City University of Hong Kong
Subjects: Orthogonal polynomials -- Asymptotic theory
Riemann-Hilbert problems
Notes: CityU Call Number: QA404.5.D34 2006
Includes bibliographical references (leaves 96-100)
Thesis (Ph.D.)--City University of Hong Kong, 2006
iv, 100 leaves : ill. ; 30 cm.
Type: Thesis
Abstract: In this thesis, we study the asymptotics of orthogonal polynomials as the degree grows to infinity. Our method is based on a recent and powerful method, the Riemann-Hilbert approach, introduced by Deift and Zhou. Both continuous and discrete orthogonal polynomials are discussed. To understand this new method well, we consider two specific examples, the Laguerre polynomials and the Krawtchouk polynomials. First, for the continuous case, we study the asymptotic behavior of the Laguerre polynomials Ln(αn)(nz) as n→∞. Here, αn is a sequence of negative numbers and-αn/n tends to a limit A > 1 as n→∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane C+ = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half plane C- = {z : Im z ≤ 0}. For the discrete case, we study the asymptotics of the Krawtchouk polynomials KNn (z; p, q) as the degree n becomes large. Asymptotic expansions are obtained, when the ratio n/N tends to a limit c Є (0, 1) as n→∞. The results are valid in one or two regions in the complex plane depending on the values of c and p. Some modifications and improvements are made to this method. For instance, we do not require deformation of contours. Moreover, our results hold globally in the complex plane, particularly in regions containing the curve on which these polynomials are orthogonal. These are not available in the precious work using this method.
Online Catalog Link: http://lib.cityu.edu.hk/record=b2146998
Appears in Collections:MA - Doctor of Philosophy

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