数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (4): 1005-1034.doi: 10.1016/S0252-9602(09)60084-9

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ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA THE RIEMANN-HILBERT APPROACH

R. Wong,赵育求   

  1. Department of Mathematics, City University of Hong Kong, Hong Kong, China|Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
  • 收稿日期:2009-01-29 出版日期:2009-07-20 发布日期:2009-07-20
  • 基金资助:

    The research of Zhao was supported in part by the National Natural Science Foundation of China (10471154 and 10871212)

ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA THE RIEMANN-HILBERT APPROACH

 R. Wong, DIAO Yo-Qiu   

  1. Department of Mathematics, City University of Hong Kong, Hong Kong, China|Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
  • Received:2009-01-29 Online:2009-07-20 Published:2009-07-20
  • Supported by:

    The research of Zhao was supported in part by the National Natural Science Foundation of China (10471154 and 10871212)

摘要:

In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Deift & Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal
polynomials, and certain orthogonal polynomials with singular behavior.

关键词: orthogonal polynomials, asymptotic methods, Riemann-Hilbert approach

Abstract:

In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Deift & Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal
polynomials, and certain orthogonal polynomials with singular behavior.

Key words: orthogonal polynomials, asymptotic methods, Riemann-Hilbert approach

中图分类号: 

  • 41A60