A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

P. Deift, T. Kriecherbauer, K. R. McLaughlin, S. Venakides, X. Zhou

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23 Scopus citations

Abstract

A few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials. In this paper we give an overview of our method and review the results which have been obtained in Deift et al. (Internat. Math. Res. Notices (1997) 759, Comm. Pure Appl. Math. 52 (1999) 1491, 1335), Deift (Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, Vol. 3, New York University, 1999), Kriecherbauer and McLaughlin (Internat. Math. Res. Notices (1999) 299) and Baik et al. (J. Amer. Math. Soc. 12 (1999) 1119). We mainly consider orthogonal polynomials with respect to weights on the real line which are either (1) Freud-type weights dα(x) = e-Q(x) dx (Q polynomial or Q(x) = xβ, β>0), or (2) varying weights dαn(x) = e-nV(x) dx (V analytic, limx→∞V(x)/logx = ∞). We obtain Plancherel-Rotach-type asymptotics in the entire complex plane as well as asymptotic formulae with error estimates for the leading coefficients, for the recurrence coefficients, and for the zeros of the orthogonal polynomials. Our proof starts from an observation of Fokas et al. (Comm. Math. Phys. 142 (1991) 313) that the orthogonal polynomials can be determined as solutions of certain matrix valued Riemann-Hilbert problems. We analyze the Riemann-Hilbert problems by a steepest descent type method introduced by Deift and Zhou (Ann. Math. 137 (1993) 295) and further developed in Deift and Zhou (Comm. Pure Appl. Math. 48 (1995) 277) and Deift et al. (Proc. Nat. Acad. Sci. USA 95 (1998) 450). A crucial step in our analysis is the use of the well-known equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials.

Original languageEnglish (US)
Pages (from-to)47-63
Number of pages17
JournalJournal of Computational and Applied Mathematics
Volume133
Issue number1-2
DOIs
StatePublished - Aug 1 2001

Keywords

  • Asymptotic analysis
  • Freud weight
  • Orthogonal polynomials
  • Riemann-hilbert problem

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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