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A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

Publication ,  Journal Article
Deift, P; Kriecherbauer, T; McLaughlin, KR; Venakides, S; Zhou, X
Published in: Journal of Computational and Applied Mathematics
August 1, 2001

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. © 2001 Elsevier Science B.V. All rights reserved.

Duke Scholars

Published In

Journal of Computational and Applied Mathematics

DOI

ISSN

0377-0427

Publication Date

August 1, 2001

Volume

133

Issue

1-2

Start / End Page

47 / 63

Related Subject Headings

  • Numerical & Computational Mathematics
  • 0906 Electrical and Electronic Engineering
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
 

Citation

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Deift, P., Kriecherbauer, T., McLaughlin, K. R., Venakides, S., & Zhou, X. (2001). A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. Journal of Computational and Applied Mathematics, 133(1–2), 47–63. https://doi.org/10.1016/S0377-0427(00)00634-8
Deift, P., T. Kriecherbauer, K. R. McLaughlin, S. Venakides, and X. Zhou. “A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials.” Journal of Computational and Applied Mathematics 133, no. 1–2 (August 1, 2001): 47–63. https://doi.org/10.1016/S0377-0427(00)00634-8.
Deift P, Kriecherbauer T, McLaughlin KR, Venakides S, Zhou X. A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. Journal of Computational and Applied Mathematics. 2001 Aug 1;133(1–2):47–63.
Deift, P., et al. “A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials.” Journal of Computational and Applied Mathematics, vol. 133, no. 1–2, Aug. 2001, pp. 47–63. Scopus, doi:10.1016/S0377-0427(00)00634-8.
Deift P, Kriecherbauer T, McLaughlin KR, Venakides S, Zhou X. A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. Journal of Computational and Applied Mathematics. 2001 Aug 1;133(1–2):47–63.
Journal cover image

Published In

Journal of Computational and Applied Mathematics

DOI

ISSN

0377-0427

Publication Date

August 1, 2001

Volume

133

Issue

1-2

Start / End Page

47 / 63

Related Subject Headings

  • Numerical & Computational Mathematics
  • 0906 Electrical and Electronic Engineering
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics