Math @ Duke

Publications [#244298] of Xin Zhou
Papers Published
 with Deift, P; Kriecherbauer, T; McLaughlin, KR; Venakides, S; Zhou, X, A riemannHilbert approach to asymptotic questions for orthogonal polynomials,
Journal of Computational and Applied Mathematics, vol. 133 no. 12
(2001),
pp. 4763, ISSN 03770427 [doi]
(last updated on 2018/07/16)
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 RiemannHilbert 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) Freudtype weights dα(x) = eQ(x) dx (Q polynomial or Q(x) = xβ, β>0), or (2) varying weights dαn(x) = enV(x) dx (V analytic, limx→∞V(x)/logx = ∞). We obtain PlancherelRotachtype 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 RiemannHilbert problems. We analyze the RiemannHilbert 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 wellknown equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials. © 2001 Elsevier Science B.V. All rights reserved.


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