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Zbl 1026.42024
Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X.
Strong asymptotics of orthogonal polynomials with respect to exponential weights. (English)
[J] Commun. Pure Appl. Math. 52, No.12, 1491-1552 (1999). ISSN 0010-3640

In this excellent paper the authors study asymptotics for orthogonal polynomials with respect to the weights $$w(x)dx=e^{-Q(x)}dx$$ on the real line, with $Q(x)$ an even polynomial of degree $2m$ with positive leading coefficients \par The main results cover \par a. asymptotics for the leading and recurrence coefficients (Theorem 2.1), \par b. Plancherel-Rotach asymptotics on the whole complex plane (Theorem 2.2), \par c. asymptotic location of the zeros (Theorem 2.3). \par The deep results are derived through recently developed methods and a reformulation as a Riemann-Hilbert problem due to {\it A. S. Fokas, A. R. Its} and {\it A. V. Kitaev} [Commun. Math. Phys. 142, 313-344 (1991; Zbl 0742.35047); ibid. 147, 395-430 (1992; Zbl 0760.35051)]. \par The solution of this Riemann-Hilbert problem is then subjected to a series of transformations, leading to deep asymptotic results. \par The technical and delicate operations are described in detail and give the reader a good insight in the different techniques needed. In view of the intricacies of the methods, the length of the paper is just about right.
[Marcel G.de Bruin (Delft)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
30E25 Boundary value problems, complex analysis
35Q15 Riemann-Hilbert problems

Keywords: orthogonal polynomials; Plancherel-Rotach asymptotics; Riemann-Hilbert problem; zeros; leading coefficients; recurrence coefficients

Citations: Zbl 0760.35051; Zbl 0742.35047

Cited in: Zbl 1213.33013 Zbl 1158.15021 Zbl 1157.42007 Zbl 1136.82021 Zbl 1129.42010 Zbl 1061.30035 Zbl 0997.47033

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