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Strong asymptotics of orthogonal polynomials with respect to exponential weights

✍ Scribed by P. Deift; T. Kriecherbauer; K. T-R McLaughlin; S. Venakides; X. Zhou

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 344 KB
Volume: 52
Category: Article
ISSN: 0010-3640
DOI: 10.1002/(sici)1097-0312(199912)52:12<1491::aid-cpa2>3.0.co;2-#

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✦ Synopsis

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e -Q(x) dx on the real line, where Q(x) = ∑ 2m k=0 q k x k , q 2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22,23].

We employ the steepest-descent-type method introduced in [18] and further developed in [17,19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials.

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