Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics |
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| Authors: | A Martinez-Finkelshtein KT-R McLaughlin EB Saff |
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| Institution: | (1) Departamento de Estadistica y Matematica Aplicada, Universidad de Almeria, La Canada, 04129 Almeria, Spain;(2) Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721, USA;(3) Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA |
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| Abstract: | We provide a representation in terms of certain canonical functions
for a sequence of polynomials orthogonal with respect to a weight
that is strictly positive and analytic on the unit circle. These
formulas yield a complete asymptotic expansion for these
polynomials, valid uniformly in the whole complex plane. As a
consequence, we obtain some results about the distribution of zeros
of these polynomials. The main technique is the steepest descent
analysis of Deift and Zhou, based on the matrix Riemann-Hilbert
characterization proposed by Fokas, Its, and Kitaev. |
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| Keywords: | Orthogonal polynomials Unit circle Uniform asymptotics Verblunsky coefficients Scattering function Cauchy transform |
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