Open AccessSzego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics
Author(s) -
Andrei Martínez–Finkelshtein,
K. T-R McLaughlin,
Edward B. Saff
Publication year - 2006
Publication title -
constructive approximation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.921
H-Index - 51
eISSN - 1432-0940
pISSN - 0176-4276
DOI - 10.1007/s00365-005-0617-6
Subject(s) - mathematics , orthogonal polynomials , mehler–heine formula , complex plane , method of steepest descent , discrete orthogonal polynomials , representation (politics) , classical orthogonal polynomials , sequence (biology) , pure mathematics , wilson polynomials , difference polynomials , hahn polynomials , mathematical analysis , gegenbauer polynomials , politics , political science , law , biology , genetics
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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