Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics

Details Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics

2005-02-15

arxiv.org/abs/math/0502300v1

Mathematics

Classical Analysis and ODEs

49 pages, 19 figures

Scientific paper

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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