Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics

By Andrei Martínez-Finkelshtein, K. Mclaughlin, Edward B. Saff. 2005

57 citations recorded by the source corpus.

AbstractWe 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f208cd25d0346b159d6a399a5b47cce26c58e295

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In Algebra and Number Theory

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices (2005, methodology): “[ 22 ], the authors found a series expansion for π(z).”
Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities (2009, background): “carried out in [34], and for the case of a weight with αj-singularities but without jumps, in [35].”
CMV matrices: Five years after (2006, methodology): “We note that this method of computing zeros was used in the recent paper of Mart´onez-Finkelshtein, McLaughlin, and Saff [ 57 ].”