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

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Constructive Approximation Aims and scope

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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Authors and Affiliations

  1. Departamento de Estadistica y Matematica Aplicada, Universidad de Almeria, La Canada, 04129 Almeria, Spain

    A. Martinez-Finkelshtein

  2. Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721, USA

    K.T.-R. McLaughlin

  3. Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA

    E.B. Saff

Authors
  1. A. Martinez-Finkelshtein
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  2. K.T.-R. McLaughlin
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  3. E.B. Saff
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Corresponding authors

Correspondence to A. Martinez-Finkelshtein, K.T.-R. McLaughlin or E.B. Saff.

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Martinez-Finkelshtein, A., McLaughlin, KR. & Saff, E. Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics. Constr Approx 24, 319–363 (2006). https://doi.org/10.1007/s00365-005-0617-6

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  • DOI: https://doi.org/10.1007/s00365-005-0617-6

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