Szego orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics

A. Martínez-Finkelshtein, K. T.R. McLaughlin, E. B. Saff

Research output: Contribution to journalArticle

29 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)319-363
Number of pages45
JournalConstructive Approximation
Volume24
Issue number3
DOIs
StatePublished - Nov 1 2006

Keywords

  • Cauchy transform
  • Orthogonal polynomials
  • Scattering function
  • Uniform asymptotics
  • Unit circle
  • Verblunsky coefficients

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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