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

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

doi:10.1007/s00365-005-0617-6

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

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

Mathematics

Research output: Contribution to journal › Article › peer-review

32 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 language	English (US)
Pages (from-to)	319-363
Number of pages	45
Journal	Constructive Approximation
Volume	24
Issue number	3
DOIs	https://doi.org/10.1007/s00365-005-0617-6
State	Published - Nov 2006

Keywords

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

ASJC Scopus subject areas

Analysis
General Mathematics
Computational Mathematics

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10.1007/s00365-005-0617-6

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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.",

keywords = "Cauchy transform, Orthogonal polynomials, Scattering function, Uniform asymptotics, Unit circle, Verblunsky coefficients",

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T1 - Szego orthogonal polynomials with respect to an analytic weight

T2 - Canonical representation and strong asymptotics

AU - Martínez-Finkelshtein, A.

AU - McLaughlin, K. T.R.

AU - Saff, E. B.

PY - 2006/11

Y1 - 2006/11

N2 - 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.

AB - 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.

KW - Cauchy transform

KW - Orthogonal polynomials

KW - Scattering function

KW - Uniform asymptotics

KW - Unit circle

KW - Verblunsky coefficients

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SN - 0176-4276

VL - 24

SP - 319

EP - 363

JO - Constructive Approximation

JF - Constructive Approximation

IS - 3

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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