Szego orthogonal polynomials with respect to an analytic weight

Canonical representation and strong asymptotics

A. Martínez-Finkelshtein, Kenneth D T Mclaughlin, E. B. Saff

Research output: Contribution to journalArticle

28 Citations (Scopus)

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 2006

Fingerprint

Szegö Polynomials
Canonical Representation
Distribution of Zeros
Zeros of Polynomials
Steepest Descent
Strictly positive
Unit circle
Orthogonal Polynomials
Argand diagram
Hilbert
Asymptotic Expansion
Polynomials
Valid
Polynomial

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis

Cite this

Szego orthogonal polynomials with respect to an analytic weight : Canonical representation and strong asymptotics. / Martínez-Finkelshtein, A.; Mclaughlin, Kenneth D T; Saff, E. B.

In: Constructive Approximation, Vol. 24, No. 3, 11.2006, p. 319-363.

Research output: Contribution to journalArticle

@article{54acfe7d92f74eea92c517a36943782b,
title = "Szego orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics",
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",
author = "A. Mart{\'i}nez-Finkelshtein and Mclaughlin, {Kenneth D T} and Saff, {E. B.}",
year = "2006",
month = "11",
doi = "10.1007/s00365-005-0617-6",
language = "English (US)",
volume = "24",
pages = "319--363",
journal = "Constructive Approximation",
issn = "0176-4276",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - Szego orthogonal polynomials with respect to an analytic weight

T2 - Canonical representation and strong asymptotics

AU - Martínez-Finkelshtein, A.

AU - Mclaughlin, Kenneth D T

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

UR - http://www.scopus.com/inward/record.url?scp=33747637122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747637122&partnerID=8YFLogxK

U2 - 10.1007/s00365-005-0617-6

DO - 10.1007/s00365-005-0617-6

M3 - Article

VL - 24

SP - 319

EP - 363

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 3

ER -