Asymptotics of polynomials orthogonal with respect to a logarithmic weight

Thomas Oliver Conway, Percy Deift

Research output: Contribution to journalArticle

Abstract

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight (Formula Presented) on (−1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at x = 1.

Original languageEnglish (US)
Article number056
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume14
DOIs
StatePublished - Jun 12 2018

Fingerprint

Orthogonal Polynomials
Logarithmic
Steepest Descent Method
Riemann-Hilbert Problem
Coefficient
Recurrence
Hilbert
Asymptotic Behavior
Singularity
Verify
Standards

Keywords

  • Orthogonal polynomials
  • Recurrence coefficients
  • Riemann-Hilbert problems
  • Steepest descent method

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

Cite this

Asymptotics of polynomials orthogonal with respect to a logarithmic weight. / Conway, Thomas Oliver; Deift, Percy.

In: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), Vol. 14, 056, 12.06.2018.

Research output: Contribution to journalArticle

@article{2f4c84b1320640b4b8f033b002bc9cec,
title = "Asymptotics of polynomials orthogonal with respect to a logarithmic weight",
abstract = "In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight (Formula Presented) on (−1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at x = 1.",
keywords = "Orthogonal polynomials, Recurrence coefficients, Riemann-Hilbert problems, Steepest descent method",
author = "Conway, {Thomas Oliver} and Percy Deift",
year = "2018",
month = "6",
day = "12",
doi = "10.3842/SIGMA.2018.056",
language = "English (US)",
volume = "14",
journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",
issn = "1815-0659",
publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",

}

TY - JOUR

T1 - Asymptotics of polynomials orthogonal with respect to a logarithmic weight

AU - Conway, Thomas Oliver

AU - Deift, Percy

PY - 2018/6/12

Y1 - 2018/6/12

N2 - In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight (Formula Presented) on (−1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at x = 1.

AB - In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight (Formula Presented) on (−1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at x = 1.

KW - Orthogonal polynomials

KW - Recurrence coefficients

KW - Riemann-Hilbert problems

KW - Steepest descent method

UR - http://www.scopus.com/inward/record.url?scp=85050344342&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050344342&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2018.056

DO - 10.3842/SIGMA.2018.056

M3 - Article

AN - SCOPUS:85050344342

VL - 14

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 056

ER -