Rational functions with a general distribution of poles on the real line orthogonal with respect to varying exponential weights: I

Kenneth D T Mclaughlin, A. H. Vartanian, X. Zhou

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of 'model' matrix Riemann-Hilbert problems which are amenable to asymptotic analysis via the Deift-Zhou non-linear steepest-descent method.

Original languageEnglish (US)
Pages (from-to)187-364
Number of pages178
JournalMathematical Physics Analysis and Geometry
Volume11
Issue number3-4
DOIs
StatePublished - Nov 2008

Fingerprint

Exponential Weights
Real Line
Rational function
Pole
Riemann-Hilbert Problem
Energy Minimization
Minimization Problem
Orthogonal Rational Functions
Equilibrium Measure
Steepest Descent Method
Regularity Properties
Matrix Models
Asymptotic Analysis
Existence and Uniqueness
Family

Keywords

  • Asymptotics
  • Equilibrium measures
  • Orthogonal rational functions
  • Riemann-Hilbert problems
  • Variational problems

ASJC Scopus subject areas

  • Mathematical Physics
  • Geometry and Topology

Cite this

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abstract = "Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of 'model' matrix Riemann-Hilbert problems which are amenable to asymptotic analysis via the Deift-Zhou non-linear steepest-descent method.",
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AU - Vartanian, A. H.

AU - Zhou, X.

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N2 - Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of 'model' matrix Riemann-Hilbert problems which are amenable to asymptotic analysis via the Deift-Zhou non-linear steepest-descent method.

AB - Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of 'model' matrix Riemann-Hilbert problems which are amenable to asymptotic analysis via the Deift-Zhou non-linear steepest-descent method.

KW - Asymptotics

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KW - Riemann-Hilbert problems

KW - Variational problems

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