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Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I

  • K. T.-R. McLaughlin
  • A. H. Vartanian
  • X. Zhou
Article

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.

Keywords

Asymptotics Equilibrium measures Orthogonal rational functions Riemann–Hilbert problems Variational problems 

Mathematics Subject Classifications (2000)

Primary 42C05; Secondary 30E20 30E25 30C15 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • K. T.-R. McLaughlin
    • 1
  • A. H. Vartanian
    • 2
  • X. Zhou
    • 3
  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA
  2. 2.Department of MathematicsCollege of CharlestonCharlestonUSA
  3. 3.Department of MathematicsDuke UniversityDurhamUSA

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