### 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 language | English (US) |
---|---|

Pages (from-to) | 187-364 |

Number of pages | 178 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 11 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 2008 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematical Physics
- Geometry and Topology

### Cite this

*Mathematical Physics Analysis and Geometry*,

*11*(3-4), 187-364. https://doi.org/10.1007/s11040-008-9042-y

**Rational functions with a general distribution of poles on the real line orthogonal with respect to varying exponential weights : I.** / Mclaughlin, Kenneth D T; Vartanian, A. H.; Zhou, X.

Research output: Contribution to journal › Article

*Mathematical Physics Analysis and Geometry*, vol. 11, no. 3-4, pp. 187-364. https://doi.org/10.1007/s11040-008-9042-y

}

TY - JOUR

T1 - Rational functions with a general distribution of poles on the real line orthogonal with respect to varying exponential weights

T2 - I

AU - Mclaughlin, Kenneth D T

AU - Vartanian, A. H.

AU - Zhou, X.

PY - 2008/11

Y1 - 2008/11

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

KW - Equilibrium measures

KW - Orthogonal rational functions

KW - Riemann-Hilbert problems

KW - Variational problems

UR - http://www.scopus.com/inward/record.url?scp=56049090504&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56049090504&partnerID=8YFLogxK

U2 - 10.1007/s11040-008-9042-y

DO - 10.1007/s11040-008-9042-y

M3 - Article

AN - SCOPUS:56049090504

VL - 11

SP - 187

EP - 364

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

SN - 1385-0172

IS - 3-4

ER -