Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach

We investigate the asymptotic behavior of polynomials orthogonal over a symmetric arc of the unit circle with respect to a generalized Jacobi-type weight. Full asymptotic expansions for the orthogonal polynomials are obtained at every point of the complex plane. Our method of proof is based on a characterization of the orthogonal polynomials as solutions of a 2X2 matrix Riemann-Hilbert problem, which extends to the unit circle the original Riemann-Hilbert characterization for orthogonal polynomials on the real line, first discovered by Fokas, Its, and Kitaev. In order to extricate the behavior of the polynomials from its Riemann-Hilbert matrix representation, we follow the steepest descent method of matrix transformations developed by Deift and Zhou.

Name thesis:degree_name

Ph.D. in Mathematics

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2017

Author dc:creator

• Naugle, Lynsey Cargile

Contributors dc:contributor

• Erwin Mina Diaz
• Cecille Labuda
• Qingying Bu

• Mathematics

Repository record dc:identifier

https://egrove.olemiss.edu/etd/679

OAI identifier oai:identifier

oai:egrove.olemiss.edu:etd-1678