Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach
Date of Award
Ph.D. in Mathematics
Erwin Mina Diaz
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.
Naugle, Lynsey Cargile, "Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach" (2017). Electronic Theses and Dissertations. 679.