Orthogonal Polynomials, Paraorthogonal Polynomials, and Point Perturbation

Citation

Wong, Manwah Lilian (2009) Orthogonal Polynomials, Paraorthogonal Polynomials, and Point Perturbation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/E55S-4A69. https://resolver.caltech.edu/CaltechETD:etd-06092009-004344

Abstract

This thesis consists of three parts.

Part 1 starts with an introduction to orthogonal polynomials, to be followed by some well-known theorems pertinent to the results we shall discuss. It also states the new results that are going to be proven in Parts 2 and 3.

In Part 2, we consider a sequence of paraorthogonal polynomials and investigate their zeros. Then we introduce paraorthogonal polynomials of the second kind and prove that zeros of first and second kind paraorthogonal polynomials interlace.

In Part 3, we consider the point mass problem. First, we give the point mass formula for the perturbed Verblunsky coefficients. Then we investigate the asymptotics of orthogonal polynomials on the unit circle and apply the results to the point mass formula to compute the perturbed Verblunsky coefficients. Finally, we present two examples, one on the unit circle and one on the real line, such that adding a point mass will generate non-exponential perturbations of the recursion coefficients.

Thesis Files