A PhD thesis @ NALAG

TW 2005_02

Patrick Van gucht
Orthogonal rational functions: identification, realization and computation

Advisor(s): Adhemar Bultheel

Abstract

In this thesis we explore the natural generalization of orthogonal polynomials (OPs) to the orthogonal rational functions (ORFs) with prescribed poles.

In the first place, we consider some theoretical generalizations. We describe the recurrence relations for these (strictly) proper ORFs and deduce a moment-based algorithm to compute the reflection coefficients that occur in the recurrence relations. Asymptotic behaviour of the ORFs on the support of the measure is described and therefrom we deduce a Bernstein equiconvergence theorem that states that the ordinary Fourier series and the general ORF-Fourier series converge uniformly to each other. A rational generalization of Fejér's Theorem is a nice corollary. As a result of a relation between the ORFs on the unit circle and the interval, we only need to explore the circular case.

Next we deduce a linear algebraic approach to ORFs in the form of an inverse eigenvalue problem, which involves semiseparable and orthogonal matrices. By imposing some natural similarity in interpolation points and the poles of the ORFs all computations can be done using real numbers.

The numerical benefits of the orthogonality of the rational functions is pointed out and a state space description of the general ORFs is given.