Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

TL;DR

This paper studies orthogonal rational functions on the unit circle with prescribed poles outside or inside the circle, exploring their properties, matrix relations, and connections to eigenvalue algorithms.

Contribution

It characterizes properties of ORF with various pole placements and links them to matrix transformations and eigenvalue problem algorithms.

Findings

Properties of matrices from elementary unitary transformations

Relations between ORF and eigenvalue algorithms

Characterization of ORF with poles outside or inside the unit circle

Abstract

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.