Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients

TL;DR

This paper extends classical results on orthogonal polynomials to the case of asymptotically periodic coefficients, providing a new characterization of the isospectral torus to analyze perturbations.

Contribution

It introduces a novel approach to study perturbations of orthogonal polynomials with periodic recursion coefficients using a characterization of the isospectral torus.

Findings

Extended classical results to asymptotically periodic cases

Developed a new characterization of the isospectral torus

Provided insights into the spectral properties of perturbed polynomials

Abstract

We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.