Asymptotics of orthogonal polynomials and point perturbation on the unit circle

TL;DR

This paper investigates how adding a point to the spectrum affects orthogonal polynomials on the unit circle, providing asymptotic formulas and convergence results for various classes of Verblunsky coefficients.

Contribution

It extends existing results by deriving explicit asymptotics and convergence properties for perturbations of Verblunsky coefficients in both asymptotically constant and periodic cases.

Findings

Explicit asymptotic formulas for orthogonal polynomials in the gap

Perturbation convergence and bounded variation proved

Rate of convergence characterized for real Verblunsky coefficients

Abstract

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically periodic case and prove similar results. In the last two sections, we show that the bounded variation condition can be removed if a certain symmetry condition is satisfied. Finally, we consider the special case when the Verblunsky coefficients are real with the rate of convergence being c_n . We prove that the rate of convergence of the perturbation is in fact O(c_n). In particular, the special case c_n = 1/n will serve as a counterexample to the possibility that the convergence of the perturbed Verblunsky coefficients should be exponentially fast when a point is added to a gap.