Applications of a New Formula for OPUC with Periodic Verblunsky Coefficients

TL;DR

This paper introduces a new formula for orthonormal polynomials with periodic Verblunsky coefficients, utilizing Chebyshev polynomials and discriminants, leading to applications in spectral analysis and universality results.

Contribution

It provides a novel explicit formula for OPUC with periodic coefficients, addressing a problem on singular points and universality in spectral measures.

Findings

Resolved a problem on the existence of singular points in measure support.

Established universality results at all points of the essential support.

Connected Chebyshev polynomials with spectral properties of periodic measures.

Abstract

We find a new formula for the orthonormal polynomials corresponding to a measure mu on the unit circle whose Verblunsky coefficients are periodic. The formula is presented using the Chebyshev polynomials of the second kind and the discriminant of the periodic sequence. We present several applications including a resolution of a problem suggested by Simon in 2006 regarding the existence of singular points in the bands of the support of the measure and a universality result at all points of the essential support of mu.