Title: On the spacing of zeros of paraorthogonal polynomials for singular measures

Abstract:

Given a measure supported on the unit circle and a sequence of numbers on the unit circle, one may derive their corresponding sequence of paraorthogonal polynomials. The zeros of paraorthogonal polynomials naturally appear in various areas of mathematics, such as the spectral analysis of unitary operators, random matrix theory, and numerical analysis. Of the many interesting properties the zeros possess, we are especially interested in the asymptotics of the gaps between consecutive zeros as they relate to the continuity of the underlying measure. In this talk we discuss several results pertaining absolutely continuous and singular continuous measures, as well as further refining the approach to these questions by characterizing continuity based on Hausdorff dimensions. Our results are connected to a larger conjecture, and they suggest its fine details are not as simple as one might expect. This is a joint work with J. Breuer.

Abstract:

Given a measure supported on the unit circle and a sequence of numbers on the unit circle, one may derive their corresponding sequence of paraorthogonal polynomials. The zeros of paraorthogonal polynomials naturally appear in various areas of mathematics, such as the spectral analysis of unitary operators, random matrix theory, and numerical analysis. Of the many interesting properties the zeros possess, we are especially interested in the asymptotics of the gaps between consecutive zeros as they relate to the continuity of the underlying measure. In this talk we discuss several results pertaining absolutely continuous and singular continuous measures, as well as further refining the approach to these questions by characterizing continuity based on Hausdorff dimensions. Our results are connected to a larger conjecture, and they suggest its fine details are not as simple as one might expect. This is a joint work with J. Breuer.

## Date:

Wed, 06/11/2019 - 12:00 to 13:00

## Location:

Ross 70