Regularity and the Cesaro-Nevai class

TL;DR

This paper explores the implications of measure regularity in orthogonal polynomials on the behavior of their recurrence coefficients, establishing new asymptotic results for Jacobi and Verblunsky parameters.

Contribution

It demonstrates how measure regularity in the sense of Ullman-Stahl-Totik influences the asymptotic behavior of recurrence coefficients in orthogonal polynomials.

Findings

Regularity on [-2,2] implies the average squared deviation of recurrence coefficients tends to zero.

Established asymptotic relations between measure regularity and Jacobi/Verblunsky parameters.

Provided new insights into the Cesaro-Nevai class for orthogonal polynomials.

Abstract

We consider OPRL and OPUC with measures regular in the sense of Ullman-Stahl-Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on $[-2,2]$ implies $\lim_{N\to\infty} N^{-1} [\sum_{n=1}^N (a_n-1)^2 + b_n^2] =0$.