Weak limits of the measures of maximal entropy for Orthogonal polynomials

TL;DR

This paper investigates the behavior of measures of maximal entropy associated with orthonormal polynomials, showing their weak* limits are contained within the polynomial hull of the support and converge to equilibrium measures under certain conditions.

Contribution

It establishes the containment of weak* limit measures within the polynomial hull and proves convergence to equilibrium measures for root regular measures.

Findings

Weak* limits of maximal entropy measures are contained in the polynomial hull of the support.

For root regular measures, these measures converge to the equilibrium measure.

The support of limit measures is related to the polynomial convex hull of the original measure's support.

Abstract

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of maximal entropy $\omega_n$ for $P_n$ is contained in the polynomial-convex hull of $S(\mu)$. And for $n$-th root regular measures the $\omega_n$ converge weak* to the equilibrium measure on $S(\mu)$.