Weak limits for weighted means of orthogonal polynomials

TL;DR

This paper develops weak limit formulas for weighted means of orthogonal polynomials, introducing a new mean Nevai class that ensures convergence to an equilibrium measure, with applications to ultraspherical polynomials.

Contribution

It introduces a new mean Nevai class that guarantees the existence of an equilibrium measure for weighted means of orthogonal polynomials.

Findings

Means of Christoffel-Darboux kernels converge weakly to the equilibrium measure.

The equilibrium measure for ultraspherical polynomials is characterized.

Weak limit formulas serve as asymptotic weak addition formulas.

Abstract

This article is a first attempt to obtain weak limit formulas for weighted means of orthogonal polynomials. For this, we introduce a new mean Nevai class that guarantees the existence of an equilibrium measure for the limit of the means. We show that for a family of measures in this mean Nevai class also the means of the Christoffel-Darboux kernels and the asymptotic distribution of the roots converge weakly to the same equilibrium measure. As a main example, we study the mean Nevai classes in which the equilibrium measure is the orthogonality measure of the ultraspherical polynomials. The respective weak limit formula can be regarded as an asymptotic weak addition formula for the corresponding class of measures.