Weak limits for weighted means of orthogonal polynomials

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 a limiting measure for the weighted 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 limiting measure. As a main example, we study the mean Nevai classes in which the limiting 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.