Weak limits of zeros of orthogonal polynomials

Abstract

Letμ be a positive unit Borel measure with infinite support on the interval [−1, 1]. LetP _n(x, μ) denote the monic orthogonal polynomial of degreen associated withμ, and letv _n(μ) denote the unit measure with mass 1/n at each zero ofP _n(x, μ). A carrier is a Borel subset of the support ofμ having unitμ-measure, and a measurev is carrier related toμ when it has the same carriers asμ. We demonstrate that for each carrierB of positive capacity there is a measurev, which is carrier related toμ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v _n(v)} ^=1/∞ _n .