Weak convergence of CD kernels and applications

Abstract

We prove a general result on equality of the weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dμ) and (1/n)Kn(x, x)dμ(x). By combining this with the asymptotic upper bounds of Máté and Nevai [16] and Totik [33] on nλn(x), we prove some general results on ∫ Ι(1/n)Kn(x, x)dμs → 0 for the singular part of dμ and ∫ Ι |ρE(x) − (w(x)/n)Kn(x, x)| dx → 0, where ρE is the density of the equilibrium measure and w(x) the density of dμ.

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