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Published February 1, 2009 | Published
Journal Article Open

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.

Additional Information

© 2009 Duke University Press. Received 19 December 2007. Revision received 1 May 2008; publication date 1 February 2009. It is a pleasure to thank Jonathan Breuer, Yoram Last, and especially Vilmos Totik for useful conversations. I also thank Ehud de Shalit and Yoram Last for the hospitality of the Einstein Institute of Mathematics of the Hebrew University during part of the preparation of this article.

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Created:
August 21, 2023
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October 17, 2023