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Weak convergence of CD kernels and applications

Simon, Barry (2009) Weak convergence of CD kernels and applications. Duke Mathematical Journal, 146 (2). pp. 305-330. ISSN 0012-7094.

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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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Simon, Barry0000-0003-2561-8539
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.
Funding AgencyGrant Number
Binational Science Foundation (USA-Israel)2002068
Subject Keywords:Orthogonal polynomials; unit-circle; paraorthogonal polynomials; zeros; asymptotics; quadrature; intervals; operators; matrices
Issue or Number:2
Record Number:CaltechAUTHORS:SIMdmj09
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13478
Deposited By: Jason Perez
Deposited On:23 Apr 2009 18:59
Last Modified:01 May 2017 22:50

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