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Weak convergence of CD kernels: A new approach on the circle and real line

Simanek, Brian (2012) Weak convergence of CD kernels: A new approach on the circle and real line. Journal of Approximation Theory, 164 (1). pp. 204-209. ISSN 0021-9045.

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Given a probability measure μ supported on some compact set K ⊆ C and with orthonormal polynomials {pn(z)}_(n∈N), define the measures dμ_(n) = 1/(n+1) ∑^(n)_(j=0)|p_(j)(z)|^(2)dμ(z) and let ν_n be the normalized zero counting measure for the polynomial p_n. If μ is supported on a compact subset of the real line or on the unit circle, we provide a new proof of a 2009 theorem due to Simon that for any fixed k ∈ N the kth moment of ν_(n+1) and μ_n differ by at most O(n^(−1)) as n → ∞.

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Additional Information:© 2011 Elsevier Inc. Received 14 March 2011; received in revised form 16 September 2011; accepted 13 October 2011. Available online 20 October 2011. It is a pleasure to thank my advisor Barry Simon for many useful comments and suggestions. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-0703267.
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-0703267
Subject Keywords:Orthogonal polynomials; Reproducing Kernel; Balayage
Record Number:CaltechAUTHORS:20120124-111404598
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Official Citation:Brian Simanek, Weak convergence of CD kernels: A new approach on the circle and real line, Journal of Approximation Theory, Volume 164, Issue 1, January 2012, Pages 204-209, ISSN 0021-9045, 10.1016/j.jat.2011.10.001.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28940
Deposited By: Jason Perez
Deposited On:24 Jan 2012 22:00
Last Modified:23 Aug 2016 10:09

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