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Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

Simon, Barry (2004) Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line. Journal of Approximation Theory, 126 (2). pp. 198-217. ISSN 0021-9045. doi:10.1016/j.jat.2003.12.002. https://resolver.caltech.edu/CaltechAUTHORS:20170512-073744838

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Abstract

We study ratio asymptotics, that is, existence of the limit of Pn_(+1)(z)/P_n(z) (P_n= monic orthogonal polynomial) and the existence of weak limits of pn^2dμ(p_n=P_n/||P_n||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z_0 with Im(z_0)≠0 implies dμ is in a Nevai class (i.e., a_n→a and bn→b where a_n,b_n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove pn^2dμ has a weak limit if and only if limb_n, lima_(2n), and lima_(2n+1) all exist. In both cases, we write down the limits explicitly.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jat.2003.12.002DOIArticle
http://www.sciencedirect.com/science/article/pii/S0021904503002107PublisherArticle
ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
Additional Information:© 2003 Elsevier Inc. Received 10 July 2003, Accepted 1 December 2003, Available online 7 February 2004. Communicated by Leonid Golinskii Supported in part by NSF Grant DMS-0140592.
Funders:
Funding AgencyGrant Number
NSFDMS-0140592
Subject Keywords:Polynomials; Asymptotics; Jacobi matrices
Issue or Number:2
Classification Code:MSC: 42C05; 47B36; 81Q10
DOI:10.1016/j.jat.2003.12.002
Record Number:CaltechAUTHORS:20170512-073744838
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170512-073744838
Official Citation:Barry Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, Journal of Approximation Theory, Volume 126, Issue 2, 2004, Pages 198-217, ISSN 0021-9045, http://dx.doi.org/10.1016/j.jat.2003.12.002. (http://www.sciencedirect.com/science/article/pii/S0021904503002107)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77389
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:12 May 2017 23:52
Last Modified:15 Nov 2021 17:30

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