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 | |||||||||
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| 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. | |||||||||
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| 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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