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Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim (2012) Finite Gap Jacobi Matrices, III. Beyond the Szegő Class. Constructive Approximation, 35 (2). pp. 259-272. ISSN 0176-4276. doi:10.1007/s00365-012-9152-4.

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Let e ⊂ R be a finite union of ℓ+1 disjoint closed intervals, and denote by ω_j the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω_1,…,ω_ℓ. Let {a_nb_n}^∞_(n=−∞) be a point in the isospectral torus for e and p_n its orthogonal polynomials. Let {a_nb_n}^∞_(n=1) be a half-line Jacobi matrix with a_n=a_n+δa_n, b_n=b_n+δb_n. Suppose ∑^∞_(n=1)│δan│^2 + │δb_n│^2 < ∞ and ∑^N_n=1^e^(2πiωn), δa_n ∑^N_n=1^e^(2πiωn) δb_n have finite limits as N → ∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, p_n(z)p_n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds.

Item Type:Article
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Simon, Barry0000-0003-2561-8539
Additional Information:© 2012 Springer Science. Received: 1 July 2011. Accepted: 18 October 2011. Published online: 25 January 2012. Communicated by Vilmos Totik. J.S.C. and M.Z. gratefully acknowledge the kind invitation and hospitality of the Mathematics Department of Caltech where this work was completed. J.S.C. was supported in part by a Steno Research Grant (09-064947) from the Danish Research Council for Nature and Universe. B.S. was supported in part by NSF grant DMS-0968856. M.Z. was supported in part by NSF grant DMS-0965411.
Funding AgencyGrant Number
Danish Natural Science Research Council09-064947
Subject Keywords:Szegő asymptotics; Orthogonal polynomials; Almost periodic sequences; Slowly decaying perturbations
Issue or Number:2
Classification Code:MSC: 42C05, 39A11, 34L15
Record Number:CaltechAUTHORS:20120319-093342145
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Official Citation:Christiansen, J.S., Simon, B. & Zinchenko, M. Constr Approx (2012) 35: 259. doi:10.1007/s00365-012-9152-4
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29765
Deposited By: Ruth Sustaita
Deposited On:19 Mar 2012 17:48
Last Modified:09 Nov 2021 19:29

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