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Finite Gap Jacobi Matrices, II. The Szegő Class

Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim (2011) Finite Gap Jacobi Matrices, II. The Szegő Class. Constructive Approximation, 33 (3). pp. 365-403. ISSN 0176-4276.

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Let e ⊂ R be a finite union of disjoint closed intervals. We study measures whose essential support is e and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to lim sup a_1...a_n/cap e_n >0 (this includes prior results of Widom and Peherstorfer-Yuditskii). Using Remling's extension of the Denisov-Rakhmanov theorem and an analysis of Jost functions, we provide a new proof of Szego asymptotics, including L^2 asymptotics on the spectrum. We make heavy use of the covering map formalism of Sodin-Yuditskii as presented in our first paper in this series.

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Simon, Barry0000-0003-2561-8539
Additional Information:© 2010 Springer Science+Business Media, LLC. Received: 31 May 2009. Revised: 9 October 2009. Accepted: 10 November 2009. Published online: 13 April 2010. Communicated by Vilmos Totik. J.S. Christiansen was supported in part by a Steno Research Grant from FNU, the Danish Research Council. B. Simon was supported in part by NSF grant DMS-0652919. Maxim Zinchenko was supported in part by NSF grant DMS-0965411. We would like to thank F. Peherstorfer and P. Yuditskii for the private communication [14]. J.S.C. would like to thank M. Flach and A. Lange for the hospitality of Caltech where this work was completed.
Funding AgencyGrant Number
Danish Natural Science Research CouncilUNSPECIFIED
Subject Keywords:Isospectral torus; Szego asymptotics; Orthogonal polynomials
Issue or Number:3
Classification Code:MSC (2000): 42C05 · 58J53 · 14H30
Record Number:CaltechAUTHORS:20110414-085411774
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Official Citation:Christiansen, J.S., Simon, B. & Zinchenko, M. Constr Approx (2011) 33: 365. doi:10.1007/s00365-010-9094-7
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23323
Deposited By: Ruth Sustaita
Deposited On:14 Apr 2011 18:34
Last Modified:03 Oct 2019 02:45

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