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. https://resolver.caltech.edu/CaltechAUTHORS:20110414-085411774
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Abstract
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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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. | |||||||||||||||
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Subject Keywords: | Isospectral torus; Szego asymptotics; Orthogonal polynomials | |||||||||||||||
Issue or Number: | 3 | |||||||||||||||
Classification Code: | MSC (2000): 42C05 · 58J53 · 14H30 | |||||||||||||||
Record Number: | CaltechAUTHORS:20110414-085411774 | |||||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20110414-085411774 | |||||||||||||||
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 | |||||||||||||||
Collection: | CaltechAUTHORS | |||||||||||||||
Deposited By: | Ruth Sustaita | |||||||||||||||
Deposited On: | 14 Apr 2011 18:34 | |||||||||||||||
Last Modified: | 03 Oct 2019 02:45 |
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