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Published May 2010 | Submitted
Journal Article Open

Perturbations of orthogonal polynomials with periodic recursion coefficients

Abstract

The results of Denisov-Rakhmanov, Szegő-Shohat-Nevai, and Killip-Simon are extended from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.

Additional Information

© 2010 Annals of Mathematics. Received February 2, 2007. Revised September 3, 2008. Published 25 April 2010. It is a pleasure to thank Leonid Golinskii, Irina Nenciu, Leonid Pastur, and Peter Yuditskii for useful discussions. Note Added August, 2008. During the refereeing of this paper, Remling (in [94]), motivated in part by this paper, found a positive resolution of the conjecture that, in the language of our Theorem 9.5, every set in G is a Denisov-Rakhmanov set. His analysis depends on a very interesting theorem on right limits of Jacobi matrices with absolutely continuous spectrum—it provides a new approach to Denisov-Rakhmanov theorems. D.D. was supported in part by NSF grants DMS-0500910 and DMS-0653720. R.K. was supported in part by NSF grant DMS-0401277 and a Sloan Foundation Fellowship. B.S. was supported in part by NSF grant DMS-0140592 and U.S.-Israel Binational Science Foundation (BSF) Grant No. 2002068.

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August 19, 2023
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