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Orthogonal polynomials on the unit circle: New results

Simon, Barry (2004) Orthogonal polynomials on the unit circle: New results. International Mathematics Research Notices, 2004 (53). pp. 2837-2880. ISSN 1073-7928. http://resolver.caltech.edu/CaltechAUTHORS:20110818-105014767

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Abstract

We announce numerous new results in the theory of orthogonal polynomials on the unit circle, most of which involve the connection between a measure on the unit circle in the complex plane and the coefficients in the recursion relations for the polynomials known as Verblunsky coefficients. Included are several applications of the recently discovered matrix realization of Cantero, Moral, and Velázquez. In analogy with the spectral theory of Jacobi matrices, several classes of exotic Verblunsky coefficients are studied. A version of Rahkmanov's theorem is proven with a single gap with eigenvalues allowed in the gap. Analogs of Borg's theorem and the Birman-Schwinger principle are found.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1155/S1073792804141664DOIUNSPECIFIED
http://imrn.oxfordjournals.org/content/2004/53/2837PublisherUNSPECIFIED
Additional Information:© 2004 Hindawi Publishing Corporation. Received 5 May 2004. Accepted July 26, 2004. Communicated by Percy Deift. I would like to thank P. Deift, S. Denisov, L. Golinskii, S. Khruschchev, R. Killip, I. Nenciu, P. Nevai, F. Peherstorfer, V. Totik, and A. Zlatoš for useful discussions. This work was supported in part by National Scientific Foundation (NSF) Grant DMS-0140592.
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NSFDMS-0140592
Record Number:CaltechAUTHORS:20110818-105014767
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20110818-105014767
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24932
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:14 Oct 2011 18:22
Last Modified:14 Oct 2011 18:22

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