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Verblunsky coefficients and Nehari sequences


Authors: Yukio Kasahara and Nicholas H. Bingham
Journal: Trans. Amer. Math. Soc. 366 (2014), 1363-1378
MSC (2010): Primary 42C05; Secondary 42A10, 42A70
DOI: https://doi.org/10.1090/S0002-9947-2013-05874-6
Published electronically: July 18, 2013
MathSciNet review: 3145734
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Abstract: We are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle. In general, the Szegö function is determined by its modulus, while the condition in question is that it is also determined by its argument, or in terms of the function theory, that the square of the Szegö function is rigid. In prediction theory, this is known as a spectral characterization of complete nondeterminacy for stationary processes, studied by Bloomfield, Jewel and Hayashi (1983) going back to a small but important result in the work of Levinson and McKean (1964). It is also related to the cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem, and there is a one-to-one correspondence between the Verblunsky coefficients and the negatively indexed Fourier coefficients of the phase factor of the Szegö function, which we call a Nehari sequence. We present some fundamental results on the correspondence, including extensions of the strong Szegö and Baxter's theorems.


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Additional Information

Yukio Kasahara
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email: y-kasa@math.sci.hokudai.ac.jp

Nicholas H. Bingham
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ United Kingdom
Email: n.bingham@ic.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2013-05874-6
Keywords: Orthogonal polynomials on the unit circle, Verblunky coefficients, Nehari problem, rigid functions
Received by editor(s): December 3, 2011
Published electronically: July 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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