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Mathematics > Classical Analysis and ODEs

arXiv:math/0701055 (math)
[Submitted on 2 Jan 2007 (v1), last revised 24 Oct 2011 (this version, v3)]

Title:A note on Poisson brackets for orthogonal polynomials on the unit circle

Authors:Irina Nenciu
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Abstract:The connection of orthogonal polynomials on the unit circle (OPUC) to the defocusing Ablowitz-Ladik integrable system involves the definition of a Poisson structure on the space of Verblunsky coefficients. In this paper, we compute the complete set of Poisson brackets for the monic orthogonal and the orthonormal polynomials on the unit circle, as well as for the second kind polynomials and the Wall polynomials. This answers a question posed by Cantero and Simon for the case of measures with finite support. We also show that the results hold for the case of measures with periodic Verblunsky coefficients.
Comments: 10 pages; the statements and proofs have been corrected and expanded, and some further comments have been added
Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Symplectic Geometry (math.SG); Exactly Solvable and Integrable Systems (nlin.SI)
MSC classes: 42C05, 53D17
Cite as: arXiv:math/0701055 [math.CA]
  (or arXiv:math/0701055v3 [math.CA] for this version)
  https://doi.org/10.48550/arXiv.math/0701055

Submission history

From: Irina Nenciu [view email]
[v1] Tue, 2 Jan 2007 11:29:07 UTC (9 KB)
[v2] Tue, 20 Feb 2007 21:46:04 UTC (9 KB)
[v3] Mon, 24 Oct 2011 01:17:01 UTC (11 KB)
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