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Álvarez Fernández, Carlos and Mañas Baena, Manuel (2013) Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies. Advances in mathematics, 240 . pp. 132193. ISSN 00018708

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Official URL: http://dx.doi.org/10.1016/j.aim.2013.02.020
URL  URL Type 

http://www.sciencedirect.com  Publisher 
http://arxiv.org/abs/1202.2898  Organisation 
Abstract
We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Todalike integrable systems using the GaussBorel factorization of a CanteroMoralVelazquez moment matrix, that we construct in terms of a complex quasidefinite measure supported on the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials on the unit circle and the corresponding second kind functions. We obtain Jacobi operators, 5term recursion relations, ChristoffelDarboux kernels, and corresponding ChristoffelDarboux formulas from this point of view in a completely algebraic way. We generalize the Cantero MoralVelazquez sequence of Laurent monomials, recursion relations, ChristoffelDarboux kernels, and corresponding ChristoffelDarboux formulas in this extended context. We introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a Todatype integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and ZakharovShabat equations using the classical integrability theory tools. We explicitly derive the dynamical system associated with the coefficients of the orthogonal Laurent polynomials and we compare it with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szego polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, we obtain the representation of the orthogonal Laurent polynomials (and their second kind functions), using the formalism of Miwa shifts in terms of taufunctions and the subsequent bilinear equations.
Item Type:  Article 

Additional Information:  ©2013 Elsevier Inc. All rights reserved. 
Uncontrolled Keywords:  Differentialdifference equations; Kadomtsevpetviashvili equation; Discrete hphierarchy; Szego polynomials; Multicomponent kp; Moment problem; Schur flows; Matrices; Zeros; Asymptotics 
Subjects:  Sciences > Physics > PhysicsMathematical models Sciences > Physics > Mathematical physics 
ID Code:  31459 
Deposited On:  17 Jul 2015 08:28 
Last Modified:  10 Dec 2018 15:09 
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