Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Álvarez-Fernández, Carlos; Mañas, Manuel

Publication:
Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

dc.affiliation.dpto	UC3M. Departamento de Matemáticas	es
dc.affiliation.grupoinv	UC3M. Grupo de Investigación: Análisis Aplicado	es
dc.contributor.author	Álvarez-Fernández, Carlos
dc.contributor.author	Mañas, Manuel
dc.date.accessioned	2016-07-18T10:14:04Z
dc.date.available	2016-07-18T10:14:04Z
dc.date.issued	2013-06-20
dc.description.abstract	We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Toda-like integrable systems using the Gauss–Borel factorization of a Cantero–Moral–Velázquez moment matrix, that we construct in terms of a complex quasi-definite 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, 5-term recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas from this point of view in a completely algebraic way. We generalize the Cantero–Moral–Velázquez sequence of Laurent monomials, recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux 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 Toda-type integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and Zakharov–Shabat 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 Szegő 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 τ-functions and the subsequent bilinear equations.	en
dc.description.sponsorship	MM thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200 and from the Spanish Ministerio de Economía y Competitividad MTM2012-36732-C03-01.	en
dc.format.extent	62
dc.format.mimetype	application/pdf
dc.identifier.bibliographicCitation	Advances in Mathematics, 2013, 240, pp. 132-193	en
dc.identifier.doi	10.1016/j.aim.2013.02.020
dc.identifier.issn	0001-8708
dc.identifier.publicationfirstpage	132
dc.identifier.publicationlastpage	193
dc.identifier.publicationtitle	Advances in Mathematics	en
dc.identifier.publicationvolume	240
dc.identifier.uri	https://hdl.handle.net/10016/23359
dc.language.iso	eng
dc.publisher	Elsevier
dc.relation.projectID	Gobierno de España. MTM2012-36732-C03-01	es
dc.relation.publisherversion	http://dx.doi.org/10.1016/j.aim.2013.02.020
dc.rights	© Elsevier 2013
dc.rights	Atribución-NoComercial-SinDerivadas 3.0 España	*
dc.rights.accessRights	open access	en
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/es/	*
dc.subject.eciencia	Matemáticas	es
dc.subject.other	Orthogonal Laurent polynomials in the unit circle	en
dc.subject.other	CMV ordering	en
dc.subject.other	Gauss–Borel factorization	en
dc.subject.other	Christoffel– Darboux	en
dc.subject.other	Integrable systems	en
dc.title	Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies	en
dc.type	research article	*
dc.type.hasVersion	AM	*
dspace.entity.type	Publication