A CMV connection between orthogonal polynomials on the unit circle and the real line

Cantero, M.J.; Moral, L.; Velázquez, L.; Marcellán Español, Francisco José

Publication:
A CMV connection between orthogonal polynomials on the unit circle and the real line

dc.affiliation.dpto	UC3M. Departamento de Matemáticas	es
dc.affiliation.grupoinv	UC3M. Grupo de Investigación: Análisis Aplicado	es
dc.contributor.author	Cantero, M.J.
dc.contributor.author	Moral, L.
dc.contributor.author	Velázquez, L.
dc.contributor.author	Marcellán Español, Francisco José
dc.contributor.funder	Ministerio de Economía y Competitividad (España)	es
dc.date.accessioned	2022-07-15T09:31:48Z
dc.date.available	2022-07-15T09:31:48Z
dc.date.issued	2021-03-31
dc.description.abstract	M. Derevyagin, L. Vinet and A. Zhedanov introduced in Derevyagin et al. (2012) a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real parameter λ. In Derevyagin et al. (2012) the authors prove that this map yields a natural link between the Jacobi polynomials on the unit circle and the little and big −1 Jacobi polynomials on the real line. They also provide explicit expressions for the measure and orthogonal polynomials associated with the Jacobi matrix in terms of those related to the CMV matrix, but only for the value λ = 1 which simplifies the connection –basic DVZ connection–. However, similar explicit expressions for an arbitrary value of λ –(general) DVZ connection– are missing in Derevyagin et al. (2012). This is the main problem overcome in this paper. This work introduces a new approach to the DVZ connection which formulates it as a two-dimensional eigenproblem by using known properties of CMV matrices. This allows us to go further than Derevyagin et al. (2012), providing explicit relations between the measures and orthogonal polynomials for the general DVZ connection. It turns out that this connection maps a measure on the unit circle into a rational perturbation of an even measure supported on two symmetric intervals of the real line, which reduce to a single interval for the basic DVZ connection, while the perturbation becomes a degree one polynomial. Some instances of the DVZ connection are shown to give new one-parameter families of orthogonal polynomials on the real line.	en
dc.description.sponsorship	The work of the first, third and fourth authors has been supported in part by the research project MTM2017-89941-P from Ministerio de Economía, Industria y Competitividad of Spain and the European Regional Development Fund (ERDF), by project UAL18-FQM-B025-A (UAL/CECEU/FEDER) and by projects E26 17R and E48 20R of Diputación General de Aragón (Spain) and the ERDF 2014–2020 “Construyendo Europa desde Aragón”. The work of the second author has been partially supported by the research project PGC2018–096504-B-C33 supported by Agencia Estatal de Investigación of Spain.	en
dc.format.extent	22
dc.identifier.bibliographicCitation	Cantero, M. J., Marcellán, F., Moral, L., & Velázquez, L. (2021). A CMV connection between orthogonal polynomials on the unit circle and the real line. In Journal of Approximation Theory (Vol. 266, p. 105579). Elsevier BV.	en
dc.identifier.doi	https://doi.org/10.1016/j.jat.2021.105579
dc.identifier.issn	0021-9045
dc.identifier.publicationfirstpage	1
dc.identifier.publicationlastpage	22
dc.identifier.publicationtitle	JOURNAL OF APPROXIMATION THEORY	en
dc.identifier.publicationvolume	266
dc.identifier.uri	https://hdl.handle.net/10016/35473
dc.identifier.uxxi	AR/0000028739
dc.language.iso	eng	en
dc.publisher	Elsevier BV.	en
dc.relation.projectID	Gobierno de España. MTM2017-89941-P	es
dc.relation.projectID	Gobierno de España. UAL18-FQM-B025-A	es
dc.relation.projectID	Gobierno de España. PGC2018–096504-B-C33	es
dc.rights	© 2021 The Authors. Published by Elsevier Inc.	en
dc.rights	Atribución 3.0 España
dc.rights.accessRights	open access	en
dc.rights.uri	http://creativecommons.org/licenses/by/3.0/es/
dc.subject.eciencia	Matemáticas	es
dc.subject.other	Orthogonal Polynomials	en
dc.subject.other	Szego Connection	en
dc.subject.other	Jacobi Matrices	en
dc.subject.other	Cmv Matrices	en
dc.subject.other	Verblunsky Coefficients	en
dc.title	A CMV connection between orthogonal polynomials on the unit circle and the real line	en
dc.type	research article	*
dc.type.hasVersion	VoR	*
dspace.entity.type	Publication