RT Journal Article
T1 A CMV connection between orthogonal polynomials on the unit circle and the real line
A1 Cantero, M.J.
A1 Moral, L.
A1 Velázquez, L.
A1 Marcellán Español, Francisco José
AB M. Derevyagin, L. Vinet and A. Zhedanov introduced in Derevyagin et al. (2012) a new connectionbetween orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into aJacobi one depending on a real parameter λ. In Derevyagin et al. (2012) the authors prove that this mapyields a natural link between the Jacobi polynomials on the unit circle and the little and big −1 Jacobipolynomials on the real line. They also provide explicit expressions for the measure and orthogonalpolynomials associated with the Jacobi matrix in terms of those related to the CMV matrix, but onlyfor the value λ = 1 which simplifies the connection –basic DVZ connection–. However, similar explicitexpressions 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-dimensionaleigenproblem by using known properties of CMV matrices. This allows us to go further than Derevyaginet al. (2012), providing explicit relations between the measures and orthogonal polynomials for thegeneral DVZ connection. It turns out that this connection maps a measure on the unit circle into arational perturbation of an even measure supported on two symmetric intervals of the real line, whichreduce to a single interval for the basic DVZ connection, while the perturbation becomes a degree onepolynomial. Some instances of the DVZ connection are shown to give new one-parameter families oforthogonal polynomials on the real line.
PB Elsevier BV.
SN 0021-9045
YR 2021
FD 2021-03-31
LK https://hdl.handle.net/10016/35473
UL https://hdl.handle.net/10016/35473
LA eng
NO The work of the first, third and fourth authors has been supported in part by the researchproject MTM2017-89941-P from Ministerio de Economía, Industria y Competitividad of Spainand 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 deAragón (Spain) and the ERDF 2014–2020 “Construyendo Europa desde Aragón”.The work of the second author has been partially supported by the research projectPGC2018–096504-B-C33 supported by Agencia Estatal de Investigación of Spain.
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RD 1 sept. 2024