A CMV connection between orthogonal polynomials on the unit circle and the real line
| Autor: | L. Moral, M. J. Cantero, Luis Velázquez, Francisco Marcellán |
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| Přispěvatelé: | Ministerio de Economía y Competitividad (España) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Polynomial 42C05 Matemáticas General Mathematics 010103 numerical & computational mathematics Verblunsky Coefficients 01 natural sciences Orthogonal Polynomials Cmv Matrices symbols.namesake Matrix (mathematics) Jacobi Matrices Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics Szego Connection Real line Mathematics Numerical Analysis Applied Mathematics Orthogonal polynomials on the unit circle 010102 general mathematics Connection (mathematics) Unit circle Mathematics - Classical Analysis and ODEs Orthogonal polynomials symbols Jacobi polynomials Analysis |
| Zdroj: | e-Archivo. Repositorio Institucional de la Universidad Carlos III de Madrid instname |
| Popis: | 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. 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. |
| Databáze: | OpenAIRE |
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