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

publication date

  • June 2021

start page

  • 1

end page

  • 22

volume

  • 266

International Standard Serial Number (ISSN)

  • 0021-9045

Electronic International Standard Serial Number (EISSN)

  • 1096-0430

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.

keywords

  • orthogonal polynomials; szego connection; jacobi matrices; cmv matrices; verblunsky coefficients