Publication: A CMV connection between orthogonal polynomials on the unit circle and the real line
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2021-03-31
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Elsevier BV.
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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.
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Keywords
Orthogonal Polynomials, Szego Connection, Jacobi Matrices, Cmv Matrices, Verblunsky Coefficients
Bibliographic citation
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.