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

TL;DR

This paper extends the DVZ connection between orthogonal polynomials on the unit circle and the real line, providing explicit relations for all parameter values using a new two-dimensional eigenproblem approach.

Contribution

It introduces a novel method to generalize the DVZ connection for arbitrary parameters, linking measures and polynomials explicitly through a two-dimensional eigenproblem framework.

Findings

Explicit relations for measures and polynomials for all DVZ parameters.

The connection maps measures to rational perturbations of symmetric measures.

New families of orthogonal polynomials on the real line are identified.

Abstract

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Constr. Approx. 36 (2012) 513-535 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 $\lambda$. These 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 $\lambda=1$ which simplifies the connection -basic DVZ connection-. However, similar explicit expressions for an arbitrary value of $\lambda$ -- (general) DVZ connection -- are missing. 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, 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.