From OPRL to OPUC. A matrix approach beyond the Szegő transformations

M. Derevyagin, L. Vinet and A. Zhedanov introduced in [1] a new connection between orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line(OPRL). It maps any real CMV matrix into a Jacobi one depending on a real parameter $\lambda$. In such a contribution 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 $\lambda=1$ which simplifies the connection –basic DVZ connection–. However, similar explicit expressions for an arbitrary value of $\lambda$ –(general) DVZ connection– are missing in the above mentioned paper. In this problem we will focus our presentation.

First of all, we will summarize the state of the art, with a special emphasis in the so called Szegő transformations between measures supported on the interval $[-1,1]$ and some measures supported on the unit circle. Next, we will discuss 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 the DVZ approach 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. We will follow the approach presented in our recent paper [2].

This is a joint work with M. J. Cantero, L. Moral. L. Velázquez (Universidad de Zaragoza and IUMA, Spain).