ENG 
ESP Session S32 - Special functions and orthogonal polynomials
Monday, July 12, 13:00 ~ 14:00 UTC-3
A CMV connection between orthogonal polynomials on the unit circle and the real line
María-José Cantero
Universidad de Zaragoza, España - This email address is being protected from spambots. You need JavaScript enabled to view it.
It is very well known the connection between orthogonal polynomials on the unit circle and orthogonal polynomials on the real line, given by Szeg\H{o}. This connection, based on a map which transforms the unit circle onto certain interval of the real line, induces a one-to-one correspondence between symmetric measures on the unit circle and measures on the mentioned interval.
A new relation between these two kinds of polynomials has been recently discovered by Derevyagin, Vinet and Zhedanov (DVZ). The DVZ connection starts from a factorization of real CMV matrices ${\cal C}$ (unitary analogue of Jacobi matrices) into two tridiagonal factors, whose linear combination yields a Jacobi matrix ${\cal K}$ depending on a real parameter $\lambda$ ({\it general DVZ connection)}. The main result of the authors refers to the Jacobi matrix $\cal K$ built out of the Jacobi polynomials on the unit circle, which leads to a connection with the so called big $-1$ Jacobi polynomials. They also obtain the relation between the orthogonal polynomials and orthogonality measures for an arbitrary CMV matrix $\cal C$ and the corresponding Jacobi matrix $\cal K$, but only for the value $\lambda=1$, which simplifies the connection ({\it basic DVZ connection}). However, for generalized DVZ, such a general relation is missing.
We will present a different approach to this connection which allows us to go further than DVZ. We start by using CMV tools to obtain directly the orthogonal polynomials associated with ${\cal K}$ in terms of the basis related to ${\cal C}$, and then we use this to discover the relation between the corresponding orthogonality measures. The advantages of our approach are more evident for the generalized DVZ connection, where we obtain explicit formulas for the relation between the orthogonal polynomials and orthogonality measures associated with $\cal K$ and $\cal C$. The utility of these results will be illustrated with some examples providing new families of orthogonal polynomials on the real line.
Joint work with Francisco Marcellán (Universidad Carlos III de Madrid, España), Leandro Moral (Universidad de Zaragoza, España) and Luis Velázquez (Universidad de Zaragoza, España).