Dual Szegö pairs of sequences of rational matrix-valued functions

Abstract

We study certain sequences of rational matrix-valued functions with poles outside the unit circle. These sequences are recursively constructed based on a sequence of complex numbers with norm less than one and a sequence of strictly contractive matrices. We present some basic facts on the rational matrix-valued functions belonging to such kind of sequences and we will see that the validity of some Christoffel-Darboux formulae is an essential property. Furthermore, we point out that the considered dual pairs consist of orthogonal systems. In fact, we get similar results as in the classical theory of Szegö's orthogonal polynomials on the unit circle of the first and second kind.