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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 23723, 37 pages
http://dx.doi.org/10.1155/IJMMS/2006/23723

Dual Szegö pairs of sequences of rational matrix-valued functions

Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, Heverlee (Leuven) 3001, Belgium

Received 5 January 2005; Revised 9 January 2006; Accepted 12 March 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.