Build up of a Szegő theory for rational matrix functions (Q2706817)

This PhD thesis gives an introduction to the matrix version of the orthogonal rational functions on the unit circle as treated in [\textit{A. Bultheel, P. González-Vera, E. Hendriksen} and \textit{O. Njåstad}, ``Orthogonal rational functions'' (1999; Zbl 0923.42017)]. Matrix-valued rational functions with prescribed poles are orthogonalized with respect to a matrix-valued measure on the unit circle. Like for the polynomial case (all poles at infinity) treated for example in [\textit{Ph. Delsarte, Y. V. Genin} and \textit{Y. G. Kamp}, ``Orthogonal polynomial matrices on the unit circle'', IEEE Trans. Circuits Syst. 25, 149--160 (1978; Zbl 0408.15018)], there is a left and a right matrix-valued inproduct.NEWLINENEWLINEThe approach followed is by using some (matrix-valued) optimality properties that are satisfied by the orthogonal rational functions and the associated reproducing kernels. This leads to matrix versions of the Gram-Schmidt algorithm, the generalization of the three-term recurrence relation, Christoffel-Darboux relations and a Favard theorem, and matrix versions of generalized Szegő parameters.