Para-orthogonal rational matrix-valued functions on the unit circle

Abstract

Abstract. In this paper, we continue previous investigations with the ultimate goal being a Szegő theory for orthogonal rational matrix functions. We implement here the concept of paraorthogonal functions on the unit circle in the context of rational matrix functions and present some fundamental properties of the para-orthogonal functions in question. We discuss, among other things, the relationship between these functions and orthogonal rational matrix functions as well as existence criteria and some para-orthogonal functions of particular interest.Mathematics subject classification (2010): Primary 47A56, 42C05; Secondary 30E05.

Matrix methods for quadrature formulas on the unit circle. A survey

“…We should mention here that we have chosen the α n to be all in D for convenience, but that it is possible to choose them anywhere outside T on condition that α i αj = 1 for all i and j. This was used in the context of matrix valued ORF in the papers by Lasarow and coworkers (e.g., [27][28][29]47]). A different approach to introduce poles in D and E for a balanced situation is explained in Section 7 where the CMV representation is derived.…”

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

“…In the papers by Lasarow and coworkers (e.g., [23,24,25,34]) matrix versions of the ORF are considered. In these papers also an arbitrary choice of the poles is allowed but only with the restrictive condition that if α is used as a pole, then 1/α cannot be used anymore.…”

“…We relate the resulting ORF with the ORF having all their poles outside or all their poles inside the disk, and study the corresponding recurrence relations. With respect to this new orthogonal rational basis, the shift operator will be represented by a matrix Möbius transformation of a snake-shaped matrix.In the papers by Lasarow and coworkers (e.g., [23,24,25,34]) matrix versions of the ORF are considered. In these papers also an arbitrary choice of the poles is allowed but only with the restrictive condition that if α is used as a pole, then 1/α cannot be used anymore.…”