Dual Szegö pairs of sequences of rational matrix-valued functions

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

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

AbstractThe main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz–Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved

Einige grundlegende Aussagen zur Szegő-Theorie orthogonaler rationaler Matrixfunktionen

status: publishe

On canonical solutions of the truncated trigonometric matrix moment problem

The truncated trigonometric matrix moment problem is a well studied object. The main theme of the talk is the discussion of some distinguished solutions of that moment problem. Roughly speaking, we discuss certain solutions which are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle with a special structure. We give some general information on this type of solutions, but we will focus on the so-called nondegenerate case. In that case, the measures in question form a family of solutions which can be parametrized by the set of unitary matrices. In particular, we will show that each member of this family offers an extremal property in the solution set of the moment problem concerning the weight assigned to some point of the open unit disk.Talk in the "Seminario del Cuerpo Academico Ecuaciones de Fisica Matematicas" at the "Universidad Michoacana de San Nicolas de Hidalgo" in Morelia (Mexico)status: publishe

On canonical solutions of the truncated trigonometric matrix moment problem

The main theme of the talk is the discussion of some distinguished solutions of the truncated trigonometric matrix moment problem. Roughly speaking, these solutions are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle with a special structure. We give some general information on this type of solutions, but we will focus on the so-called nondegenerate situation. In that case, these molecular measures form a family of solutions which can be parametrized by the set of unitary matrices. In particular, we will show that each member of this family offers an extremal property within the solution set of the moment problem in question concerning the weight assigned to some point of the open unit disk. In doing so, an application of the theory of orthogonal matrix polynomials on the unit circle is used to get that insight.status: publishe

On the existence of para-orthogonal rational functions on the unit circle

Similar as in the classical case of polynomials, as is known, para-orthogonal rational functions on the unit circle can be used to obtain quadrature formulas of Szegő-type to approximate some integrals. In the present talk we carry out a thorough discussion of the existence of such rational functions in terms of the underlying Borel measure on the unit circle. The talk is based on a joint work with Adhemar Bultheel.status: publishe

Solution of a multiple Nevanlinna-Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna-Pick type for matrix-valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. The problem can be regarded as a generalization of the classical Carathéodory coefficient problem on the one hand and on the other hand of the original Nevanlinna-Pick problem. One can find several approaches to the solution of such kind of interpolation problems in the literature. We will explain, how one can use the theory of orthogonal rational matrix functions to study the interpolation problem in question. In particular, for the non-degenerate case, i.e. in the situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, we present descriptions of the solution set of this problem in terms of orthogonal rational matrix functions. These rational matrix functions play here a similar role as Szegö's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem.status: publishe

Some basic facts on orthogonal rational matrix functions on the unit circle

We consider spaces of (square) matrix functions each entry of which is a rational (complex-valued) function with prescribed poles. In particular, the poles of these rational functions are not located on the unit circle of the complex plane. This fact will be necessary in a way, since based on a given nonnegative Hermitian matrix Borel measure on the unit circle, the spaces will be equipped simultaneously with left and right matrix inner products via integration. Essentially, we study some special systems of orthogonal rational matrix functions in that context and some basics on the systems in question will be presented. The bottom line is, we will see that larger parts of the classical theory of orthogonal polynomials on the unit circle can be extended to this more general situation. Among other things, we point out a construction of orthogonal rational matrix functions via reproducing kernels, the important role of Christoffel-Darboux formulae, and a characterization of the orthogonal systems via specific recurrence relations. In doing so, an essential feature is marked by an inherent (but far from self-evident) interplay between objects associated with the left and right versions of matrix inner products. The talk is based on a joint work with Bernd Fritzsche and Bernd Kirstein.Talk in the "Coloquio del Posgrado en Ciencias Matematicas" at the "Universidad Michoacana de San Nicolas de Hidalgo" in Morelia (Mexico)status: publishe

On canonical solutions of the truncated trigonometric matrix moment problem

The main theme of the talk is the discussion of some distinguished solutions of the truncated trigonometric matrix moment problem. Roughly speaking, these solutions are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle with a special structure. We give some general information on this type of solutions, but we will focus on the so-called nondegenerate situation. In that case, these molecular measures form a family of solutions which can be parametrized by the set of unitary matrices. In particular, we will show that each member of this family offers an extremal property within the solution set of the moment problem in question concerning the weight assigned to some point of the open unit disk. In doing so, an application of the theory of orthogonal matrix polynomials on the unit circle is used to get that insight.status: publishe