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

The 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.

[1]  Y. Kamp,et al.  Orthogonal polynomial matrices on the unit circle , 1978 .

[2]  A. Bultheel,et al.  An indeterminate rational moment problem and Carathéodory functions , 2008 .

[3]  On a class of extremal solutions of the nondegenerate matricial Carathéodory problem , 2007 .

[4]  B. Fritzsche,et al.  On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I , 2010, J. Comput. Appl. Math..

[5]  Extensions of Matrix-valued Inner Products on Modules and the Inversion Formula for Block Toeplitz Matrices , 2001 .

[6]  Aad Dijksma,et al.  Operator Theory: Adv. Appl. , 2005 .

[7]  A. Bultheel,et al.  Determinacy of a rational moment problem , 2001 .

[8]  Bernd Kirstein,et al.  An Extension Problem for Non‐Negative Hermitian Block Toeplitz Matrices, IV , 1987 .

[9]  D. Arov,et al.  Problem of search of the minimum of entropy in indeterminate extension problems , 1981 .

[10]  B. Fritzsche,et al.  Orthogonal rational matrix‐valued functions on the unit circle: Recurrence relations and a Favard‐type theorem , 2006 .

[11]  Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes , 1974 .

[12]  M. Rosenberg.  Spectral integrals of operator-valued functions. II. From the study of stationary processes , 1976 .

[13]  A. Bultheel.  Inequalities in Hilbert modules of matrix-valued functions , 1982 .

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  Dual pairs of orthogonal systems of rational matrix-valued functions on the unit circle , 2006 .

[16]  Szegő Pairs of Orthogonal Rational Matrix-valued Functions on the Unit Circle , 2005 .

[17]  Adhemar Bultheel,et al.  A rational Stieltjes moment problem , 2002, Appl. Math. Comput..

[18]  On a moment problem for rational matrix-valued functions , 2003 .

[19]  Israel Gohberg,et al.  Orthogonal polynomials over Hilbert modules , 1994 .

[20]  A. Bultheel,et al.  Moment problems and orthogonal functions , 1993 .

[21]  Orthogonal rational matrix‐valued functions on the unit circle , 2005 .

[22]  H. Dym.  J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation , 1989 .

[23]  On Hilbert modules of rational matrix-valued functions and related inverse problems , 2005 .

[24]  Infinite analogues of block toeplitz matrices and related orthogonal functions , 1995 .

[25]  B. Fritzsche,et al.  On rank invariance of moment matrices of nonnegative Hermitian‐valued Borel measures on the unit circle , 2004 .

[26]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[27]  M. Rosenberg.  The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure , 1964 .

[28]  S. Itoh.  REPRODUCING KERNELS IN MODULES OVER C^*-ALGEBRAS AND THEIR APPLICATIONS , 1990 .