On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

Abstract

We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (sj) ∞ j=0 of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case. Mathematics Subject Classification (2000). Primary 44A60, 47A57; Secondary 30E05.

On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

“…Moreover, we will review essential facts about them which will later be of use. This material is taken from [10].…”

“…The importance of the set H ≥ q,2n comes from the fact that problem P(R, 2n, ≤) has a solution if and only if the sequence (s j ) 2n j=0 belongs to H ≥ q,2n (see, e.g. [10,Theorem 4.16]). If n ∈ N 0 and if (s j ) 2n j=0 ∈ H ≥ q,2n (resp.…”

“…Furthermore, if n ∈ N 0 , then we will denote by H ≥,e q,2n+1 the set of all sequences (s j ) 2n+1 j=0 of complex q × q matrices for which there is a complex q × q matrix s 2n+2 such that (s j ) 2(n+1) j=0 belongs to H ≥ q, 2(n+1) . Observe that if a nonnegative integer m and a sequence (s j ) m j=0 of complex q × q matrices are given, then problem P(R, m, =) has a solution if and only if (s j ) m j=0 belongs to H ≥,e q,m (see [10,Theorem 4.17]). Obviously, H ≥,e q,2k ⊆ H ≥ q,2k for each k ∈ N 0 and H ≥,e q,0 = H ≥ q,0 .…”

“…Obviously, if n ∈ N 0 and if (s j ) n j=0 ∈ H ≥,e q,n , then (s j ) m j=0 ∈ H ≥,e q,m for each m ∈ N 0,n . In view of Dyukarev et al [10,Remark 2.7], for each n ∈ N 0 , the inclusion H > q,2n ⊆ H ≥,e q,2n holds true. An important topic in [10] was the study of the intrinsic structure of sequences belonging to one of the sets H ≥ q,2n , H > q, 2n and H ≥,e q,2n .…”

Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes

“…It explicits connections between interpolation, Leech's factorization theorem (see [9,10]) and the state space method. Next, the paper On a simultaneous approach to the even and odd truncated matricial Hamburger moment by Bernd Fritzsche, Bernd Kirstein and Conrad Mädler, continues the former investigations of the authors on matricial versions of power moment problems (see [4,5,7] and the papers in the volume [1]). The approach is based on Schur analysis, The main tool consists of an appropriate adaptation of the classical algorithm due to I. Schur and R. Nevanlinna to the moment problems under consideration.…”