A Truncated Matricial Moment Problem on a Finite Interval. The Case of an Odd Number of Prescribed Moments.

The main goal of this paper is to study the truncated matricial moment problem on a finite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an effective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions. [ABSTRACT FROM AUTHOR]