On matrix integration of matrix polynomials (Q5946631)

This paper proposes a general framework for the concept of integration, orthogonal polynomials and quadrature formulas for (in general, complex, but in some cases, real) matrix-valued functions. Starting from a reproducing kernel Hilbert space of positive definite forms, the author gives a sketchy proof of the abstract version of the solution of the Hamburger moment problem and the existence of a quadrature rule, well-known in the scalar situation. By means of a process of ``semi-orthonormalization'' (analogue of the Gram-Schmidt algorithm) orthogonal polynomials are introduced, which give to the quadrature formula a more ``standard'' look. The author discusses also some results, appeared previously in literature, related to this set-up.