Matricial Gaussian quadrature rules: singular case

TL;DR

This paper characterizes the existence of minimal matrix-valued measures with prescribed atoms and ranks, generalizing previous results and providing a constructive proof for the strong truncated Hamburger matrix moment problem.

Contribution

It generalizes prior work by characterizing minimal representing measures with prescribed atoms and ranks, extending to positive semidefinite moment matrices.

Findings

Characterization of minimal matrix-valued measures with prescribed atoms and ranks

Generalization of previous results to positive semidefinite moment matrices

Constructive linear-algebraic proof of the strong truncated Hamburger matrix moment problem

Abstract

Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $\mu$. Any finitely atomic representing measure with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure that contains a prescribed atom with a prescribed rank of the corresponding mass, thereby generalizing our recent result, which addresses the same problem in the case where the moment matrix is positive definite. As a corollary, we obtain a constructive, linear-algebraic proof of the strong truncated Hamburger matrix moment problem.