Matricial Gaussian quadrature rules: singular case
Aljaž Zalar, Igor Zobovič
TL;DR
The paper addresses the truncated univariate matrix moment problem in the singular, positive semidefinite setting, aiming to characterize when a minimal matrix-valued representing measure with a prescribed atom at $t$ and prescribed mass rank $m$ exists. It develops a constructive, linear-algebraic framework using block-structured matrices and column-dependency relations to capture the propagation of relations in the moment matrix, extending the positive definite results to the semidefinite case. Key contributions include an equivalence that ties the existence of such a measure to combinatorial data from these matrices, a uniqueness criterion under certain kernel conditions, and a corollary resolving a strong truncated Hamburger matrix moment problem, all accompanied by a numerical example. The results provide explicit criteria and procedures for obtaining matricial Gaussian quadrature rules in singular settings, enriching the matrix moment literature and offering practical tools for relevant applications.
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 $μ$. 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.