Toeplitz, Hankel, de Branges and two truncated matrix moment problems

TL;DR

This paper explores truncated matrix moment problems using reproducing kernel Hilbert spaces, providing new formulas and methods for solving these problems in both the Hamburger and trigonometric cases.

Contribution

It introduces a novel approach employing reproducing kernels to analyze truncated matrix moment problems, simplifying computations and deriving new formulas.

Findings

Reproducing kernel methods facilitate easier computation of projections.

New formulas for truncated matrix moment problems are derived.

The approach unifies treatment of Hamburger and trigonometric cases.

Abstract

This paper deals with (1) the truncated matrix Hamburger moment problem from the point of view of reproducing kernel Hilbert spaces of vector valued entire functions of the kind introduced and extensively studied by Louis de Branges and (2) the truncated matrix trigonometric moment problem viewed through an analogous class of spaces that are formulated with respect to the open unit disc rather than the open upper half-plane. In this approach projections are computed via appropriately chosen reproducing kernels instead of orthogonal bases. This approach eases the bookkeeping and leads to pleasing formulas.