A determinant characterization of moment sequences with finitely many mass-points

TL;DR

This paper characterizes moment sequences with finitely many mass points using Hankel determinants, showing that specific determinant conditions imply the sequence is a moment sequence of a discrete measure with finite support.

Contribution

It provides a determinant-based criterion for identifying when a sequence is a finite discrete moment sequence, extending classical moment problem results.

Findings

Hankel determinants D_n are positive for n<n_0 and zero for n greater or equal to n_0.

Sequences with these determinant conditions correspond to moments of discrete measures with n_0 points.

Hankel matrices are positive semi-definite under the given conditions.

Abstract

To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive semi-definite, and in particular (s_n) is the sequence of moments of a discrete measure concentrated in n_0 points on the real line. We stress that the conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of the Hankel matrices.