Truncated moment problems on positive-dimensional algebraic varieties

TL;DR

This paper extends Prony's method to measures supported on algebraic varieties of any dimension, showing that finitely many moments determine the support's Zariski closure, which can be computed via moment matrices.

Contribution

It generalizes Prony's method from finitely-supported measures to those supported on positive-dimensional algebraic varieties, providing a new way to determine support structure.

Findings

Support's Zariski closure is determined by finitely many moments.

Support can be recovered from the kernel of moment matrices.

Method applies to signed and non-negative measures on algebraic varieties.

Abstract

This manuscript transfers the main aspects of Prony's method from finitely-supported measures to the classes of signed or non-negative measures supported on algebraic varieties of any dimension. In particular, we show that the Zariski closure of the support of these measures is determined by finitely many moments and can be computed from the kernel of certain moment matrices.