# Prony's method under an almost sharp multivariate Ingham inequality

## TL;DR

This paper extends Prony's method to multivariate signals by establishing a sharper Ingham inequality, improving parameter identifiability conditions and refining the theoretical understanding of multivariate moment-based reconstruction.

## Contribution

It introduces a new multivariate Ingham inequality with a logarithmic constant and refines identifiability conditions for multivariate Prony methods.

## Key findings

- Improved dimension-dependent constant from square-root to logarithmic in Ingham inequality.
- Enhanced understanding of parameter identifiability in multivariate Prony's method.
- Connections to flat extension principle and Hilbert function stagnation.

## Abstract

The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.