Reconstructing sparse exponential polynomials from samples: Stirling numbers and Hermite interpolation

TL;DR

This paper explores the reconstruction of sparse exponential polynomials from samples, highlighting the role of Stirling numbers and Hermite interpolation, and analyzing the relationship between coefficients and multiplicity spaces.

Contribution

It extends Prony's method to handle exponential polynomials with multiplicities, connecting coefficients to multiplicity spaces using Stirling numbers and Hermite interpolation.

Findings

Relationship between coefficients and multiplicity spaces clarified

Extension of Prony's method to exponential polynomials with multiplicities

Use of Stirling numbers and Hermite interpolation in reconstruction

Abstract

Prony's method, in its various concrete algorithmic realizations, is concerned with the reconstruction of a sparse exponential sum from integer samples. In several variables, the reconstruction is based on finding the variety for a zero dimensional radical ideal. If one replaces the coefficients in the representation by polynomials, i.e., tries to recover sparse exponential polynomials, the zeros associated to the ideal have multiplicities attached to them . The precise relationship between the coefficients in the exponential polynomial and the multiplicity spaces are pointed out in this paper.