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

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.