Reconstructing sparse exponential polynomials from samples: Stirling numbers and Hermite interpolation
Provides theoretical foundations for reconstructing sparse exponential polynomials with polynomial coefficients, relevant to signal processing and numerical analysis.
The paper establishes the precise relationship between coefficients in sparse exponential polynomials and multiplicity spaces in Prony-type reconstruction, enabling recovery from integer samples. No concrete numerical results are provided.
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.