Learning algebraic decompositions using Prony structures

We propose an algebraic framework generalizing several variants of Prony’s method and explaining their relations. This includes Hankel and Toeplitz variants of Prony’s method for the decomposition of multivariate exponential sums, polynomials (with respect to the monomial and Chebyshev bases), Gau{\ss}ian sums, spherical harmonic sums, taking also into account whether they have their support on an algebraic set.