Toward a structural theory of learning algebraic decompositions

We propose a framework generalizing several variants of Prony's method and explaining their relations. These methods are suitable for determining the support of linear combinations in particular in vector spaces of functions from evaluations. They are based on suitable sequences of linear maps resp. their matrices and include Hankel and Toeplitz variants of Prony's method for the decomposition of multivariate exponential sums, polynomials (w.r.t. the monomial and Chebyshev bases), Gaußian sums, spherical harmonic sums, taking also into account whether they have their support on an algebraic set.