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Modifications of Prony's Method for the Recovery and Sparse Approximation of Generalized Exponential Sums

In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be recovered from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and show, that this approach can be directly used to reconstruct classical exponential sums from non-equispaced data. Furthermore, we derive a modification of Prony's method for sparse approximation with exponential sums which leads to a non-linear least-squares problem.