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

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