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We derive a new generalization of Prony's method to reconstruct M-sparse expansions of (generalized) eigenfunctions of linear operators from only O(M) suitable values in a deterministic way. The proposed method covers the wellknown reconstruction methods for M-sparse sums of exponentials as well as for the interpolation of M-sparse polynomials by using special linear operators in C(R). Further, we can derive new reconstruction formulas for M-sparse expansions of orthogonal polynomials using the Sturm-Liouville operator. The method is also applied to the recovery of M-sparse vectors in finite-dimensional vector spaces.

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In this survey, we describe the classical Prony method and whose relatives. We sketch a frequently used Prony–like method for equispaced sampled data, namely the ESPRIT method. The case of nonequispaced sampled data is discussed too. For the reconstruction of a sparse eigenfunction expansion, a generalized Prony method is presented. The Prony methods are applied to the recovery of structured functions (such as exponential sums and extended exponential sums) and of sparse vectors. The recovery of spline functions with arbitrary knots from Fourier data is also based on Prony methods. Finally, some numerical examples are given. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)