Efficient reconstruction of functions using eigenfunctions of linear operators

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In this project, we aim to unify and to generalize nonlinear reconstruction methods that go back to the Prony method. For this purpose, we will use the new insight that both the classical Prony method with a lot of applications in system identification and the Ben-Or and Tiwari algorithm for multivariate M-term polynomial interpolation can be understood as reconstruction methods for M-term expansions of eigenfunctions of special linear operators. This new perspective allows us to derive new generalized reconstruction algorithms for structured functions that can be represented for example as M-term expansions of trigonometric functions, orthogonal polynomials or Bessel functions. Thereby the generalized Prony methods will be applicable in a significantly larger range of applications e.g. in signal processing as well as for approximation of solutions of partial differential equations.The project especially focusses on the construction and analysis of efficient and numerically stable reconstruction algorithms and on deriving new error estimates for the reconstructed parameters in case of noisy input data. We want to apply the generalized Prony method to nonlinear approximation of Green's functions and to the reconstruction of structured functions from Fourier data in Magnetic Resonance Imaging.