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In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y (t) = ∑ j = 1 M (∑ m = 0 n j γ j , m t m) e 2 π λ j t , where the frequency parameters λ j ∈ ℂ are pairwise distinct. In order to reconstruct y(t) we employ a finite set of classical Fourier coefficients of y with regard to a finite interval (0 , P) ⊂ ℝ with P > 0. For our method, 2N + 2 Fourier coefficients c k (y) are sufficient to recover all parameters of y, where N : = ∑ j = 1 M (1 + n j) denotes the order of y(t). The recovery is based on the observation that for λ j ∉ i P ℤ the terms of y(t) possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput.40(3) (2018) A1494A1522]. If a sufficiently large set of L Fourier coefficients of y is available (i.e. L ≥ 2 N + 2), then our recovery method automatically detects the number M of terms of y, the multiplicities n j for j = 1 , ... , M , as well as all parameters λ j , j = 1 , ... , M , and γ j , m , j = 1 , ... , M , m = 0 , ... , n j , determining y(t). Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. [ABSTRACT FROM AUTHOR]

We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions. [ABSTRACT FROM AUTHOR]