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In this paper we derive a new recovery procedure for the reconstruction ofextended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left(\sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_jt}$, where the frequency parameters $\lambda_{j} \in {\mathbb C}$ are pairwisedistinct. For the reconstruction we employ a finite set of classical Fouriercoefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbbR}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients$c_{k}(y)$ to recover all parameters of $y$, where $N:=\sum_{j=1}^{M}(1+n_{j})$ denotes the order of $y$. The recovery is based on the observationthat for $\lambda_{j} \not\in \frac{{\mathrm i}}{P} {\mathbb Z}$ the terms of$y$ possess Fourier coefficients with rational structure. We employ a recentlyproposed stable iterative rational approximation algorithm in [12]. If asufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e.,$L > 2N+2$), then our recovery method automatically detects the number $M$ ofterms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as allparameters $\lambda_{j}$, $j=1, \ldots , M$ and $ \gamma_{j,m}$ $j=1, \ldots ,M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a newstable alternative to the known numerical approaches for the recovery ofexponential sums that are based on Prony's method.

Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients