  In this paper we derive a new recovery procedure for the reconstruction of
extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left(
\sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_j
t}$, where the frequency parameters $\lambda_{j} \in {\mathbb C}$ are pairwise
distinct. For the reconstruction we employ a finite set of classical Fourier
coefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbb
R}$ 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 observation
that for $\lambda_{j} \not\in \frac{{\mathrm i}}{P} {\mathbb Z}$ the terms of
$y$ possess Fourier coefficients with rational structure. We employ a recently
proposed stable iterative rational approximation algorithm in [12]. If a
sufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e.,
$L > 2N+2$), then our recovery method automatically detects the number $M$ of
terms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as all
parameters $\lambda_{j}$, $j=1, \ldots , M$ and $ \gamma_{j,m}$ $j=1, \ldots ,
M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a new
stable alternative to the known numerical approaches for the recovery of
exponential sums that are based on Prony's method.

Numerical Analysis;Numerical Analysis