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

In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} , \gamma_{j,m} , t^{m} \right) {\mathrm e}^{2\pi \lambda_j t}$,wherethefrequencyparameters$\lambda_{j} \in {\mathbb C}$ 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$P>0$\mathrm{.}Ourmethodrequiresatmost$2N+2$Fouriercoefficients$c_{k}(y)$torecoverallparametersof$y$,where$N:=\sum_{j=1}^{M} (1+n_{j})$denotestheorderof$y$. The recovery is based on the observation that for ${\lambda }_j\in ̸\frac{\mathrm{i}}{P}\mathbb{Z}$ 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 $j=1,\dots ,M$, as well as all parameters ${\lambda }_j$, $j=1,\dots ,M$ 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.