Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

Abstract

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] 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 [Formula: see text] with [Formula: see text]. For our method, 2N + 2 Fourier coefficients [Formula: see text] are sufficient to recover all parameters of y, where [Formula: see text] denotes the order of y(t). The recovery is based on the observation that for [Formula: see text] 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. [Formula: see text]), then our recovery method automatically detects the number M of terms of y, the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], 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.