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 form [Formula: see text], where frequency parameters text] are pairwise distinct. order to reconstruct y(t) employ finite set classical Fourier coefficients y with regard interval text]. For our method, 2N + 2 sufficient recover all y, denotes y(t). The is based on observation that terms possess rational structure. We recently proposed stable iterative approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, AAA Algorithm approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If sufficiently large L available (i.e. text]), then method automatically detects number M multiplicities as well determining Therefore, provides alternative known numerical approaches Prony’s method.