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

    https://doi.org/10.1142/S0219530521500196Cited by:5 (Source: Crossref)

    In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y(t)=j=1M(m=0njγj,mtm)e2πλjt, where the frequency parameters λj 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 (0,P) with P>0. For our method, 2N + 2 Fourier coefficients ck(y) are sufficient to recover all parameters of y, where N:=j=1M(1+nj) denotes the order of y(t). The recovery is based on the observation that for λjiP 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. L2N+2), then our recovery method automatically detects the number M of terms of y, the multiplicities nj for j=1,,M, as well as all parameters λj, j=1,,M, and γj,m, j=1,,M, m=0,,nj, 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.

    AMSC: 41A20, 42A16, 42C15, 65D15, 94A12

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