Open AccessExact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients
Author(s) -
Nadiia Derevianko,
Gerlind Plonka
Publication year - 2021
Publication title -
analysis and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.715
H-Index - 24
eISSN - 1793-6861
pISSN - 0219-5305
DOI - 10.1142/s0219530521500196
Subject(s) - mathematics , fourier series , order (exchange) , exponential function , fourier transform , interval (graph theory) , pairwise comparison , combinatorics , rational number , discrete mathematics , mathematical analysis , statistics , finance , economics
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 [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] 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 [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], 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 [Formula: see text]. 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.