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 [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.

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

“…In step J = 1 we start with r0 (z) := g k 1 , where k 1 := argmax k∈I |g k |, and set S 1 := {k 1 }, Γ 1 := I \ {k 1 }. At the iteration step J > 1, we proceed as follows to compute a rational function rJ−1 of type (J − 1, J − 1), see also [15,8,18]. Let S J ∪ Γ J = I be the partition of index sets found in the (J − 1)-th iteration step, with |S J | = J and |Γ J | = N − J.…”

“…The problem is closely related with the recovery of and approximation by sums of exponentials of the form 2M j=1 γ j e iφ j t , which has been extensively studied within the last years, see e.g. [16,17,28,2,23,22,25,19,29,18,8,9]. Moreover, there is a close connection to the question of extrapolation of the given sequence of input values (f k ) N −1 k=0 , see e.g.…”

Fourier Series Approximation in Besov Spaces

“…However, the use of structural properties can lead to better results. For the latest developments in the feld of approximation by Fourier series and linear operators, one can see [17,18]. Te Besov spaces, being on the top of the L p -spaces, are good at encoding the smoothness properties of their functions.…”

Rational Functions for the Reconstruction of Exponential Sums from their Fourier Coefficients

“…It has been shown in [1] and [2] that the first step of the algorithm can be performed in a stable way using a modification of the AAA algorithm [3] if |I| ≥ 2K + 1.…”