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Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients.
- Source :
-
Analysis & Applications . May2022, Vol. 20 Issue 3, p543-577. 35p. - Publication Year :
- 2022
-
Abstract
- In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y (t) = ∑ j = 1 M (∑ m = 0 n j γ j , m t m) e 2 π λ j t , where the frequency parameters λ j ∈ ℂ 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) ⊂ ℝ with P > 0. For our method, 2N + 2 Fourier coefficients c k (y) are sufficient to recover all parameters of y, where N : = ∑ j = 1 M (1 + n j) denotes the order of y(t). The recovery is based on the observation that for λ j ∉ i P ℤ 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. L ≥ 2 N + 2), then our recovery method automatically detects the number M of terms of y, the multiplicities n j for j = 1 , ... , M , as well as all parameters λ j , j = 1 , ... , M , and γ j , m , j = 1 , ... , M , m = 0 , ... , n j , 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *EXPONENTIAL sums
*PRONY analysis
*APPROXIMATION algorithms
Subjects
Details
- Language :
- English
- ISSN :
- 02195305
- Volume :
- 20
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 156836298
- Full Text :
- https://doi.org/10.1142/S0219530521500196