• Search
•

×

### System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
Open Access

# 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 $λj∈ℂ$ 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 $ck(y)$ 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 $λj∉iPℤ$ 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. $L≥2N+2$), 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

## References

• 1. V. M. Adamjan, D. Z. Arov and M. G. Krein , Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur–Takagi problem, Mat. Sb. 86(128) (1971) 34–75. Google Scholar
• 2. R. Badeau, B. David and G. Richard , High-resolution spectral analysis of mixtures of complex exponentials modulated by polynomials, IEEE Trans. Signal Process. 54(4) (2006) 1341–1350. Crossref, Web of Science
• 3. D. Batenkov , Accurate solution of near-colliding Prony systems via decimation and homotopy continuation, Theoret. Comput. Sci. 681 (2017) 27–40. Crossref, Web of Science
• 4. D. Batenkov and Y. Yomdin , On the accuracy of solving confluent Prony systems, SIAM J. Appl. Math. 73(1) (2013) 134–154. Crossref, Web of Science
• 5. L. Berg , Lineare Gleichungssysteme mit Bandstruktur und ihr Asymptotisches Verhalten (Deutscher Verlag der Wissenschaften, Berlin, 1986). Google Scholar
• 6. G. Beylkin and L. Monzón , On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (2005) 17–48. Crossref, Web of Science
• 7. G. Beylkin and L. Monzón , Nonlinear inversion of a band-limited Fourier transform, Appl. Comput. Harmon. Anal. 27 (2009) 351–366. Crossref, Web of Science
• 8. D. Braess , Nonlinear Approximation Theory (Springer-Verlag, Berlin, 1986). Crossref
• 9. E. J. Candes, J. Romberg and T. Tao , Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59 (2006) 1207–1223. Crossref, Web of Science
• 10. A. Cuyt and W.-S. Lee , How to get high resolution results from sparse and coarsely sampled data, Appl. Comput. Harmon. Anal. 48(3) (2020) 1066–1087. Crossref, Web of Science
• 11. N. Derevianko, G. Plonka and M. Petz, From ESPRIT to ESPIRA: Estimation of signal parameters by iterative rational approximation, preprint (2021), arXiv:2106.15140. Google Scholar
• 12. F. Filbir, H. N. Mhaskar and J. Prestin , On the problem of parameter estimation in exponential sums, Constr. Approx. 35(3) (2012) 323–343. Crossref, Web of Science
• 13. I. S. Gradshteyn and I. M. Ryzhik , Table of Integrals, Series, and Products, 5th edn. (Academic Press, London, 1994), translated from the Russian by Scripta Technica, Inc., Boston. Google Scholar
• 14. W. Hackbusch , Computation of best $L∞$ exponential sums for $1/x$ by Remez’ algorithm, Comput. Vis. Sci. 20 (2019) 1–11. Crossref, Web of Science
• 15. G. Heinig and K. Rost , Algebraic Methods for Toeplitz-Like Matrices and Operators (Birkhäuser, Basel, 1984). Crossref
• 16. D. W. Kammler , Approximation with sums of exponentials in $Lp[0,∞)$, J. Approx. Theory 16 (1976) 384–408. Crossref, Web of Science
• 17. B. Mourrain , Polynomial–exponential decomposition from moments, Found. Comput. Math. 18 (2018) 1435–1492. Crossref, Web of Science
• 18. Y. Nakatsukasa, O. Sète and L. N. Trefethen , The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494–A1522. Crossref, Web of Science
• 19. V. V. Peller , An excursion into the theory of Hankel operators, Holomorphic spaces, Math. Sci. Res. Inst. Publ. 33 (1998) 65–120. Google Scholar
• 20. V. PereyraG. J. Scherer (eds.), Exponential Data Fitting and its Applications (Bentham Science Publishers, 2010). Google Scholar
• 21. T. Peter and G. Plonka , A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Probl. 29 (2013) 025001. Crossref, Web of Science
• 22. T. Peter, D. Potts and M. Tasche , Nonlinear approximation by sums of exponentials and translates, SIAM J. Sci. Comput. 33(4) (2011) 1920–1947. Crossref, Web of Science
• 23. M. Petz, G. Plonka and N. Derevianko , Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients, Sampl. Theory Signal Process. Data Anal. 19 (2021) 7. Crossref
• 24. G. Plonka and V. Pototskaia , Computation of adaptive Fourier series by sparse approximation of exponential sums, J. Fourier Anal. Appl. 25(4) (2019) 1580–1608. Crossref, Web of Science
• 25. G. Plonka, D. Potts, G. Steidl and M. Tasche , Numerical Fourier Analysis (Birkhäuser, Basel, 2018). Crossref
• 26. G. Plonka, K. Stampfer and I. Keller , Reconstruction of stationary and non-stationary signals by the generalized Prony method, Anal. Appl. 17(2) (2019) 179–210. Link, Web of Science
• 27. G. Plonka and M. Tasche , Prony methods for recovery of structured functions, GAMM-Mitt. 37(2) (2014) 239–258. Crossref
• 28. D. Potts and M. Tasche , Parameter estimation for exponential sums by approximate Prony method, Signal Process. 90(5) (2010) 1631–1642. Crossref, Web of Science
• 29. D. Potts and M. Tasche , Parameter estimation for multivariate exponential sums, Electron. Trans. Numer. Anal. 40 (2013) 204–224. Web of Science
• 30. J. R. Rice , Chebyshev approximation by exponentials, SIAM J. Appl. Math. 10(1) (1962) 149–161. Crossref, Web of Science
• 31. A. Sidi , Interpolation at equidistant points by a sum of exponential functions, J. Approx. Theory 34 (1982) 194–210. Crossref, Web of Science
• 32. A. Sidi , Interpolation by a sum of exponential functions when some exponents are preassigned, J. Math. Anal. Appl. 112 (1985) 151–164. Crossref, Web of Science
• 33. J. A. Tropp and A. C. Gilbert , Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inform. Theory 53(12) (2008) 4655–4666. Crossref, Web of Science
• 34. M. Vetterli, P. Marziliano and T. Blu , Sampling signals with finite rate of innovation, IEEE Trans. Signal Process. 50(6) (2002) 1417–1428. Crossref, Web of Science
• 35. G. Welker , Approximation mit einer erweiterten klasse von exponentialsummen, J. Approx. Theory 33 (1981) 281–287. Crossref, Web of Science
• 36. H. Wilber, A. Damle and A. Townsend, Data-driven algorithms for signal processing with rational functions, preprint (2021), arXiv:2105.07324. Google Scholar
Remember to check out the Most Cited Articles!

Check out our Differential Equations and Mathematical Analysis books in our Mathematics 2021 catalogue
Featuring authors such as Ronen Peretz, Antonio Martínez-Abejón & Martin Schechter