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Modifications of Prony’s Method for the Recovery and Sparse Approximation with Generalized Exponential Sums

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Approximation Theory XVI (AT 2019)

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Abstract

In this survey we describe some modifications of Prony’s method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order. We show, how these expansions can be reconstructed from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and illustrate, how the method can be simplified if some frequency parameters are known beforehand. Furthermore, we present a modification of Prony’s method for sparse approximation with exponential sums which leads to a non-linear least-squares problem.

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Acknowledgements

The authors gratefully acknowledge support by the German Research Foundation in the framework of the RTG 2088. Further, the authors thank the reviewers for many helpful comments to improve the presentation of the results in this paper.

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Authors and Affiliations

  1. University of Göttingen, Institute for Numerical and Applied Mathematics, Göttingen, Germany

    Ingeborg Keller & Gerlind Plonka

Authors
  1. Ingeborg Keller
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  2. Gerlind Plonka
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Correspondence to Gerlind Plonka .

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Editors and Affiliations

  1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA

    Gregory E. Fasshauer

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Marian Neamtu

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Larry L. Schumaker

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Keller, I., Plonka, G. (2021). Modifications of Prony’s Method for the Recovery and Sparse Approximation with Generalized Exponential Sums. In: Fasshauer, G.E., Neamtu, M., Schumaker, L.L. (eds) Approximation Theory XVI. AT 2019. Springer Proceedings in Mathematics & Statistics, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-030-57464-2_7

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