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Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

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Abstract

In this paper, we study the convergence of adaptive Fourier sums for real-valued \(2\pi \)-periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of \(N+1\) terms. The obtained approximation can be interpreted as an adaptive N-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.

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Acknowledgements

The authors would like to thank the reviewers for valuable comments to improve the manuscript. The authors gratefully acknowledge the funding of this work by the DFG in the framework of the GRK 2088 and the project PL 170/16-1.

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Authors and Affiliations

  1. Institute for Numerical and Applied Mathematics, Göttingen University, Lotzestr. 16-18, 37083, Göttingen, Germany

    Gerlind Plonka & Vlada Pototskaia

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  1. Gerlind Plonka
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  2. Vlada Pototskaia
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Correspondence to Gerlind Plonka.

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Communicated by Massimo Fornasier.

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Plonka, G., Pototskaia, V. Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums. J Fourier Anal Appl 25, 1580–1608 (2019). https://doi.org/10.1007/s00041-018-9635-1

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  • DOI: https://doi.org/10.1007/s00041-018-9635-1

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