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Reconstruction of sparse Legendre and Gegenbauer expansions

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Abstract

We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony-like methods. The method proposed is robust with respect to noisy sampled data. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order.

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Acknowledgments

The first named author gratefully acknowledges the support by the German Research Foundation within the project PO 711/10-2. Moreover, the authors would like to thank the anonymous referees for their valuable comments, which improved this paper.

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Authors and Affiliations

  1. Department of Mathematics, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Daniel Potts

  2. Institute of Mathematics, University of Rostock, 18051, Rostock, Germany

    Manfred Tasche

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  1. Daniel Potts
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  2. Manfred Tasche
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Correspondence to Daniel Potts.

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Communicated by Michael S. Floater.

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Potts, D., Tasche, M. Reconstruction of sparse Legendre and Gegenbauer expansions. Bit Numer Math 56, 1019–1043 (2016). https://doi.org/10.1007/s10543-015-0598-1

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  • DOI: https://doi.org/10.1007/s10543-015-0598-1

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