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Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials

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Abstract

We present a new sampling method that allows for the unique reconstruction of (sparse) multivariate trigonometric polynomials. The crucial idea is to use several rank-1 lattices as spatial discretization in order to overcome limitations of a single rank-1 lattice sampling method. The structure of the corresponding sampling scheme allows for the fast computation of the evaluation and the reconstruction of multivariate trigonometric polynomials, i.e., a fast Fourier transform. Moreover, we present a first algorithm that constructs a reconstructing sampling scheme consisting of several \(\text {rank}\text{- }1\) lattices for arbitrary, given frequency index sets. Various numerical tests indicate the advantages of the constructed sampling schemes.

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Authors and Affiliations

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

    Lutz Kämmerer

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  1. Lutz Kämmerer
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Correspondence to Lutz Kämmerer.

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Communicated by Vladimir Temlyakov.

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Kämmerer, L. Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials. J Fourier Anal Appl 24, 17–44 (2018). https://doi.org/10.1007/s00041-016-9520-8

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