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

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 rank - 1 lattices for arbitrary, given frequency index sets. Various numerical tests indicate the advantages of the constructed sampling schemes.