Reconstructing Hyperbolic Cross Trigonometric Polynomials by Sampling along Rank-1 Lattices

With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100.