Constructing spatial discretizations for sparse multivariate trigonometric polynomials that allow for a fast discrete Fourier transform

Abstract: The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on the union of several rank-1 lattices as sampling scheme and call such schemes multiple rank-1 lattices. This approach automatically makes available a fast discrete Fourier transform (FFT) on the data. The key objective of the construction of spatial discretizations is the unique reconstruction of the trigonometric polynomial using the sampling values at the sampling nodes. We develop construction methods for multiple rank-1 lattices that allow for this unique reconstruction and for estimates of the number $M$ of distinct sampling nodes within the resulting spatial discretizations. Assuming that the multivariate trigonometric polynomial under consideration is a linear combination of $T$ trigonometric monomials, the oversampling factor $M/T$ is independent of the spatial dimension and, roughly speaking, with high probability only logarithmic in $T$, which is much better than the oversampling factor that is expected when using one single rank-1 lattice. The newly developed approaches for the construction of spatial discretizations are probabilistic methods. The arithmetic complexity of these algorithms depend only linearly on the spatial dimension and, with high probability, only linearly on $T$ up to some logarithmic factors. Furthermore, we analyze the computational complexities of the resulting FFT algorithms in detail and obtain upper bounds in $\mathcal{O}\left(M\log M\right)$, where the constants depend only linearly on the spatial dimension. With high probability, we construct spatial discretizations where $M/T\le C\log T$ holds, which implies that the complexity of the corresponding FFT converts to $\mathcal{O}\left(T\log^2 T\right)$.