Reconstructing multivariate trigonometric polynomials from samples along rank-1 lattices

Abstract

The approximation of problems in d spatial dimensions by trigonometric polynomials supported on known more or less sparse frequency index sets I ⊂ ℤ d is an important task with a variety of applications. The use of rank-1 lattices as spatial discretizations offers a suitable possibility for sampling such sparse trigonometric polynomials. Given an arbitrary index set of frequencies, we construct rank-1 lattices that allow a stable and unique discrete Fourier transform. We use a component-bycomponent method in order to determine the generating vector and the lattice size.