We present algorithms for the approximation of multivariate functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. Recently an algorithm for the trigonometric interpolation on
generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers. The main advantage of our method is that the algorithm is based mainly on a one-dimensional fast Fourier transform,
and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method.
Numerical results are presented up to dimension $d=10$, which confirm the theoretical findings.