Multiple Rank-1 Lattices as Sampling Schemes for Approximation

The approximation of functions using sampling values along single rank-1 lattices leads to convergence rates of the approximation errors that are far away from optimal ones in spaces of dominating mixed smoothness. A recently published idea that uses sampling values along several rank-1 lattices in order to reconstruct multivariate trigonometric polynomials accompanied by fast methods for the construction of these sampling schemes as well as available fast Fourier transform algorithms motivates investigations on the approximation properties of the arising sampling operators applied on functions of specific smoothness, in particular functions of dominating mixed smoothness which naturally leads to hyperbolic cross approximations.