Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-1 lattice. Moreover, the main result is the fact that any (non-)linear reconstruction algorithm taking function values on any integration lattice of size M has a dimension-independent lower bound of 2 - ( α + 1 ) / 2 M - α / 2 when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness α > 0 on the d -torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a component–by–component method. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids.