  In this work, we consider the approximate reconstruction of high-dimensional
periodic functions based on sampling values. As sampling schemes, we utilize
so-called reconstructing multiple rank-1 lattices, which combine several
preferable properties such as easy constructability, the existence of
high-dimensional fast Fourier transform algorithms, high reliability, and low
oversampling factors. Especially, we show error estimates for functions from
Sobolev Hilbert spaces of generalized mixed smoothness. For instance, when
measuring the sampling error in the $L_2$-norm, we show sampling error
estimates where the exponent of the main part reaches those of the optimal
sampling rate except for an offset of $1/2+\varepsilon$, i.e., the exponent is
almost a factor of two better up to the mentioned offset compared to single
rank-1 lattice sampling. Various numerical tests in medium and high dimensions
demonstrate the high performance and confirm the obtained theoretical results
of multiple rank-1 lattice sampling.
