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In this work, we consider the approximate reconstruction of high-dimensionalperiodic functions based on sampling values. As sampling schemes, we utilizeso-called reconstructing multiple rank-1 lattices, which combine severalpreferable properties such as easy constructability, the existence ofhigh-dimensional fast Fourier transform algorithms, high reliability, and lowoversampling factors. Especially, we show error estimates for functions fromSobolev Hilbert spaces of generalized mixed smoothness. For instance, whenmeasuring the sampling error in the $L_2$-norm, we show sampling errorestimates where the exponent of the main part reaches those of the optimalsampling rate except for an offset of $1/2+\varepsilon$, i.e., the exponent isalmost a factor of two better up to the mentioned offset compared to singlerank-1 lattice sampling. Various numerical tests in medium and high dimensionsdemonstrate the high performance and confirm the obtained theoretical resultsof multiple rank-1 lattice sampling.

Approximation of multivariate periodic functions based on sampling along multiple rank-1 lattices