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We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$error for the approximation of multivariate periodic functions belonging tospecific reproducing kernel Hilbert spaces (RHKS) using approximants that aretrigonometric polynomials computed from sampling values. The used samplingschemes are suitable sets of rank-1 lattices that can be constructed in anextremely efficient way. Furthermore, the structure of the sampling schemesallows for fast Fourier transform (FFT) algorithms. We present and discuss oneFFT algorithm and analyze the worst case $L_\infty$ error for this specificapproach.  Using this general result we work out very weak requirements on the RHKS thatallow for a simple upper bound on the sampling numbers in terms ofapproximation numbers, where the approximation error is measured in the$L_\infty$ norm. Tremendous advantages of this estimate are its pre-asymptoticvalidity as well as its simplicity and its specification in full detail. Itturns out, that approximation numbers and sampling numbers differ at mostslightly. The occurring multiplicative gap does not depend on the spatialdimension $d$ and depends at most logarithmically on the number of used linearinformation or sampling values, respectively. Moreover, we illustrate thecapability of the new sampling method with the aid of specific highly popularsource spaces, which yields that the suggested algorithm is nearly optimal fromdifferent points of view. For instance, we improve tractability results for the$L_\infty$ approximation for sampling methods and we achieve almost optimalsampling rates for functions of dominating mixed smoothness.  Great advantages of the suggested sampling method are the constructivemethods for determining sampling sets that guarantee the shown error bounds,the simplicity and extreme efficiency of all necessary algorithms.

Multiple Lattice Rules for Multivariate $L_\infty$ Approximation in the Worst-Case Setting