Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations

Abstract

In this paper we consider the L q -approximation of multivariate periodic functions f with L q -bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of n sampling nodes. The general performance is measured in terms of (non-)linear sampling widths ϱ n . We conduct a systematic analysis of Smolyak type interpolation algorithms in the framework of Besov–Lizorkin–Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood–Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case 2 ≤ p < q < ∞ and r > 1/p the linear sampling widths ϱ lin (S W(T d ), L q (T d )) and ϱ lin (S ,∞ B(T d ), L q (T d )) show the asymptotic behavior of the corresponding Gelfand n-widths, whereas in case 1 < p < q ≤ 2 and r > 1/p the linear sampling widths match the corresponding linear widths. In the mentioned cases linear Smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.