Tractability of approximation in the weighted Korobov space in the worst-case setting -- a complete picture

Abstract

In this paper, we study tractability of $L_2$-approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class ${\Lambda }^{\mathrm{all}}$ consisting of all continuous linear functionals and from the class ${\Lambda }^{\mathrm{std}}$, which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and $(\sigma ,\tau )$-weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability.