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

Abstract: In this paper we consider $L_p$-approximation, $p \in {2,\infty}$, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of the information complexity on the error demand $\varepsilon$ and the dimension $d$ to the decay rate of the weight sequence $(\gamma_j){j \ge 1}$ assigned to the Korobov space. Some results have been well known since the beginning of this millennium, others have been proven quite recently. We give a survey of these findings and will add some new results on the $L\infty$-approximation problem. To conclude, we give a concise overview of results and collect a number of interesting open problems.