We consider $L_p$-approximation, $p \in \{2,\infty\}$, of periodic functions from weighted Korobov spaces in the worst-case setting. 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. In particular for the case $p=2$ we now have a very clear picture of the whole situation. Also for $p=\infty$ a lot is known but unfortunately for this case there remain some open questions. We give a concise overview of the new results and present a number of interesting open problems.
Joint work with Adrian Eberts and Peter Kritzer