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

TL;DR

This paper surveys the tractability of $L_p$-approximation in weighted Korobov spaces, analyzing how the complexity depends on error tolerance, dimension, and weight decay, and introduces new findings for the $L_$ case.

Contribution

It provides a comprehensive survey of existing results and presents new insights into the $L_$-approximation problem in weighted Korobov spaces.

Findings

Summarizes known tractability results for $L_2$ and $L_$ approximation.

Introduces new results on $L_$-approximation tractability.

Identifies open problems in the field.

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.