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

Abstract

In this paper we consider L p -approximation, p ∈ {2, ∞}, 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 ε and the dimension d to the decay rate of the weight sequence (γ j ) j≥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 ∞ -approximation problem. To conclude, we give a concise overview of results and collect a number of interesting open problems.

Exponential tractability of $L_2$-approximation with function values

“…However, under the assumption that F is Hilbert, there is often a complete knowledge about the linear/Gelfand width, and hence about n all (ε, F d ), see, e.g., [12,13,17,18,21]. Moreover, there are also some general results on the power of Λ std in this case, see [23,Chapter 26], and even more results in specific settings, see, e.g., [2,4,5,30].…”