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

TL;DR

This paper provides a comprehensive analysis of the conditions under which $L_2$-approximation in weighted Korobov spaces is tractable, covering all major notions of tractability in the worst-case setting.

Contribution

It offers necessary and sufficient conditions on weights for various tractability notions, completing the understanding of approximation complexity in this setting.

Findings

Characterizes weight conditions for quasi-polynomial tractability

Identifies conditions for uniform and weak tractability

Completes the picture with known results on polynomial tractability

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 $\Lambda^{{\rm all}}$ consisting of all continuous linear functionals and from the class $\Lambda^{{\rm std}}$, 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.