Tractability of $L_2$-approximation and integration in weighted Hermite spaces of finite smoothness

TL;DR

This paper investigates the computational complexity of integration and $L_2$-approximation in weighted Hermite spaces of finite smoothness, providing characterizations of tractability based on the weights that reflect coordinate importance.

Contribution

It compares various weighted Hermite spaces and characterizes the tractability of integration and approximation problems in these spaces based on the weights.

Findings

Tractability depends on the decay of weights as dimension increases.

Characterizations of tractability are provided in terms of weight conditions.

Results inform the design of efficient algorithms for high-dimensional problems.

Abstract

In this paper we consider integration and $L_2$-approximation for functions over $\RR^s$ from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and $L_2$-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold $\varepsilon$ tends to 0 and the problem dimension $s$ grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.