Integration and approximation in cosine spaces of smooth functions

TL;DR

This paper investigates multivariate integration and approximation in cosine-based smooth function spaces, establishing conditions for exponential convergence and analyzing how errors depend on dimension through tractability concepts.

Contribution

It provides new conditions on weight sequences ensuring exponential convergence and characterizes the tractability of high-dimensional problems in cosine spaces.

Findings

Exponential convergence is achieved under specific weight decay conditions.

Necessary and sufficient conditions for various tractability notions are established.

Error dependence on dimension is thoroughly analyzed.

Abstract

We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend on two sequences of real numbers and decay exponentially. As a consequence the functions are infinitely often differentiable, and therefore it is natural to expect exponential convergence of the worst-case error. We give conditions on the weight sequences under which we have exponential convergence for the integration as well as the approximation problem. Furthermore, we investigate the dependence of the errors on the dimension by considering various notions of tractability. We prove sufficient and necessary conditions to achieve these tractability notions.