Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension

Abstract

In this paper we study the worst-case error of numerical integration on the unit sphere ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$, $d\ge 2$, for certain spaces of continuous functions on ${\mathbb{S}}^d$. For the classical Sobolev spaces ${\mathbb{H}}^s({\mathbb{S}}^d)$ ($s>\frac{d}{2}$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. We investigate the behaviour for $s\to \frac{d}{2}$ by introducing spaces ${\mathbb{H}}^{\frac{d}{2},\gamma }({\mathbb{S}}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error.

References in corpus (1)

• QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere

Cited by in corpus (1)

• New lower bounds for the integration of periodic functions