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

TL;DR

This paper investigates the worst-case error bounds for numerical integration on high-dimensional spheres, extending classical Sobolev space results to include spaces with logarithmic weights at critical smoothness levels.

Contribution

It introduces new function spaces with logarithmic weights and derives upper and lower bounds for integration errors in these spaces, extending existing results.

Findings

Established bounds for spaces with logarithmic weights at critical smoothness

Extended classical Sobolev space error estimates to new weighted spaces

Provided insights into error behavior as smoothness approaches critical value

Abstract

In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) 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 $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error.