A lower bound for the worst-case cubature error on spheres of arbitrary dimension

Hesse, Kerstin (2006) A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numerische Mathematik, 103 (3). pp. 413-433. ISSN 0029-599X

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Abstract

This paper is concerned with numerical integration on the unit sphere $S^r$ of dimension $r\\geq 2$ in the Euclidean space $\\mathbb{R}^{r+1}$. We consider the worst-case cubature error, denoted by $E(Q_m;H^s(S^r))$, of an arbitrary $m$-point cubature rule $Q_m$ for functions in the unit ball of the Sobolev space $H^s(S^r)$, where $s>r/2$, and show that $E(Q_m;H^s(S^r))\\geq c_{s,r} m^{-s/r}$. The positive constant $c_{s,r}$ in the estimate depends only on the sphere dimension $r\\geq 2$ and the index $s$ of the Sobolev space $H^s(S^r)$. This result was previously only known for $r=2$, in which case the estimate is order optimal. The method of proof is constructive: we construct for each $Q_m$ a `bad' function $f_m$, that is, a function which vanishes in all nodes of the cubature rule and for which $\\|f_m\\|_{s,r}^{-1} |\\int_{S^r} f_m(\\mathbf{x}) d\\omega_r(\\mathbf{x})| \\geq c_{s,r} m^{-s/r}$. Our proof uses a packing of the sphere $S^r$ with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Kerstin Hesse
Date Deposited: 06 Feb 2012 20:31
Last Modified: 10 Jul 2012 10:50
URI: http://sro.sussex.ac.uk/id/eprint/26326
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