# Optimal lower bounds for cubature error on the sphere S2

Hesse, Kerstin and Sloan, Ian H (2005) Optimal lower bounds for cubature error on the sphere S2. Journal of Complexity, 21 (6). pp. 790-803. ISSN 0885-064X

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## Abstract

We show that the worst-case cubature error $E(Q_m;H^s)$ of an $m$-point cubature rule $Q_m$ for functions in the unit ball of the Sobolev space $H^s=H^s(S^2)$, $s&gt;1$, has the lower bound $E(Q_m;H^s) \\geq c_s m^{-s/2}$, where the constant $c_s$ is independent of $Q_m$ and $m$. This lower bound result is optimal, since we have established in previous work that there exist sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of cubature rules for which $E(Q_{m(n)};H^s) \\leq \\tilde{c}_s (m(n))^{s/2}$ with a constant $\\tilde{c}_s$ independent of $n$. The method of proof is constructive: given the cubature rule $Q_m$, we construct explicitly a bad function $f_m\\in H^s$, which is a function for which $Q_m f_m=0$ and $\\|f_m\\|_{H^s}^{-1} | \\int_{S^2} f_m(\\mathbf{x}) d\\omega(\\mathbf{x})| \\geq c_s m^{-s/2}$. The construction uses results about packings of spherical caps on the sphere.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Kerstin Hesse 06 Feb 2012 20:04 04 Apr 2012 11:50 http://sro.sussex.ac.uk/id/eprint/23919