# Numerical integration over spheres of arbitrary dimension

Brauchart, Johann S and Hesse, Kerstin (2007) Numerical integration over spheres of arbitrary dimension. Constructive Approximation, 25 (1). pp. 41-71. ISSN 0176-4276

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## Abstract

In this paper, we study the worst-case error (of numerical integration) on the unit sphere $\\mathbb{S}^{d}$, $d\\geq 2$, for all functions in the unit ball of the Sobolev space $\\mathbb{H}^s(\\mathbb{S}^d)$, where $s&gt;d/2$. More precisely, we consider infinite sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of $m(n)$-point numerical integration rules $Q_{m(n)}$, where (i) $Q_{m(n)}$ is exact for all spherical polynomials of degree $\\leq n$, and (ii) $Q_{m(n)}$ has positive weights or, alternatively to (ii), the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) $E(Q_{m(n)};\\mathbb{H}^s(\\matbb{S}^d))$ in $\\mathbb{H}^s(\\mathbb{S}^d)$ has the upper bound $c n^{-s}$, where the constant $c$ depends on $s$ and $d$ (and possibly the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$). This extends the recent results for the sphere $\\mathbb{S}^2$ by K.Hesse and I.H.Sloan to spheres $\\mathbb{S}^d$ of arbitrary dimension $d\\geq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of numerical integration rules satisfies $m(n)=\\mathcal{O}(n^d)$ an order-optimal rate of convergence is achieved.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Kerstin Hesse 06 Feb 2012 19:40 04 Apr 2012 10:27 http://sro.sussex.ac.uk/id/eprint/21705