Cubature over the sphere S2 in Sobolev spaces of arbitrary order

Hesse, Kerstin and Sloan, Ian H (2006) Cubature over the sphere S2 in Sobolev spaces of arbitrary order. Journal of Approximation Theory, 141 (2). pp. 118-133. ISSN 0021-9045

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This paper studies numerical integration (or cubature) over the unit sphere $S^2\\subset\\mathbb{R}^3$ for functions in arbitrary Sobolev spaces $H^s(S^2)$, $s>1$. We discuss sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of cubature rules, where (i) the rule $Q_{m(n)}$ uses $m(n)$ points and is assumed to integrate exactly all (spherical) polynomials of degree $\\leq n$, and (ii) the sequence $(Q_{m(n)})$ satisfies a certain local regularity property. This local regularity property is automatically satisfied if each $Q_{m(n)}$ has positive weights. It is shown that for functions in the unit ball of the Sobolev space $H^s(S^2)$, $s>1$, the worst-case cubature error has the order of convergence $O(n^{-s})$, a result previously known only for the particular case $s=3/2$. The crucial step in the extension to general $s>1$ is a novel representation of $\\sum_{\\ell=n+1}^\\infty (\\ell+\\frac{1}{2})^{-2s+1} P_\\ell(t)$, where $P_\\ell$ is the Legendre polynomial of degree $\\ell$, in which the dominant term is a polynomial of degree $n$, which is therefore integrated exactly by the rule $Q_{m(n)}$. The order of convergence $O(n^{-s})$ is optimal for sequences $(Q_{m(n)})$ of cubature rules with properties (i) and (ii) if $Q_{m(n)}$ uses $m(n)=O(n^2)$ points.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Kerstin Hesse
Date Deposited: 06 Feb 2012 19:07
Last Modified: 03 Apr 2012 15:03
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