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Numerical Integration over Spheres of Arbitrary Dimension

Authors:	Johann S Brauchart Kerstin Hesse

Institution:	(1) Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA;(2) School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia

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

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