Numerical Integration over Spheres of Arbitrary Dimension
Numerical Integration over Spheres of Arbitrary Dimension
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任意维球面上的数值积分
DOI：
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发表时间：
2007
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通讯作者：
Kerstin Hesse
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作者：
J. Brauchart;Kerstin Hesse
In this paper we study the worst-case error (of numerical integration) on the unit sphere ${Bbb S}^{d}, dgeq 2,$ for all functions in the unit ball of the Sobolev space ${Bbb H}^s({Bbb S}^d),$ where $s>{d}/{2}.$ More precisely, we consider infinite sequences $(Q_{m(n)})_{nin{Bbb N}}$ 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 $(Q_{m(n)})_{nin{Bbb N}}$ 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 $(Q_{m(n)})_{nin{Bbb N}}).$ 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 $dgeq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{nin{Bbb N}}$ of numerical integration rules satisfies $m(n)={cal O}(n^d)$ an order-optimal rate of convergence is achieved.
In this paper we study the worst-case error (of numerical integration) on the unit sphere ${Bbb S}^{d}, dgeq 2,$ for all functions in the unit ball of the Sobolev space ${Bbb H}^s({Bbb S}^d),$ where $s>{d}/{2}.$ More precisely, we consider infinite sequences $(Q_{m(n)})_{nin{Bbb N}}$ 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 $(Q_{m(n)})_{nin{Bbb N}}$ 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 $(Q_{m(n)})_{nin{Bbb N}}).$ 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 $dgeq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{nin{Bbb N}}$ of numerical integration rules satisfies $m(n)={cal O}(n^d)$ an order-optimal rate of convergence is achieved.