A lower bound for the worst-case cubature error on spheres of arbitrary dimension

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• 作者：Kerstin Hesse
• 关键词：Mathematics Subject Classification (1991) Primary 41A55 ; Secondary 46B70 ; 46E22 ; 46E35 ; 52C17 ; 65D30 ; 65D32
• 刊名：Numerische Mathematik
• 出版年：2006
• 出版时间：May 2006
• 年：2006
• 卷：103
• 期：3
• 页码：413-433
• 全文大小：235 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

This paper is concerned with numerical integration on the unit sphere Sr of dimension r≥2 in the Euclidean space ℝr+1. We consider the worst-case cubature error, denoted by E(Q_m;Hs(Sr)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space Hs(Sr), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space Hs(Sr). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere Sr with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.