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

  • Authors:

  • Kerstin Hesse

  • Affiliations:

  • School of Mathematics, The University of New South Wales, 2052, Sydney, NSW, Australia

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2006

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Abstract

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(Qm;Hs(Sr)), of an arbitrary m-point cubature rule Qm for functions in the unit ball of the Sobolev space Hs(Sr), where s**, and show that **. The positive constant cs,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 Qm a `bad' function fm, 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.