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

Abstract

This paper is concerned with numerical integration on the unit sphere S ^r of dimension r≥2 in the Euclidean space ℝ^{r +1}. We consider the worst-case cubature error, denoted by E(Q _m ;H ^s(S ^r)), of an arbitrary m-point cubature rule Q _m for functions in the unit ball of the Sobolev space H ^s(S ^r), 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 H ^s(S ^r). 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 S ^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.