Abstract
This paper is concerned with numerical integration on the unit sphere S r of dimension r≥2 in the Euclidean space ℝr +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.
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Hesse, K. A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numer. Math. 103, 413–433 (2006). https://doi.org/10.1007/s00211-006-0686-x
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DOI: https://doi.org/10.1007/s00211-006-0686-x