Optimal lower bounds for cubature error on the sphere S2

Authors:
Kerstin Hesse;Ian H. Sloan
Affiliations:
School of Mathematics, The University of New South Wales, Sydney, NSW, Australia;School of Mathematics, The University of New South Wales, Sydney, NSW, Australia
Venue:
Journal of Complexity
Year:
2005

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Abstract

We show that the worst-case cubature error E(Qm;Hs) of an m-point cubature rule Qm for functions in the unit ball of the Sobolev space Hs = Hs (S2), s 1, has the lower bound E(Qm; Hs) ≥ csm-s/2, where the constant cs is independent of Qm and m. This lower bound result is optimal, since we have established in previous work that there exist sequences (Qm(n))n∈N of cubature rules for which E(Qm(n); Hs)≤c˜s(m(n))-s/2, with a constant c˜s independent of n. The method of proof is constructive: given the cubature rule Qm, we construct explicitly a 'bad' function fm ∈ Hs, which is a function for which Qmfm=0 and ||fm||Hs-1|∫S2 fm(X)dω(X)| ≥ csm-s/2. The construction uses results about packings of spherical caps on the sphere.