Optimal lower bounds for cubature error on the sphere S2

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-s2, 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))-s2 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|∫S2fm(x)dω(x)|⩾csm-s2. The construction uses results about packings of spherical caps on the sphere.