Cubature over the sphere S in Sobolev spaces of arbitrary order

This paper studies numerical integration (or cubature) over the unit sphere S2 ⊂ R3 for functions in arbitrary Sobolev spaces Hs(S2), s > 1. We discuss sequences (Qm(n))n∈N of cubature rules, where (i) the rule Qm(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and (ii) the sequence (Qm(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Qm(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space Hs(S2), s > 1, the worst– case cubature error has the order of convergence O(n−s), a result previously known only for the particular case s = 3/2. The crucial step in the extension to general s > 1 is a novel representation of ∑∞ `=n+1(`+ 1 2) −2s+1P`(t), where P` is the Legendre polynomial of degree `, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Qm(n). The order of convergence O(n−s) is optimal for sequences (Qm(n)) of cubature rules with properties (i) and (ii) if Qm(n) uses m(n) = O(n2) points.