Cubature over the sphere [formula omitted] in Sobolev spaces of arbitrary order

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This paper studies numerical integration (or cubature) over the unit sphere S 2 ⊂ R 3 for functions in arbitrary Sobolev spaces H s ( S 2 ) , s > 1 . We discuss sequences ( Q m ( n ) ) n ∈ N of cubature rules, where (i) the rule Q m ( n ) uses m ( n ) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤ n and (ii) the sequence ( Q m ( n ) ) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Q m ( n ) has positive weights. It is shown that for functions in the unit ball of the Sobolev space H s ( S 2 ) , 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 ∑ ℓ = n + 1 ∞ ( ℓ + 1 2 ) - 2 s + 1 P ℓ ( t ) , where P ℓ is the Legendre polynomial of degree ℓ , in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Q m ( n ) . The order of convergence O ( n - s ) is optimal for sequences ( Q m ( n ) ) of cubature rules with properties (i) and (ii) if Q m ( n ) uses m ( n ) = O ( n 2 ) points.