Worst-case errors in a Sobolev space setting for cubature over the sphere S2

Abstract

This paper studies the problem of numerical integration over the unit sphere S2 ⊆ ℝ3 for functions in the Sobolev space H3/2(S2). We consider sequences Qm(n), n ∈ ℕ, of cubature (or numerical integration) rules, where Qm(n) is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and to use m = m(n) values of f. The cubature weights of all rules Qm(n) are assumed to be positive, or alternatively the sequence Qm(n), n ∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rules Qm(n), n ∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted by E3/2 (Qm(n)), for all functions in the unit ball of the Hilbert space H3/2 (S2) satisfies the estimate E3/2 (Qm(n)) ≤ c n−3/2, where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Qm(n), n ∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Qm(n), n ∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.