Abstract
This paper studies the problem of numerical integration over the unit sphereS2⊆ ℝ3for functions in the Sobolev spaceH3/2(S2). We consider sequencesQm(n),n∈ ℕ, of cubature (or numerical integration) rules, whereQm(n)is assumed to integrate exactly all (spherical) polynomials of degree ≤n, and to usem = m(n)values off. The cubature weights of all rulesQm(n)are assumed to be positive, or alternatively the sequenceQm(n),n∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rulesQm(n),n∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted byE3/2(Qm(n)), for all functions in the unit ball of the Hilbert spaceH3/2(S2) satisfies the estimateE3/2(Qm(n)) ≤c n−3/2, where the constantcis a universal constant for all sequences of positive weight cubature rules. For a sequenceQm(n),n∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constantcmay depend on the sequenceQm(n),n∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.
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