Worst-case errors in a Sobolev space setting for cubature over the sphere $S^2$

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