Quasi-Monte Carlo methods can be efficient for integration over products of spheres

We study the worst-case error of quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined on the product of d copies of the unit sphere Ss⊆Rs+1. The space is a tensor product of d reproducing kernel Hilbert spaces defined in terms of uniformly bounded 'weight' parameters γd,j for j=1,2,…,d. We prove that strong QMC tractability holds (i.e. the number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded independently of d) if and only if limsupd→∞∑j=1dγd,j<∞; and tractability holds (i.e. the number of function evaluations grows at most polynomially in d) if and only if limsupd→∞∑j=1dγd,j/log(d+1)<∞. The arguments are not constructive.