Equidistribution on the Sphere
SIAM Journal on Scientific Computing
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Uniform distribution, discrepancy, and reproducing kernel Hilbert spaces
Journal of Complexity
A lower bound for the worst-case cubature error on spheres of arbitrary dimension
Numerische Mathematik
LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres
Journal of Approximation Theory
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Quantifying uniformity of a configuration of points on the sphere is an interesting topic that is receiving growing attention in numerical analysis. An elegant solution has been provided by Cui and Freeden [J. Cui, W. Freeden, Equidistribution on the sphere, SIAM J. Sci. Comput. 18 (2) (1997) 595-609], where a class of discrepancies, called generalized discrepancies and originally associated with pseudodifferential operators on the unit sphere in R^3, has been introduced. The objective of this paper is to extend to the sphere of arbitrary dimension this class of discrepancies and to study their numerical properties. First we show that generalized discrepancies are diaphonies on the hypersphere. This allows us to completely characterize the sequences of points for which convergence to zero of these discrepancies takes place. Then we discuss the worst-case error of quadrature rules and we derive a result on tractability of multivariate integration on the hypersphere. At last we provide several versions of Koksma-Hlawka type inequalities for integration of functions defined on the sphere.