Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces

Abstract

We prove that the covering radius of an N-point subset XN of the unit sphere Sd⊂Rd+1 is bounded above by a power of the worst-case error for equal weight cubature 1N∑x∈XNf(x)≈∫Sdfdσd for functions in the Sobolev space Wps(Sd), where σd denotes normalized area measure on Sd. These bounds are close to optimal when s is close to d/p. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for Wps(Sd), which have previously been introduced only in the Hilbert space setting p=2. We say that a sequence (XN) of N-point configurations is a QMC-design sequence for Wps(Sd) with s>d/p provided the worst-case equal weight cubature error for XN has order N−s/d as N→∞, a property that holds, in particular, for a sequence of spherical t-designs in which each design has order td points. For the case p=1, we deduce that any QMC-design sequence (XN) for W1s(Sd) with s>d has the optimal covering property; i.e., the covering radius of XN has order N−1/d as N→∞. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of XN. As a consequence we prove that any QMC-design sequence for Wps(Sd) is also a QMC-design sequence for Wp′s(Sd) for all 1≤p<p′≤∞ and, furthermore, if (XN) is a quasi-uniform QMC-design sequence for Wps(Sd), then it is also a QMC-design sequence for Wps′(Sd) for all s>s′>d/p.