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 $X_N$ of the unit sphere $S^d\subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}{\sum }_{\mathbf{x}\in X_N}f(\mathbf{x})\approx {\int }_{S^d}f\mathrm{d}{\sigma }_d$ for functions in the Sobolev space ${\mathbb{W}}_p^s(S^d)$, where ${\sigma }_d$ denotes normalized area measure on $S^d.$ 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 ${\mathbb{W}}_p^s(S^d)$, which have previously been introduced only in the Hilbert space setting $p=2$. We say that a sequence $(X_N)$ of $N$-point configurations is a QMC-design sequence for ${\mathbb{W}}_p^s(S^d)$ with $s>d/p$ provided the worst-case equal weight cubature error for $X_N$ has order $N^{-s/d}$ as $N\to \infty$, a property that holds, in particular, for a sequence of spherical $t$-designs in which each design has order $t^d$ points. For the case $p=1$, we deduce that any QMC-design sequence $(X_N)$ for ${\mathbb{W}}_1^s(S^d)$ with $s>d$ has the optimal covering property; i.e., the covering radius of $X_N$ has order $N^{-1/d}$ as $N\to \infty$. 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 $X_N$. As a consequence we prove that any QMC-design sequence for ${\mathbb{W}}_p^s(S^d)$ is also a QMC-design sequence for ${\mathbb{W}}_{p^{\prime }}^s(S^d)$ for all $1\le p<p^{\prime }\le \infty$ and, furthermore, if $(X_N)$ is a quasi-uniform QMC-design sequence for ${\mathbb{W}}_p^s(S^d)$, then it is also a QMC-design sequence for ${\mathbb{W}}_p^{s^{\prime }}(S^d)$ for all $s>s^{\prime }>d/p$.

32 pages, with changes of accepted paper, added supporting organization

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