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

Johann S. Brauchart, Josef Dick, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, Robert S. Womersley

Published 2014-07-31Version 1

We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$ is bounded above by a power of the worst-case error for equal-weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx \int_{\mathbb{S}^d} f \, \mathrm{d} \sigma_d$ for functions in the Sobolev space $\mathbb{W}_p^s(\mathbb{S}^d)$, where $\sigma_d$ denotes normalized area measure on $\mathbb{S}^d.$ 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 QMC design sequences for $\mathbb{W}_p^s(\mathbb{S}^d)$, 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 $\mathbb{W}_p^s(\mathbb{S}^d)$ with $s > d/p$ provided the worst-case cubature error for $X_N$ has order $N^{-s/d}$ as $N \to \infty$, a property that holds, in particular, for optimal order spherical design sequences. For the case $p = 1$, we deduce that any QMC design sequence $(X_N)$ for $\mathbb{W}_1^s(\mathbb{S}^d)$ 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 $\mathbb{W}_p^s(\mathbb{S}^d)$ is also a QMC design sequence for $\mathbb{W}_{p^\prime}^s(\mathbb{S}^d)$ for all $1 \leq p < p^\prime \leq \infty$ and, furthermore, if $(X_N)$ is a quasi-uniform QMC design sequence for $\mathbb{W}_p^s(\mathbb{S}^d)$, then it is also a QMC design sequence for $\mathbb{W}_p^{s^\prime}(\mathbb{S}^d)$ for all $s > s^\prime > d/p$.