{
  "id": "1407.8311",
  "version": "v1",
  "published": "2014-07-31T08:37:28.000Z",
  "updated": "2014-07-31T08:37:28.000Z",
  "title": "Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces",
  "authors": [
    "Johann S. Brauchart",
    "Josef Dick",
    "Edward B. Saff",
    "Ian H. Sloan",
    "Yu Guang Wang",
    "Robert S. Womersley"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "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 $s > s^\\prime > d/p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-31T08:37:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D30",
      "65D32",
      "52C17",
      "41A55"
    ],
    "keywords": [
      "qmc design sequence",
      "worst-case error",
      "equal weight cubature",
      "sobolev space",
      "spherical caps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.8311B"
    }
  }
}