{
  "id": "1707.08929",
  "version": "v1",
  "published": "2017-07-27T16:49:47.000Z",
  "updated": "2017-07-27T16:49:47.000Z",
  "title": "A Koksma-Hlawka-Potential Identity on the $d$ Dimensional Sphere and its Applications to Discrepancy",
  "authors": [
    "S. B. Damelin"
  ],
  "comment": "Version: 7.27.17",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $d\\geq 2$ be an integer, $S^d\\subset {\\mathbb R}^{d+1}$ the unit sphere and $\\sigma$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the quantity $\\left|\\int_{S^d}fd\\sigma\\right|$, for a class of harmonic functions $f:\\mathbb R^{d+1}\\to \\mathbb R$. Our error estimate involves 2 sided bounds for a Newtonian potential with respect to $\\sigma$ away from its support. In particular, our main result allows us to study quadrature errors, for scatterings on the sphere with given mesh norm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-27T16:49:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K36",
      "65D32",
      "41A35",
      "41A63",
      "26D10",
      "11E12",
      "31A05",
      "31B15",
      "33C55"
    ],
    "keywords": [
      "dimensional sphere",
      "koksma-hlawka-potential identity",
      "applications",
      "discrepancy",
      "error estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}