{
  "id": "1312.1135",
  "version": "v2",
  "published": "2013-12-04T12:22:50.000Z",
  "updated": "2014-03-12T00:41:22.000Z",
  "title": "Numerical integration of Hölder continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series",
  "authors": [
    "Josef Dick"
  ],
  "comment": "Added literature review and tractability discussion; Minor corrections",
  "categories": [
    "math.NA"
  ],
  "abstract": "We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \\ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable and $f$ satisfies a H\\\"older condition of order $\\alpha$, for some $0 < \\alpha \\le 1$. We show a convergent rate of the integration error of order $\\max((s-1) N^{-1/2}, s^{\\alpha/2} N^{-\\alpha} )$. The construction of the quadrature points is explicit and is based on Weil sums.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-12T00:41:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D30",
      "65D32",
      "65C05",
      "65C10"
    ],
    "keywords": [
      "numerical integration",
      "walsh series",
      "absolutely convergent",
      "hölder continuous",
      "quasi-monte carlo rules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.1135D"
    }
  }
}