{
  "id": "1203.2445",
  "version": "v3",
  "published": "2012-03-12T10:14:05.000Z",
  "updated": "2012-07-31T12:45:38.000Z",
  "title": "On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity",
  "authors": [
    "Shuhuang Xiang",
    "Folkmar Bornemann"
  ],
  "comment": "7 pages, the figure of the revision has an unsymmetric example, to appear in SIAM J. Numer. Anal",
  "journal": "SIAM J. Numer. Anal. 50 (2012) 2581-2587",
  "categories": [
    "math.NA"
  ],
  "abstract": "We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw-Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n^{-s-1}) for some s > 0, Clenshaw-Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-07-31T12:45:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D32",
      "41A25",
      "41A55"
    ],
    "keywords": [
      "convergence rate",
      "limited regularity",
      "clenshaw-curtis quadrature",
      "polynomial best approximation",
      "gauss quadrature inherit"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.2445X"
    }
  }
}