{
  "id": "1001.0509",
  "version": "v3",
  "published": "2010-01-04T13:31:16.000Z",
  "updated": "2010-10-05T17:14:45.000Z",
  "title": "Approximation error of the Lagrange reconstructing polynomial",
  "authors": [
    "G. A. Gerolymos"
  ],
  "comment": "31 pages, 1 table; revised version to appear in J. Approx. Theory",
  "journal": "J. Approx. Theory 163 (2011) 267-305",
  "doi": "10.1016/j.jat.2010.09.007",
  "categories": [
    "math.NA",
    "physics.comp-ph"
  ],
  "abstract": "The reconstruction approach [Shu C.W.: {\\em SIAM Rev.} {\\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\\tfrac{1}{2}\\Delta x,x+\\tfrac{1}{2}\\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\\em J. Comp. Phys.} {\\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\\tau(x)$, related with the reconstruction pair of ${\\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-10-05T17:14:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D99",
      "65D05",
      "65D25",
      "65M06",
      "65M08"
    ],
    "keywords": [
      "lagrange reconstructing polynomial",
      "approximation error",
      "homogeneous grid",
      "sliding averages",
      "introducing rational numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}