{
  "id": "1609.05515",
  "version": "v1",
  "published": "2016-09-18T17:09:11.000Z",
  "updated": "2016-09-18T17:09:11.000Z",
  "title": "Best polynomial approximation on the unit ball",
  "authors": [
    "Miguel Pinar",
    "Yuan Xu"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $E_n(f)_\\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\\varpi_\\mu, \\mathbb{B}^d)$, where $\\mathbb{B}^d$ is the unit ball in $\\mathbb{R}^d$ and $\\varpi_\\mu(x) = (1-\\|x\\|^2)^\\mu$ for $\\mu > -1$. Our main result shows that, for $s \\in \\mathbb{N}$, $$ E_n(f)_\\mu \\le c n^{-2s}[E_{n-2s}(\\Delta^s f)_{\\mu+2s} + E_{n}(\\Delta_0^s f)_{\\mu}], $$ where $\\Delta$ and $\\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-18T17:09:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C50",
      "42C10"
    ],
    "keywords": [
      "best polynomial approximation",
      "unit ball",
      "right hand side contains odd",
      "side contains odd order derivatives",
      "hand side contains odd order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}