{
  "id": "1801.01092",
  "version": "v1",
  "published": "2018-01-03T17:41:51.000Z",
  "updated": "2018-01-03T17:41:51.000Z",
  "title": "Rational approximation of $x^n$",
  "authors": [
    "Yuji Nakatsukasa",
    "Lloyd N. Trefethen"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $x^n$ on $[\\kern .3pt 0,1]$ by rational functions of type $(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)} \\sim 2\\kern .3pt H^{k+1/2}$ independently of $n$, where $H \\approx 1/9.28903$ is Halphen's constant. This is the same formula as for minimax approximation of $e^x$ on $(-\\infty,0\\kern .3pt]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-03T17:41:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A20"
    ],
    "keywords": [
      "rational approximation",
      "best supremum norm",
      "appropriate limit",
      "minimax approximation",
      "rational functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}