{
  "id": "1407.5061",
  "version": "v1",
  "published": "2014-07-18T17:20:42.000Z",
  "updated": "2014-07-18T17:20:42.000Z",
  "title": "On the leading coefficient of polynomials orthogonal over domains with corners",
  "authors": [
    "Erwin Miña-Díaz"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\\{p_n\\}_{n=0}^\\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading coefficient $\\lambda_n>0$. N. Stylianopoulos has recently proven that the asymptotic behavior of $\\lambda_n$ as $n\\to\\infty$ is given by \\[ \\frac{n+1}{\\pi}\\frac{\\gamma^{2n+2}}{ \\lambda_n^{2}}=1-\\alpha_n, \\] where $\\alpha_n=O(1/n)$ as $n\\to\\infty$ and $\\gamma$ is the reciprocal of the logarithmic capacity of the boundary $\\partial G$. In this paper, we prove that the $O(1/n)$ estimate for the error term $\\alpha_n$ is, in general, best possible, by exhibiting an example for which \\[ \\liminf_{n\\to\\infty}\\,n\\alpha_n>0. \\] The proof makes use of the Faber polynomials, about which a conjecture is formulated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-18T17:20:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "30E10",
      "30E15",
      "30C10",
      "30C15"
    ],
    "keywords": [
      "leading coefficient",
      "polynomials orthogonal",
      "piecewise analytic jordan curve",
      "error term",
      "faber polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5061M"
    }
  }
}