{
  "id": "0911.5407",
  "version": "v1",
  "published": "2009-11-28T17:18:54.000Z",
  "updated": "2009-11-28T17:18:54.000Z",
  "title": "Asymptotic behavior and zero distribution of Carleman orthogonal polynomials",
  "authors": [
    "Peter Dragnev",
    "Erwin Miña-Díaz"
  ],
  "comment": "23 pages, 4 figures",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $L$ be an analytic Jordan curve and let $\\{p_n(z)\\}_{n=0}^\\infty$ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of $L$. A well-known result of Carleman states that \\label{eq12} \\lim_{n\\to\\infty}\\frac{p_n(z)}{\\sqrt{(n+1)/\\pi} [\\phi(z)]^{n}}= \\phi'(z) locally uniformly on certain open neighborhood of the closed exterior of $L$, where $\\phi$ is the canonical conformal map of the exterior of $L$ onto the exterior of the unit circle. In this paper we extend the validity of (\\ref{eq12}) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the $p_n$'s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-28T17:18:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "30E10",
      "30E15",
      "30C10",
      "30C15"
    ],
    "keywords": [
      "carleman orthogonal polynomials",
      "zero distribution",
      "asymptotic behavior",
      "analytic jordan curve",
      "maximal open set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.5407D"
    }
  }
}