{
  "id": "1703.04243",
  "version": "v1",
  "published": "2017-03-13T04:33:27.000Z",
  "updated": "2017-03-13T04:33:27.000Z",
  "title": "Jacobi polynomials on the Bernstein ellipse",
  "authors": [
    "Haiyong Wang",
    "Lun Zhang"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NA",
    "math.CA"
  ],
  "abstract": "In this paper, we are concerned with Jacobi polynomials $P_n^{(\\alpha,\\beta)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(\\alpha,\\beta)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\\left|P_n^{(\\alpha,\\beta)}(z)\\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $\\alpha+\\beta\\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $\\alpha=\\beta$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-13T04:33:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N35",
      "65D05",
      "41A05",
      "41A25"
    ],
    "keywords": [
      "bernstein ellipse",
      "jacobi polynomials",
      "minimum value",
      "spectral interpolation",
      "maximum value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}