{
  "id": "1403.6927",
  "version": "v2",
  "published": "2014-03-27T06:43:55.000Z",
  "updated": "2014-03-28T00:50:32.000Z",
  "title": "On a Theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials",
  "authors": [
    "Vanessa G. Paschoa",
    "Dilcia Pérez",
    "Yamilet Quintana"
  ],
  "comment": "2 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\{Q^{(\\alpha)}_{n,\\lambda}\\}_{n\\geq 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$\\langle f,g\\rangle_{S}:=\\int_{-1}^{1}f(x)g(x)(1-x^{2})^{\\alpha-\\frac{1}{2}}dx+\\lambda \\int_{-1}^{1}f'(x)g'(x)(1-x^{2})^{\\alpha-\\frac{1}{2}} dx,$$ where $\\alpha>-\\frac{1}{2}$ and $\\lambda\\geq 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov \\cite{BN2010}, in order to study the maximization of a local extremum of the $k$th derivative $\\frac{d^k}{dx^k}Q^{(\\alpha)}_{n,\\lambda}$ in $[-M_{n,\\lambda}, M_{n,\\lambda}]$, where $M_{n,\\lambda}$ is a suitable value such that all zeros of the polynomial $Q^{(\\alpha)}_{n,\\lambda}$ are contained in $[-M_{n,\\lambda}, M_{n,\\lambda}]$ and the function $\\left|Q^{(\\alpha)}_{n,\\lambda}\\right|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-28T00:50:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "41A17"
    ],
    "keywords": [
      "gegenbauer-sobolev polynomials",
      "monic orthogonal polynomials",
      "gegenbauer-sobolev inner product",
      "local extremum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.6927P"
    }
  }
}