{
  "id": "0805.1618",
  "version": "v1",
  "published": "2008-05-12T12:17:15.000Z",
  "updated": "2008-05-12T12:17:15.000Z",
  "title": "Bernstein operators for exponential polynomials",
  "authors": [
    "J. M. Aldaz",
    "O. Kounchev",
    "H. Render"
  ],
  "comment": "A very similar version is to appear in Constructive Approximation",
  "journal": "Constr. Approx. 29 (2009), no. 3, 345--367",
  "doi": "10.1007/s00365-008-9010-6",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\\lambda_{0},...,\\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b] $ is smaller than $\\pi /M_{n}$, where $M_{n}:=\\max \\left\\{| \\text{Im}% \\lambda_{j}| :j=0,...,n\\right\\} $, then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $% p_{n,k}$ is positive on the open interval $(a,b) .$ Under the additional assumption that $\\lambda_{0}$ and $\\lambda_{1}$ are real and distinct, our first main result states that there exist points $% a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $\\alpha_{0},..,\\alpha_{n}$%, such that the operator \\begin{equation*} B_{n}f:=\\sum_{k=0}^{n}\\alpha_{k}f(t_{k}) p_{n,k}(x) \\end{equation*} satisfies $B_{n}e^{\\lambda_{j}x}=e^{\\lambda_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\\in C[ a,b] $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-05-12T12:17:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bernstein operators",
      "exponential polynomials",
      "first main result states",
      "linear differential operator",
      "second main result"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.1618A"
    }
  }
}