{
  "id": "1311.1157",
  "version": "v1",
  "published": "2013-11-05T18:50:41.000Z",
  "updated": "2013-11-05T18:50:41.000Z",
  "title": "Extremal functions with vanishing condition",
  "authors": [
    "Friedrich Littmann",
    "Mark Spanier"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We determine the optimal majorant $M^+$ and minorant $M^-$ of exponential type for the truncation of $x\\mapsto (x^2+a^2)^{-1}$ with respect to general de Branges measures. We prove that \\[ \\int_\\mathbb{R} (M^+ - M^-) |E(x)|^{-2}dx = \\frac{1}{a^2 K(0,0)} \\] where $K$ is the reproducing kernel for $\\mathcal{H}(E)$. As an application we determine the optimal majorant and minorant for the Heaviside function that vanish at a fixed point $\\alpha = ia$ on the imaginary axis. We show that the difference of majorant and minorant has integral value $(\\pi a - \\tanh(\\pi a))^{-1} \\pi a$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-05T18:50:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A30",
      "41A52",
      "41A05",
      "41A44",
      "42A82"
    ],
    "keywords": [
      "extremal functions",
      "vanishing condition",
      "optimal majorant",
      "branges measures",
      "heaviside function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.1157L"
    }
  }
}