{
  "id": "0809.4050",
  "version": "v1",
  "published": "2008-09-23T23:29:14.000Z",
  "updated": "2008-09-23T23:29:14.000Z",
  "title": "Some extremal functions in Fourier analysis, II",
  "authors": [
    "Emanuel Carneiro",
    "Jeffrey D. Vaaler"
  ],
  "comment": "40 pages. Accepted for publication in Trans. Amer. Math. Soc",
  "journal": "Transactions of the American Mathematical Society, v. 362, p. 5803-5803, 2010",
  "doi": "10.1090/S0002-9947-2010-04886-X",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We obtain extremal majorants and minorants of exponential type for a class of even functions on $\\R$ which includes $\\log |x|$ and $|x|^\\alpha$, where $-1 < \\alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erd\\\"{o}s-Tur\\'{a}n-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-23T23:29:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A30",
      "41A52",
      "42A05"
    ],
    "keywords": [
      "fourier analysis",
      "extremal functions",
      "dimensional hardy-littlewood-sobolev inequalities",
      "exponential type",
      "periodic versions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.4050C"
    }
  }
}