{
  "id": "1406.5456",
  "version": "v1",
  "published": "2014-06-20T16:31:30.000Z",
  "updated": "2014-06-20T16:31:30.000Z",
  "title": "Extremal functions in de Branges and Euclidean spaces",
  "authors": [
    "Emanuel Carneiro",
    "Friedrich Littmann"
  ],
  "journal": "Adv. Math. 260 (2014), 281-349",
  "doi": "10.1016/j.aim.2014.04.007",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on $\\mathbb{R}^N$. These extremal functions minimize the $L^1(\\mathbb{R}^N, |x|^{2\\nu + 2 - N}dx)$-distance to the original function, where $\\nu >-1$ is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function $\\mathcal{F}_{\\lambda}(x) = e^{-\\lambda|x|}$, where $\\lambda >0$, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter $\\lambda >0$ against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere $\\mathbb{S}^{N-1}$ with an axis of symmetry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-20T16:31:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A30",
      "46E22",
      "41A05",
      "41A63"
    ],
    "keywords": [
      "extremal functions",
      "euclidean spaces",
      "extremal one-sided approximations",
      "radial functions",
      "exponential function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5456C"
    }
  }
}