{
  "id": "1210.5102",
  "version": "v3",
  "published": "2012-10-18T11:44:36.000Z",
  "updated": "2014-11-02T15:21:42.000Z",
  "title": "Composition in ultradifferentiable classes",
  "authors": [
    "Armin Rainer",
    "Gerhard Schindl"
  ],
  "comment": "28 pages, mistake in Lemma 2.9 and ramifications corrected, Theorem 6.3 improved; to appear in Studia Math",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $\\omega$, and, more generally, by weight matrices $\\mathfrak{M}$, and investigate continuity of composition $(g,f) \\mapsto f \\circ g$. In addition, we represent the Beurling space $\\mathcal{E}^{(\\omega)}$ and the Roumieu space $\\mathcal{E}^{\\{\\omega\\}}$ as intersection and union of spaces $\\mathcal{E}^{(M)}$ and $\\mathcal{E}^{\\{M\\}}$ for associated weight sequences, respectively.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-25T12:25:46.000Z",
      "abstract": "We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $\\omega$, and, more generally, by weight matrices $\\mathfrak{M}$, and investigate continuity of composition $(g,f) \\mapsto f \\circ g$. Thereby we represent the Beurling space $\\mathcal{E}^{(\\omega)}$ and the Roumieu space $\\mathcal{E}^{\\{\\omega\\}}$ as intersection and union of spaces $\\mathcal{E}^{(M)}$ and $\\mathcal{E}^{\\{M\\}}$ for associated weight sequences, respectively.",
      "comment": "28 pages, mistake in Lemma 2.9 and ramifications corrected, Theorem 6.3 improved",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-11-02T15:21:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E10",
      "30D60",
      "46E10",
      "47B33"
    ],
    "keywords": [
      "ultradifferentiable classes",
      "composition",
      "associated weight sequences",
      "weight functions",
      "roumieu space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.5102R"
    }
  }
}