{
  "id": "2307.09037",
  "version": "v1",
  "published": "2023-07-18T07:46:57.000Z",
  "updated": "2023-07-18T07:46:57.000Z",
  "title": "A Fréchet-Lie group on distributions",
  "authors": [
    "Manon Ryckebusch"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that the following operation, $(f \\star g)(x,y)= \\int_{-\\infty}^{+\\infty} f (x,t) g(t,y) dt$, is well defined on the weak closure of the space of functions that are smooth over a compact of $\\mathbb{R}^2$. We establish that a subset of this weak closure has the structure of a Fr\\'echet space $\\mathcal{D}$ on distributions. Invertible elements of $\\mathcal{D}$ form a dense subset of it and a Fr{\\'e}chet-Lie group for the operation $\\star$. This product generalizes the convolution, Volterra compositions of first and second type and Schwartz's bracket.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-18T07:46:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fréchet-lie group",
      "distributions",
      "weak closure",
      "schwartzs bracket",
      "frechet space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}