{
  "id": "1909.03945",
  "version": "v1",
  "published": "2019-09-09T15:49:12.000Z",
  "updated": "2019-09-09T15:49:12.000Z",
  "title": "The Fourier transform of thick distributions",
  "authors": [
    "Ricardo Estrada",
    "Jasson Vindas",
    "Yunyun Yang"
  ],
  "comment": "21 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We first construct a space of test functions $\\mathcal{W}\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) $ defined in $\\mathbb{R}_{\\text{c}} ^{n}=\\mathbb{R}^{n}\\cup\\left\\{ \\mathbf{\\infty}\\right\\} ,$ the one point compactification of $\\mathbb{R}^{n},$ that have a thick type behavior at infinity of special logarithmic type and its dual space $\\mathcal{W}^{\\prime }\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) ,$ the space of $sl-$thick distributions. We show that there is a canonical projection of $\\mathcal{W} ^{\\prime}\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) $ onto $\\mathcal{S} ^{\\prime}\\left( \\mathbb{R}^{n}\\right) .$ We study several $sl-$thick distributions and consider operations in $\\mathcal{W}^{\\prime}\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) .$ We define and study the Fourier transform of thick test functions of $\\mathcal{S}_{\\ast}\\left( \\mathbb{R}^{n}\\right) $ and thick tempered functions of $\\mathcal{S}_{\\ast}^{\\prime}\\left( \\mathbb{R}^{n}\\right) .$ We construct isomorphisms \\[ \\mathcal{F}_{\\ast}:\\mathcal{S}_{\\ast}^{\\prime}\\left( \\mathbb{R}^{n}\\right) \\longrightarrow\\mathcal{W}^{\\prime}\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) \\,, \\] \\[ \\mathcal{F}^{\\ast}:\\mathcal{W}^{\\prime}\\left( \\mathbb{R}_{\\text{c}} ^{n}\\right) \\longrightarrow\\mathcal{S}_{\\ast}^{\\prime}\\left( \\mathbb{R} ^{n}\\right) \\,, \\] that extend the Fourier transform of tempered distributions, namely, $\\Pi\\mathcal{F}_{\\ast}=\\mathcal{F}\\Pi$ and $\\Pi\\mathcal{F}^{\\ast} =\\mathcal{F}\\Pi,$ where $\\Pi$ are the canonical projections of $\\mathcal{S} _{\\ast}^{\\prime}\\left( \\mathbb{R}^{n}\\right) $ or $\\mathcal{W}^{\\prime }\\left( \\mathbb{R}_{\\text{c}}^{n}\\right) $ onto $\\mathcal{S}^{\\prime}\\left( \\mathbb{R}^{n}\\right) .$ We find the Fourier transform of several finite part regularizations and of general thick delta functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-09T15:49:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46F10",
      "42B10"
    ],
    "keywords": [
      "fourier transform",
      "thick distributions",
      "general thick delta functions",
      "thick type behavior",
      "special logarithmic type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}