{
  "id": "1003.0689",
  "version": "v2",
  "published": "2010-03-02T21:37:19.000Z",
  "updated": "2010-12-19T15:56:05.000Z",
  "title": "On the Clifford-Fourier transform",
  "authors": [
    "H. De Bie",
    "Y. Xu"
  ],
  "comment": "Some small changes, 30 pages, accepted for publication in IMRN",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) := e^{\\frac{i \\pi}{2} \\Gamma_{y}}e^{-i <x,y>}$, replacing the kernel $e^{i <x,y>}$ of the ordinary Fourier transform, where $\\Gamma_{y} := - \\sum_{j<k} e_{j}e_{k} (y_{j} \\partial_{y_{k}} - y_{k}\\partial_{y_{j}})$. An explicit formula of $K(x,y)$ is derived, which can be further simplified to a finite sum of Bessel functions when $m$ is even. The closed formula of the kernel allows us to study the Clifford-Fourier transform and prove the inversion formula, for which a generalized translation operator and a convolution are defined and used.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-19T15:56:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30G35",
      "42B10"
    ],
    "keywords": [
      "clifford-fourier transform",
      "ordinary fourier transform",
      "kernel function",
      "explicit formula",
      "clifford algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.0689D"
    }
  }
}