{
  "id": "1706.09047",
  "version": "v1",
  "published": "2017-06-21T16:19:10.000Z",
  "updated": "2017-06-21T16:19:10.000Z",
  "title": "On harmonic analysis of spherical convolutions on semisimple Lie groups",
  "authors": [
    "Olufemi O. Oyadare"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group $G,$ with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element $e \\in G$ is a weighted Fourier transforms of the Abel transform at $0.$ Being a function on $G,$ the restriction of this integral of its spherical Fourier transforms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite distribution on $G,$ which is tempered and invariant on $G=SL(2,\\mathbb{R}).$ These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-21T16:19:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "22E30",
      "22E46"
    ],
    "keywords": [
      "spherical convolutions",
      "harmonic analysis",
      "spherical functions",
      "connected semisimple lie group",
      "non-zero constant multiple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}