{
  "id": "math/0304360",
  "version": "v1",
  "published": "2003-04-23T18:52:20.000Z",
  "updated": "2003-04-23T18:52:20.000Z",
  "title": "Continuous action of Lie groups on $\\mathbb{R}^n$ and Frames",
  "authors": [
    "Gestur Olafsson"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\\R^n)$ using the action of closed subgroups $H\\subset \\mathrm{GL}(n,\\mathbb{R})$ such that $H$ has an open orbit $\\cO$ in $\\R^n$ under the action $(h,\\omega)\\mapsto (h^{-1})^T(\\omega)$. If $H$ has the form $ANR$, where $A$ is simply connected and abelian, $N$ contains a co-compact discrete subgroup and $R$ is compact containing the stabilizer group of $\\omega\\in\\cO$ then we construct a frame for the space $L^2_{\\cO}(\\R^n)$ of $L^2$-functions whose Fourier transform is supported in $\\cO$. We apply this to the case where $H^T=H$ and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in a paper by Fabec and Olafsson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-04-23T18:52:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40",
      "43A85"
    ],
    "keywords": [
      "lie groups",
      "continuous action",
      "co-compact discrete subgroup",
      "fourier transform",
      "stabilizer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......4360O"
    }
  }
}