{
  "id": "1203.1509",
  "version": "v1",
  "published": "2012-03-07T15:56:10.000Z",
  "updated": "2012-03-07T15:56:10.000Z",
  "title": "Zak Transform for Semidirect Product of Locally Compact Groups",
  "authors": [
    "Arash Ghaani Farashahi",
    "Ali Akbar Arefijamaal"
  ],
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $H$ be a locally compact group and $K$ be an LCA group also let $\\tau:H\\to Aut(K)$ be a continuous homomorphism and $G_\\tau=H\\ltimes_\\tau K$ be the semidirect product of $H$ and $K$ with respect to $\\tau$. In this article we define the Zak transform $\\mathcal{Z}_L$ on $L^2(G_\\tau)$ with respect to a $\\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\\mathrm{SL}(2,\\mathbb{Z})\\ltimes_\\tau\\mathbb{R}^2$ and also the Weyl-Heisenberg groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-07T15:56:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally compact group",
      "semidirect product group",
      "zak transform satisfies",
      "invariant uniform lattice",
      "plancherel formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.1509G"
    }
  }
}