{
  "id": "1905.06827",
  "version": "v1",
  "published": "2019-05-16T15:08:28.000Z",
  "updated": "2019-05-16T15:08:28.000Z",
  "title": "The Balian-Low theorem for locally compact abelian groups and vector bundles",
  "authors": [
    "Ulrik Enstad"
  ],
  "comment": "40 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Let $\\Lambda$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $\\Lambda^{\\perp} \\subseteq \\widehat{G}$. We investigate the validity of the following statement: For every $\\eta$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\\{ M_{\\tau} T_{\\lambda} \\eta \\}_{\\lambda \\in \\Lambda, \\tau \\in \\Lambda^{\\perp}}$ is not a frame for $L^2(G)$. When $G = \\mathbb{R}$ and $\\Lambda = \\alpha \\mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\\Lambda)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\\Lambda) \\times (\\widehat{G}/\\Lambda^{\\perp})$. We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert $C^*$-modules. As an application, we prove a new Balian-Low theorem for the group $\\mathbb{R} \\times \\mathbb{Q}_p$, where $\\mathbb{Q}_p$ denotes the $p$-adic numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-16T15:08:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally compact abelian group",
      "balian-low theorem",
      "vector bundle",
      "adic numbers",
      "statement generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}