{
  "id": "2001.07080",
  "version": "v1",
  "published": "2020-01-20T12:36:37.000Z",
  "updated": "2020-01-20T12:36:37.000Z",
  "title": "Deformations of Gabor frames on the adeles and other locally compact abelian groups",
  "authors": [
    "Ulrik Enstad",
    "Mads S. Jakobsen",
    "Franz Luef",
    "Tron Omland"
  ],
  "comment": "37 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group $G$. More precisely, we show that Gabor frames over lattices in the time-frequency plane of $G$ with windows in the Feichtinger algebra are stable under small deformations of the lattice by an automorphism of ${G}\\times \\widehat{G}$. The topology we use on the automorphisms is the Braconnier topology. We characterize the groups in which the Balian-Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component. This generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform on locally compact abelian groups. We apply our results to a class of number-theoretic groups, including the adele group associated to a global field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-20T12:36:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "11R56",
      "43A70",
      "46E30"
    ],
    "keywords": [
      "locally compact abelian group",
      "uniform gabor frames",
      "noncompact identity component",
      "feichtinger algebra holds",
      "global field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}