{
  "id": "0906.2302",
  "version": "v2",
  "published": "2009-06-12T11:31:52.000Z",
  "updated": "2010-01-20T09:46:36.000Z",
  "title": "From exact systems to Riesz bases in the Balian-Low theorem",
  "authors": [
    "Shahaf Nitzan",
    "Jan-Fredrik Olsen"
  ],
  "comment": "v1: 16 pages. v2: 35 pages. Reorganized the presentation, corrected some typos, and added some additional results. In particular, added proof that main theorem is sharp",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-01-20T09:46:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "42A38",
      "46E35"
    ],
    "keywords": [
      "riesz bases",
      "exact systems",
      "establish balian-low type theorems",
      "lattice gabor systems",
      "localisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.2302N"
    }
  }
}