{
  "id": "0804.4613",
  "version": "v1",
  "published": "2008-04-29T14:03:29.000Z",
  "updated": "2008-04-29T14:03:29.000Z",
  "title": "Gabor (Super)Frames with Hermite Functions",
  "authors": [
    "Karlheinz Gröchenig",
    "Yurii Lyubarskii"
  ],
  "journal": "Math. Ann. 345(2) (2009), 267 -- 286",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be the vector of the first $n+1$ Hermite functions. We give a complete characterization of all lattices $\\Lambda \\subseteq \\bR ^2$ such that the Gabor system $\\{e^{2\\pi i \\lambda_2 t} \\boh (t-\\lambda_1): \\lambda = (\\lambda_1, \\lambda_2) \\in \\Lambda \\}$ is a frame for $L^2 (\\bR, \\bC ^{n+1})$. As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass $\\sigma $-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-29T14:03:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "33C90",
      "94A12"
    ],
    "keywords": [
      "vector-valued gabor frames",
      "single hermite function",
      "lower frame bound",
      "gabor superframes",
      "sufficient conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.4613G"
    }
  }
}