{
  "id": "2304.11322",
  "version": "v1",
  "published": "2023-04-22T05:25:33.000Z",
  "updated": "2023-04-22T05:25:33.000Z",
  "title": "On Gabor frames generated by B-splines, totally positive functions, and Hermite functions",
  "authors": [
    "Riya Ghosh",
    "A. Antony Selvan"
  ],
  "comment": "31 pages, 11 figures, 2 tables",
  "categories": [
    "math.FA"
  ],
  "abstract": "The frame set of a window $\\phi\\in L^2(\\mathbb{R})$ is the subset of all lattice parameters $(\\alpha, \\beta)\\in \\mathbb{R}^2_+$ such that $\\mathcal{G}(\\phi,\\alpha,\\beta)=\\{e^{2\\pi i\\beta m\\cdot}\\phi(\\cdot-\\alpha k) : k, m\\in\\mathbb{Z}\\}$ forms a frame for $L^2(\\mathbb{R})$. In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters $\\alpha, \\beta>0$ with $\\alpha\\beta<1,$ there exists a $\\gamma>0$ depending on $\\alpha\\beta$ such that $\\mathcal{G}(\\phi(\\gamma\\cdot),\\alpha,\\beta)$ forms a frame for $L^2(\\mathbb{R})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-22T05:25:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "94A20"
    ],
    "keywords": [
      "totally positive functions",
      "hermite functions",
      "gabor frames",
      "frame set",
      "lattice parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}