{
  "id": "2408.16593",
  "version": "v1",
  "published": "2024-08-29T14:58:01.000Z",
  "updated": "2024-08-29T14:58:01.000Z",
  "title": "Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1\\leq p < q \\leq 2",
  "authors": [
    "Pu-Ting Yu"
  ],
  "comment": "Any comment on this topic or the manuscript would be greatly appreciated",
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper consists of two parts. In the first half, we solve the question raised by Heil as to whether the atom of a Gabor frame must be in $M^p(\\mathbb{R})$ for some $1<p<2$. Specifically, for each $0<\\alpha \\beta \\leq 1$ and $1<q\\leq 2$ we explicitly construct Gabor frames $\\mathcal{G}(g,\\alpha,\\beta)$ with atoms in $M^q(\\mathbb{R})$ but not in $M^{p}(\\mathbb{R})$ for any $1\\leq p<q$. To construct such Gabor frames, we use box functions as the window functions and show that $$f = \\sum_{k,n\\in \\mathbb{Z}} \\langle f,M_{\\beta n}T_{\\alpha k} \\mathcal{F}(\\chi_{[0,\\alpha]})\\rangle M_{\\beta n}T_{\\alpha k} ( \\mathcal{F}(\\chi_{[0,\\alpha]}))$$ holds for $f\\in M^{p,q}(\\mathbb{R})$ with unconditional convergence of the series for any $0<\\alpha\\beta \\leq 1$, $1<p<\\infty$ and $1\\leq q<\\infty$. In the second half of this paper, we study two questions related to unconditional convergence of Gabor expansions in modulation spaces. Under the assumption that the window functions are chosen from $M^p(\\mathbb{R})$ for some $1\\leq p\\leq 2,$ we will prove several equivalent statements that the equation $f = \\sum_{k,n\\in \\mathbb{Z}} \\langle f, M_{\\beta n}T_{\\alpha k} \\gamma \\rangle M_{\\beta n}T_{\\alpha k} g$ can be extended from $L^2(\\mathbb{R})$ to $M^q(\\mathbb{R})$ for all $f\\in M^q(\\mathbb{R})$ and all $p\\leq q\\leq p'$ with unconditional convergence of the series. Finally, we characterize all Gabor systems $\\{M_{\\beta n}T_{\\alpha k}g\\}_{n,k\\in \\mathbb{Z}}$ in $M^{p,q}(\\mathbb{R})$ for any $1\\leq p,q<\\infty$ for which $f = \\sum \\langle f, \\gamma_{k,n} \\rangle M_{\\beta n}T_{\\alpha k} g$ with unconditional convergence of the series for all $f$ in $M^{p,q}(\\mathbb{R})$ and all alternative duals $\\{\\gamma_{k,n}\\}_{k,n\\in \\mathbb{Z}}$ of $\\{M_{\\beta n}T_{\\alpha k} g\\}_{n,k\\in \\mathbb{Z}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-29T14:58:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unconditional convergence",
      "window functions",
      "explicitly construct gabor frames",
      "paper consists",
      "equivalent statements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}