{
  "abstract": "We prove two sufficient conditions for the frame property of Gabor systems $\\mathcal{G}(g,a,b)=\\{e^{2πi mb\\,\\cdot}g(\\cdot-ak):k,m\\in\\mathbb{Z} \\}$ generated by polynomial--Gaussian windows $g(t)=P(t)e^{-ct^2}$, where $c>0$ and $P$ is a nonzero complex polynomial of degree $n$. The conditions depend only on the product $ab$ of the lattice parameters and the degree of $P$. First, $\\mathcal{G}(g,a,b)$ is a frame whenever $ab<1$ is irrational. Second, if $ab=p/q<1$, with $p,q\\in\\mathbb{N}$ coprime, then $\\mathcal{G}(g,a,b)$ is a frame whenever $q-p>n$. Consequently, in the region $ab<1$, failure of the frame property can occur only on rational hyperbolas whose reduced denominator and numerator differ by at most $n$. The proof extends the translated-Wronskian method of Faulhuber and Petersen [\\emph{The frame set of the first Hermite function}, arXiv:2609.02610] from the first Hermite function to arbitrary polynomial multiples of a Gaussian, combining a multiplicity estimate for Wronskians with Jensen's formula.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05337",
  "author_total": 1,
  "authors": [
    {
      "name": "Jakob Lemvig"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bzpn4nmnwsxv8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05337",
  "fulltext_url": null,
  "id": "bzpn4nmnwsxv8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05337",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bzpn4nmnwsxv8.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05337"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Gabor frame set of polynomial-Gaussian windows",
  "title_text": "On the Gabor frame set of polynomial-Gaussian windows",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/bzpn4nmnwsxv8",
  "venue": "arXiv",
  "versions": []
}