{
  "abstract": "We study the Gabor frame properties of the rational window $$ g(x)=\\frac{x^2-1}{(x^2+1)(x^2+4)(x^2+9)}, $$ whose poles occur in symmetric pairs. We prove that every lattice satisfying $0<αβ<1/3$ generates a frame, and we establish an additional frame region for $β\\ge1$ and $1/3\\le αβ<0.47373$. Furthermore, we show that the rational hyperbolas $αβ=\\frac{p}{3p-1}$, for $p\\ge2,$ are entirely contained within the frame set. In contrast, we construct explicit non-frame lattice points on the hyperbolas $$αβ\\in\\left\\{\\frac{1}{3},\\frac{1}{2},\\frac{2}{3},\\frac{3}{4}\\right\\}.$$ Finally, for the density family $αβ=p/(p+1)$, we derive a symmetry reduction of the associated Zibulski-Zeevi matrix, providing numerical evidence for a richer structure of non-frame obstructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19232",
  "author_total": 1,
  "authors": [
    {
      "name": "Riya Ghosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3j6axsj89g7pq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19232",
  "fulltext_url": null,
  "id": "3j6axsj89g7pq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19232",
  "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/2609.19232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3j6axsj89g7pq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window",
  "title_text": "Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/3j6axsj89g7pq",
  "venue": "arXiv",
  "versions": []
}