{
  "abstract": "We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator $T$. Suppose that the spectrum of $T$ lies in $\\{ρe^{it}:0<ρ\\le1,\\ |t|\\le c\\}$, where $c<π$, and that its essential spectrum lies in $\\{e^{it}:|t|\\le c_{\\mathrm e}\\}$, where $0\\le c_{\\mathrm e}\\le c$. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set $Λ$, this reduction gives a frame when its one-sided lower uniform density satisfies $D_+^-(Λ)>c_{\\mathrm e}/π$ and its logarithmic block density satisfies $L(Λ)>c/π$. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At $L(Λ)>2c/π$, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length $r$, the sufficient raw-sampling conditions are $rc<π$ and $2rc_{\\mathrm e}<π$. Thus, the full range $rc<π$ holds for raw samples when the essential spectrum is contained in $\\{1\\}$; over the unrestricted class, the sharp universal range is $rc<π/2$. Divided-difference preconditioning restores $rc<π$ for all operators considered and converts exact collisions into logarithmic-derivative data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22523",
  "author_total": 3,
  "authors": [
    {
      "name": "Ilya Krishtal"
    },
    {
      "name": "Javad Mashreghi"
    },
    {
      "name": "Brendan Miller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fe5d58bzkj8n8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22523",
  "fulltext_url": null,
  "id": "fe5d58bzkj8n8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22523",
  "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",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214043053",
  "pdf_url": "https://arxiv.org/pdf/2609.22523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fe5d58bzkj8n8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sectorial Finitely Generated Dynamical Frames",
  "title_text": "Sectorial Finitely Generated Dynamical Frames",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fe5d58bzkj8n8",
  "venue": "arXiv",
  "versions": []
}