{
  "abstract": "We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14946",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmadreza Azimifard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xx07yfeeyv89.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14946",
  "fulltext_url": null,
  "id": "0xx07yfeeyv89",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14946",
  "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": "W7213317990",
  "pdf_url": "https://arxiv.org/pdf/2609.14946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xx07yfeeyv89.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization",
  "title_text": "Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0xx07yfeeyv89",
  "venue": "arXiv",
  "versions": []
}