{
  "abstract": "Let λ_0(c) > λ_1(c) > ⋯ be the eigenvalues of the time and frequency limiting operator with kernel sin c(x−y)/(π(x−y)) on [−1,1]. Fuchs' asymptotic formula 1 − λ_n(c) ~ 4√π 8^n c^{n+1/2} e^{−2c}/n! describes the defect of the top eigenvalues as c → ∞. Rigorous non-asymptotic bounds in the literature give the decay rate e^{−c}. We prove that for 0 ≤ n ≤ 4 and every c ≥ 10π, 1 − λ_n(c) ≤ C_n c^{n+1/2} e^{−2c}, with explicit constants C_n. The exponent e^{−2c} and the power c^{n+1/2} are those of Fuchs' formula. The proof combines Slepian's identity ∂_c log λ_n = 2ψ_n(1)²/c with a sharp upper bound for the mass of the band-limited extension of ψ_n outside [−1,1]: this mass is at most (ψ_n(1)²/c)(2/π)K(χ_n/c²)(1+15/c²), where K is the complete elliptic integral. The bound comes from a Bessel-type Liouville transformation at the regular singular point t = 1, in which the first-order error cancels. All constants are established with ball arithmetic. In a companion paper, the case n = 4 is used to bound the bottom of windowed Weil quadratic forms, which are related to the Riemann Hypothesis through Weil's criterion. - **Notes:** Written with extensive use of generative AI systems (Anthropic Claude, Google Gemini, OpenAI GPT) under the direction of the author; see the section \"Use of generative AI\" in the paper. Not yet reviewed by independent human experts. **Reserved DOIs (2026-10-02):** paper I 10.5281/zenodo.23092454, paper II 10.5281/zenodo.23092542. Final PDFs: release v1.0.1 of prolate-weil-bounds (paper-I/main.pdf md5 e3ebfa3387f6e59fd5946903aa0e2d09, paper-II/main.pdf md5 b1d41435d6eb19c36359a879ac70045a). Version 2 (10 October 2026): states how the value of the integral of K_0 follows from DLMF 10.32.9. Theorems, proofs and constants unchanged.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Ryota Mori"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2f9ve6b70d6yp.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.23092453",
  "fulltext_url": null,
  "id": "2f9ve6b70d6yp",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23092453",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7216283606",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/2f9ve6b70d6yp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7216283606",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23092453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical Analysis and Transform Methods",
      "subfield": "general"
    }
  ],
  "title": "A non-asymptotic bound with the sharp power of c for the eigenvalue defect 1 − λ_n(c) of time and frequency limiting",
  "title_text": "A non-asymptotic bound with the sharp power of c for the eigenvalue defect 1 − λ_n(c) of time and frequency limiting",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/2f9ve6b70d6yp",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}