A non-asymptotic bound with the sharp power of c for the eigenvalue defect 1 − λ_n(c) of time and frequency limiting
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.
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BibTeX
@misc{mori2026nonasymptotic,
title = {{A non-asymptotic bound with the sharp power of c for the eigenvalue defect 1 − λ_n(c) of time and frequency limiting}},
author = {Ryota Mori},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23092453},
url = {https://doi.org/10.5281/zenodo.23092453},
note = {CC BY},
}
RIS
TY - UNPB
TI - A non-asymptotic bound with the sharp power of c for the eigenvalue defect 1 − λ_n(c) of time and frequency limiting
AU - Ryota Mori
PY - 2026
DA - 2026/10/02
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23092453
UR - https://doi.org/10.5281/zenodo.23092453
AB - 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.
ER -