Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators
Abstract
The plunge region records the spectral transition of a time-frequency localization operator. Let A0,B0⊂ℝ be bounded measurable sets of positive measure with finite boundaries, and let ScA0,B0=PcA0QB0PcA0. We prove that, for every c>0 and 0<ε<1/2, Λε≤ C(A0,B0)L̃[1+ln+(ca/L̃)], where L̃=ln(1/[ε(1−ε)]), and ln+x=max(0,ln x). If M and K count the interval components and a and b are their maximal lengths, one may take C(A0,B0)=63MK(1+b). The estimate is uniform in c and ε and becomes O(L̃) when L̃≥ ca. On the range c≥2 and α−c<ε<1/2 covered by the parallelepiped theorem of Kulikov and Dam Larsen (2026), this recovers its d=1 order; their complementary very-small-threshold result is sharper. We give an independent direct proof with explicit geometry dependence and a single formulation for all c and ε. It acts on the off-diagonal factor of the localization operator: an exact one-dimensional oscillation factorization reduces each far-field piece to two modulated copies of the scale-free Hankel kernel 1/[2π(s+r)], Bernstein-ellipse approximation gives uniform singular-value decay, and a Taylor-rank estimate controls the boundary layer. Choosing this width at the spectral depth absorbs finer scales and leaves only ln+(ca/L̃) dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.
Related
- Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization
- An explicit tame factorization of the four-variable Anick automorphism
- Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains
- Rota-Baxter operators and factorization of Lie affgebras
- Arithmetic localization for quasiperiodic long-range operators with monotone potentials
BibTeX
@misc{azimifard2026scalefree,
title = {{Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators}},
author = {Ahmadreza Azimifard},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.23016},
eprint = {2607.23016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23016},
}
RIS
TY - UNPB
TI - Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators
AU - Ahmadreza Azimifard
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2607.23016
UR - https://arxiv.org/abs/2607.23016
AB - The plunge region records the spectral transition of a time-frequency localization operator. Let A_0,B_0⊂ℝ be bounded measurable sets of positive measure with finite boundaries, and let S_cA_0,B_0=P_cA_0Q_B_0P_cA_0. We prove that, for every c>0 and 0<ε<1/2, Λ_ε≤ C(A_0,B_0)L̃[1+ln_+(ca/L̃)], where L̃=ln(1/[ε(1−ε)]), and ln_+x=max(0,ln x). If M and K count the interval components and a and b are their maximal lengths, one may take C(A_0,B_0)=63MK(1+b). The estimate is uniform in c and ε and becomes O(L̃) when L̃≥ ca. On the range c≥2 and α^−c<ε<1/2 covered by the parallelepiped theorem of Kulikov and Dam Larsen (2026), this recovers its d=1 order; their complementary very-small-threshold result is sharper. We give an independent direct proof with explicit geometry dependence and a single formulation for all c and ε. It acts on the off-diagonal factor of the localization operator: an exact one-dimensional oscillation factorization reduces each far-field piece to two modulated copies of the scale-free Hankel kernel 1/[2π(s+r)], Bernstein-ellipse approximation gives uniform singular-value decay, and a Taylor-rank estimate controls the boundary layer. Choosing this width at the spectral depth absorbs finer scales and leaves only ln_+(ca/L̃) dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.
ER -