Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization
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.
Related
- Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators
- Joint Random Access and Localization in Cell-Free User-Centric Networks with Frequency-Selective Fading Channels
- Uniform High-Frequency Localization on Quantum Graphs
- Scaling limit of the collision measure for two-dimensional random walks
- All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping
BibTeX
@misc{azimifard2026componentwise,
title = {{Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization}},
author = {Ahmadreza Azimifard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14946},
eprint = {2609.14946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14946},
}
RIS
TY - UNPB
TI - Componentwise and Measure-Sensitive Bounds for One-Dimensional Time-Frequency Localization
AU - Ahmadreza Azimifard
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14946
UR - https://arxiv.org/abs/2609.14946
AB - 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.
ER -