Hexagonal optimality for Gaussian time-frequency sampling
Let $g(t) = 2^{1/4} e^{-\pi t^2}$. For $\delta > 1$ and a full-rank lattice $\Lambda \subset \mathbb{R}^2$ of covolume $1/\delta$, let $A_\Lambda$ and $B_\Lambda$ be the optimal frame bounds of the Gabor system of $g$ over $\Lambda$: the largest $A$ and the smallest $B$ with $$A \|f\|_2^2 \le \sum_{(x,\omega) \in \Lambda} \Bigl| \int_{\mathbb{R}} f(t)\, \overline{e^{2\pi i \omega t} g(t - x)}\, dt \Bigr|^2 \le B \|f\|_2^2 \qquad (f \in L^2(\mathbb{R})).$$ Since $\delta > 1$, they are positive and finite (Lyubarskii; Seip; Seip and Wallstén). Let $$\Lambda_{\mathrm{hex}} = \delta^{-1/2} \Bigl( \frac{2}{\sqrt{3}} \Bigr)^{1/2} \begin{pmatrix} 1 & 1/2 \\ 0 & \sqrt{3}/2 \end{pmatrix} \mathbb{Z}^2,$$ the hexagonal lattice of covolume $1/\delta$. When $\delta$ is an even integer, Faulhuber proved, using Montgomery's theorem on minimal theta functions, that among lattices of covolume $1/\delta$ the upper bound $B_\Lambda$ is minimal exactly for the hexagonal lattice and its rotations, and Bétermin, Faulhuber and Steinerberger proved that the lower bound $A_\Lambda$ is maximal for the hexagonal lattice, so that $B_\Lambda/A_\Lambda$ is minimal there.
Problem. Is $B_\Lambda / A_\Lambda \ge B_{\Lambda_{\mathrm{hex}}} / A_{\Lambda_{\mathrm{hex}}}$ for every $\delta > 1$ and every lattice $\Lambda \subset \mathbb{R}^2$ of covolume $1/\delta$?
Source: Strohmer and Beaver (2003); in this form, Faulhuber (2019), Conjecture III.1, which also asks that equality hold only for rotated hexagonal lattices.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, frame bounds, Gabor frames, hexagonal lattices
References
@article{strohmer2003,
author = {Strohmer, Thomas and Beaver, Scott},
title = {Optimal {OFDM} design for time-frequency dispersive channels},
journal = {IEEE Trans. Commun.},
volume = {51},
number = {7},
year = {2003},
pages = {1111--1122},
doi = {10.1109/TCOMM.2003.814200},
}
@inproceedings{faulhuber2019,
author = {Faulhuber, Markus},
title = {The {S}trohmer and {B}eaver conjecture for {G}aussian {G}abor systems -- a deep mathematical problem (?)},
booktitle = {2019 13th International Conference on Sampling Theory and Applications (SampTA)},
publisher = {IEEE},
year = {2019},
pages = {1--5},
doi = {10.1109/SampTA45681.2019.9030963},
eprint = {1905.05051},
archivePrefix = {arXiv},
}
@article{faulhuber2018,
author = {Faulhuber, Markus},
title = {Minimal frame operator norms via minimal theta functions},
journal = {J. Fourier Anal. Appl.},
volume = {24},
number = {2},
year = {2018},
pages = {545--559},
doi = {10.1007/s00041-017-9526-x},
}
@article{montgomery1988,
author = {Montgomery, Hugh L.},
title = {Minimal theta functions},
journal = {Glasgow Math. J.},
volume = {30},
number = {1},
year = {1988},
pages = {75--85},
doi = {10.1017/S0017089500007047},
}
@misc{betermin2026,
author = {B{\'e}termin, Laurent and Faulhuber, Markus and Steinerberger, Stefan},
title = {A variational principle for {G}aussian lattice sums},
year = {2026},
eprint = {2110.06008},
archivePrefix = {arXiv},
note = {Version 2, 5 October 2026; to appear in Forum Math. Pi},
}
@incollection{lyubarskii1992,
author = {Lyubarskii, Yu. I.},
title = {Frames in the {B}argmann space of entire functions},
booktitle = {Entire and Subharmonic Functions},
series = {Adv. Soviet Math.},
volume = {11},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1992},
pages = {167--180},
doi = {10.1090/advsov/011/08},
}
@article{seip1992,
author = {Seip, Kristian},
title = {Density theorems for sampling and interpolation in the {B}argmann--{F}ock space {I}},
journal = {J. Reine Angew. Math.},
volume = {429},
year = {1992},
pages = {91--106},
doi = {10.1515/crll.1992.429.91},
}
@article{seip1992b,
author = {Seip, Kristian and Wallst{\'e}n, Robert},
title = {Density theorems for sampling and interpolation in the {B}argmann--{F}ock space {II}},
journal = {J. Reine Angew. Math.},
volume = {429},
year = {1992},
pages = {107--114},
doi = {10.1515/crll.1992.429.107},
}
@misc{aim-gaussian-gabor-hexagonal-optimality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hexagonal optimality for Gaussian time-frequency sampling}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/449-gaussian-gabor-hexagonal-optimality.md},
note = {Entry 449 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00348, "Hexagonal optimality for Gaussian time-frequency sampling", clopen.solutions.