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arXiv · 2606.26052

On the existence problem of regular Gabor frames

Jaume de Dios Pont·Lukas Liehr·Mitchell A. Taylor
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Abstract

For every dimension $d > 1$, we establish explicit criteria on lattices $\Lambda \subset \mathbb{R}^{2d}$ with density $D(\Lambda) > 1$ such that no function with a continuous Zak transform generates a Gabor frame along $\Lambda$. In particular, this gives a negative answer to the existence problem of Gabor frames with window functions in the Schwartz space, the Feichtinger algebra, and the Fourier-invariant Wiener space. Our result is based on a characterization of when a collection of quasiperiodic functions admits a common zero, which may be of independent interest. We also include a formalization of our main result in Lean 4.

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Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor. 2026-06-24. On the existence problem of regular Gabor frames. https://arxiv.org/abs/2606.26052

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