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.
From: Lukas Liehr [view email]
[v1]
Wed, 24 Jun 2026 17:30:14 UTC (1,088 KB)
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