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arXiv · 2512.04594

Frames for compactly supported functions with irrational density

Yurii Belov·Aleksei Kulikov
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Abstract

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha,\beta)=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $\alpha < b-a, \alpha\beta < 1, \alpha\beta \notin\Q$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x) = \exp(\frac{1}{x^4-1})\chi_{(-1,1)}(x)$.

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Yurii Belov, Aleksei Kulikov. 2025-12-04. Frames for compactly supported functions with irrational density. https://arxiv.org/abs/2512.04594

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