The reason we wrote this note is twofold. First, in contrast to the (now) classical rectangular lattices αZ × βZ, not much is known about irregular ones Λ×M . The recent breakthrough related to semiregular lattices of the form Λ×βZ has been achieved in [1], where the authors considered the Gabor frames, generated by Gaussian totally positive functions of finite type. We also refer the reader to [1] for the history of the problem. Unfortunately, we are not able to apply the techniques from [1] to G(Λ,M) even in the case, when M = βZ for some β > 0. Second, the proof we suggest is very simple, much simpler than the known ones for rectangular lattices, see [2, 3]. We expect this proof can serve as a model in more general settings.

中文翻译：

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柯西核的不规则 Gabor 框架

我们写这篇笔记的原因是双重的。首先，与（现在）经典的矩形晶格 αZ × βZ 相比，对不规则晶格 Λ×M 知之甚少。最近与Λ×βZ形式的半规则晶格相关的突破已经在[1]中实现，其中作者考虑了由有限类型的高斯完全正函数生成的Gabor框架。我们还向读者推荐 [1] 以了解问题的历史。不幸的是，即使在某些 β > 0 时 M = βZ 的情况下，我们也无法将 [1] 中的技术应用于 G(Λ,M)。其次，我们建议的证明非常简单，比已知的矩形晶格，参见 [2, 3]。我们希望这个证明可以作为更一般设置的模型。