We characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

中文翻译：

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Gelfand-Shilov和调制空间上的Zak变换及其在算符理论中的应用

我们根据Zak变换的估计来表征Gelfand-Shilov空间，它们的分布空间和调制空间。我们将这些结果用于准周期函数和分布的一般研究。我们还为线性算子建立了必要和充分的条件，以便这些算子应通过Zak变换进行共轭。