In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

中文翻译：

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短时傅立叶变换和超振荡


在本文中，我们使用时频分析工具和技术（例如短时傅立叶变换（STFT）和 Zak 变换）研究超振荡理论的新结果。我们首先研究短时傅立叶变换如何作用于超振荡序列。然后，我们应用超移性质来证明短时傅立叶变换通过取极限来保留超振荡行为。事实证明，这些计算与时频分析的各种特征（例如 Gabor 空间、Gabor 核、Gabor 框架、2D 复数 Hermite 多项式和多分析函数）产生了有趣的联系。我们根据窗函数的选择来处理不同的情况，从一般情况到涉及高斯窗和埃尔米特窗的更具体情况。我们还考虑一个演化问题，其初始数据由超振荡乘以通用窗函数的时频偏移给出。最后，我们使用给定的 Hermite 窗口计算 STFT 对近似序列的作用。