通过辛算子及相关时频表示探讨调制空间
Excursus on modulation spaces via metaplectic operators and related time-frequency representations
AI总结：
本文研究了辛Wigner分布对调制空间的表征，揭示了隐藏的辛作用并证明其等价拟范数，为调制空间及相关Wiener空间提供了新的理论结果和全面视角。
摘要：
调制空间最初由Feichtinger于1983年引入。自2000年代以来，已有数千项研究将其作为正确框架，涵盖偏微分方程、伪微分算子、量子力学和信号分析。这证明了对这些空间及相关Wiener空间的深入研究是合理的。近期，辛Wigner分布（包含$τ$-Wigner分布、模糊函数和短时傅里叶变换作为特例）已被证明在适当假设下能表征调制空间。我们研究了隐藏在其构造中并保证这些空间具有等价（拟）范数的辛作用。我们在此主题上添加了新结果，并以对这些表征的全面视角作结。类似结果也适用于Wiener amalgam空间。
英文摘要：
Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the $τ$-Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.
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