发表机构
University of Torino; Università della Svizzera Italiana（都灵大学; 瑞士意大利语大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述调制空间从Feichtinger原始定义到现代辛刻画的演进，强调统一局部-全局原理，并展示亚辛Wigner分布与短时傅里叶变换的结构联系及其在非线性色散方程中的应用。
AI 中文摘要
我们呈现关于调制空间的阐述性综述，涵盖从Feichtinger在局部紧阿贝尔群上的原始构造到近期的辛表述。我们通过有界均匀单位分解、短时傅里叶变换、coorbit方法和Gabor系数追踪同一局部-全局原理，同时区分底层函数设置中加权、拟Banach和Gelfand-Shilov扩展下的等价描述。我们还讨论Gabor矩阵、卷积与嵌入估计，以及在一个非线性色散方程中的应用。最后部分致力于亚辛Wigner分布$W_{\mathcal A}$。在平移可逆情形下，这些表示在相差啁啾和相空间变量的线性变换下，是重标定的短时傅里叶变换。这一结构事实既解释了它们对调制空间和Wiener amalgam空间的刻画，也解释了它们通过亚辛Gabor框架的离散化，并将近期的辛理论自然地置于Feichtinger的调制空间框架之内。
英文摘要
We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.
