We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the submanifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces.
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Cordero et al. (2024) studied this question.