Modulation space

Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be the right kind of function spaces for time-frequency analysis. Feichtinger's algebra, while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis.

Modulation spaces are defined as follows. For ${\displaystyle 1\leq p,q\leq \infty }$, a non-negative function ${\displaystyle m(x,\omega )}$ on ${\displaystyle \mathbb {R} ^{2d}}$ and a test function ${\displaystyle g\in {\mathcal {S}}(\mathbb {R} ^{d})}$, the modulation space ${\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})}$ is defined by

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