# Modulation space

**Modulation spaces**^{[1]} 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,^{[2]} 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 , a non-negative function on and a test function , the modulation space is defined by

In the above equation, denotes the short-time Fourier transform of with respect to evaluated at . In other words, is equivalent to . The space is the same, independent of the test function chosen. The canonical choice is a Gaussian.

## Feichtinger's algebra

For and , the modulation space is known by the name Feichtinger's algebra and often denoted by for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. is a Banach space embedded in , and is invariant under the Fourier transform. It is for these and more properties that is a natural choice of test function space for time-frequency analysis.