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, 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, namely
In other words, is equivalent to. The space is the same, independent of the test function chosen. The canonical choice is a Gaussian.
We also have a Besov-type definition of modulation spaces as follows.
where is a suitable unity partition. If, then.

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. Fourier transform is an automorphism on.