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 . References

Banach spaces