Whereas the classical theory of (tempered) distributions is motivated by the application to PDEs and the description of smoothness of functions the time-frequency approach to the theory of generalized functions allows an alternative view on abstract pseudo-differential operators.

The basic function spaces (or distributions) arising in time-frequency analysis are the so-called modulation spaces, which can be characterized by the behavior of the short-time Fourier transform. In contrast, classical smoothness spaces (the family of Besov-Triebel-Lizorkin spaces) are characterized by continuous wavelet transforms.

The prototypical setting can be described by the so-called Banach Gelfand Triple, based on the Segal algebra SO(Rd), L2(Rd) and the dual space SO'(Rd), which in many cases can be used as a replacement for the Schwartz Gelfand triple. Among others one has a generalized Fourier transform and (surprisingly) a kernel theorem for this setting.