The study of function spaces in connection with time-frequency analysis led to the invention of the family of so-called modulation spaces. They share many properties with the family of Besov (and Triebel-Lizorkin) spaces. The prototype among these spaces is the Segal Algebra $S_0(Rd)$ which enjoys a minimality property within the family of all Banach spaces of distributions which are isometrically invariant under shifts in either the time or the Fourier domain. This implies that the space is Fourier invariant, but also inside all the $L^p$-spaces, and dense for $p < \infty$. By duality, the dual space is Fourier invariant as well and contains all the $L^p$-spaces.

The Banach Gelfand $(S_0,L^2,S_0')(R^d)$ can be used to describe a moderately general theory of generalized functions (or distributions), with the advantage that it makes only use of basic functional (and harmonic) analysis methods (such as Banach spaces, Hilbert spaces and bounded linear operators). Within this theory the Fourier transform, but also the Kohn-Nirenberg symbol for pseudo-differential operators have a clear meaning. Such an approach helps to emphasize the analogy between the finite-dimensional situation (FFT-based methods and linear algebra).

.. more details to be provided later on (hgfei, June 9th, 2011)