The so-called Feichtinger algebra $S_0$, resp.\ the modulation space
$M^1$ on $R^d$ can be defined by the integrability of the Wigner transform. Despite the quadratic character of the transform this defines a linear space and even a FOURIER invariant Banach space containing the Schwartz space of rapidlz decreasing functions.

Based on such a Banach space of test functions one can develop a quite general but still comparatively convenient theory of Fourier transforms, covering the continuous and discrete, the periodic and the non-periodic case.

Using the Banach Gelfand triple, consisting of $S_0$, contained
in the Hilbert space $L^2$ and this in turn embedded into the
space $SO'$ of linear functionals one has a good way to describe questions from applied fields, such as slowly varying systems,
audio signals (using spectrograms) and much more. Moreover it a key element in a variant of time-frequency analysis called Gabor Analysis, going back to the work of Dennes Gabor (published in 1946).

This Banach Gelfand Triple is a suitable tool for theoretical physics (rigged Hilbert space) and time-frequency analysis.
The spaces can also be characterized via so-called Wilson bases, which also played a significant role in the signal processing part of the
discovery of gravitational waves by the LIGO team (Phyiscs Nobel-Prize 2017).