The fact, that the theory of Fourier series is one of the oldest branches in mathematical analysis, which has even shaped the development of the development of pure mathematics (set theory, real analysis, functional analysis) has the disadvantage, that the literature supports the impression, that one has to follow the historical path in order to understand modern applications of Fourier analysis, e.g. in connection with digital signal processing or time-frequency analysis.

The topic of the talk will be a setting, which allows to treat periodic functions, discrete time signals, but also functions in $L^p$-spaces as elements of a certain space of (harmless) distributions, which arose first in time-frequency analysis. The corresponding Banach space of test functions, usually called $S_0(R^d)$ (or modulation space $M^1(R^d)$) consists of all $L^2$-functions which have an integrable short-time Fourier transform with respect to a Gaussian window. We will explain the basic
properties of the Banach Gelfand Triple $(S_0,L^2,S_0*)$,
which is many properties comparable to the usual Gelfand triple, based on the Schwartz space $S(R^d)$ of rapidly decreasing functions on $R^d$. Among others we can define the Fourier transform on $S_0*$ and have a so-called kernel theorem, i.e. operators from $S_0(R^d)$
to $S_0*(R^d)$ have a uniquely determined (matrix-like) distributional kernel $\sigma$ in $S_0*(R^{2d})$ (and vice versa). In this setting many manipulations carried out without proper justification in engineering or physics books can be well justified, using the concept of pointwise convergence in $S_0*(R^d)$.