Classical (Abstract) Harmonic Analysis is dealing with specific orthogonal expansions (Fourier series) over a specific locally compact Abelian group G. Engineers distinguish between continuous or discrete, periodic or non-periodic (meaning well decaying) functions, which is more or less the same. However, real signals are much more diverse, and thus methods of time-frequency analysis (localized Fourier transform), or Gabor Analysis (in its discretized form) has been developed in the last decades. The treatment of the natural problems arising there (convergence of double series consisting of non-orthogonal elements, in short frame theory) requires other function spaces than just the usual L^p-spaces. As it turned out the Segal algebra S_0(G) (Feichtinger's algebra) and its dual S_0(G)*, the space of mild distributions, provide the appropriate setting for many questions, even outside of time-frequency analysis. Together with the Hilbert space L^2(G) they form a chain of spaces, with S_0 in L^2 in S_0*, the so-called Banach Gelfand Triple. We will restrict our attention to G = R^d (Euclidean case).

Among others all three spaces are invariant under the (extended) Fourier transform, and there is a kernel theorem, allowing to identify the space of bounded linear operators from S_0(R^d) to S_0*(R^d) with elements (so-called ``kernels'') in S_0*(R^2d). But there is also the description of such operators via the so-called spreading representation or alternatively (connected via the symplectic Fourier transform) the Kohn-Nirenberg symbol (both belonging to S_0*(R^2d) as well).

We will concentrate on group theoretical considerations which help to understand how to approximate the Fourier transform of a function in S_0(R^d) with the help of FFT-based methods, or how one can verify that a given function g in SO(R^d) generates a (good) Gabor frame, i.e. a system that
allows to write each f in L^2(R^d) as an unconditionally convergent series of time-frequency shifted copies of the given Gabor atom g.