Time-Frequency Analysis, Wavelet Theory and in particular Gabor Analysis
have drawn a lot of attention to the theory of frames and Riesz bases.
These are the natural generalization to the notions familiar from linear algebra: spanning set resp. linear independent set. As will be shown in this talk the concentration on the Hilbert space case along is a bit too narrow-minded. Instead one should look at triples of spaces, typically the Schwartz space, the Hilbert space $L^2(Rd)$ and the tempered distributions, forming a Gelfand Triple. For example, the Fourier transform is an automorphism of this Gelfand Triple. Even more convenient and easier to describe is a Gelfand Triple based on the Segal algebra $S_0(Rd)$, because this is just a Banach space (isomorphic to $\ell^1$ using Wilson or localized Fourier bases), and hence many topological considerations are easier. It will be shown, how easy these BGTs are to use, and how the can be used to formulate basic principles of Fourier and Time-Frequency Analysis.