The Banach space $S_0(R^d)$ is the smallest Banach space which is isometrically invariant under translations and modulations. It is long known as a perfect tool for time-frequency and Gabor analysis, but also for classical Fourier analysis on $R^d$. Together with its dual space and with the Hilbert space $L^2(R^d)$ in the middle they form a Banach Gelfand triple. This setting allows to formulate a principle called ``conceptual harmonic analysis'' by the author: In which way can one transfer information (and methods) from the finite-dimensional, discrete setting to the case of the Euclidean situation. One example is the so-called kernel theorem (usually only treated for the Hilbert-Schmidt operators), which is the analogue of a matrix representation of a general linear operator on $\Cst^n$.

New characterizations allow to introduce this triple without reference to Lebesgue integration theory, but still enables us to derive the usefulness of a generalized Fourier transform. In particular, as in the finite case (FFT) one can say that the Fourier transform maps pure frequencies, i.e. complex exponential functions, into Dirac measures. Poissson's formula is valid and provides a distributional proof for the Shannon sampling theorem and much more.

The mathematical tools required are basic functional analysis and certain function spaces, called Wiener amalgam spaces defined with the help of bounded, uniform partitions of unity. The spaces used are special cases of the so-called modulation spaces which are also of interest in the theory of pseudo-differential operators and have first been presented by the author at a conference in Kiew in 1983 (see Zbl. Math. 0505.46024).