So far the setting of Banach Gelfand Triples (S0,L2,SO') over Rd was used to describe mappings such as the Fourier transform, to represent operators commuting with translation (time-invariant linear systems) as convolution operators or as Fourier multipliers, or to have a convenient setting for a kernel theorem resp. describe the transition from the kernel of an operator to its Kohn-Nirenberg symbol or the spreading distribution and back.

In the present talk we will try to explain, that at least from the functional analytic point of view this setting is also quite well suited to describe the transition between continuous and disrete setting. In contrast to the wavelet setting the context of time-frequency analysis provides a natural finite/discrete model,
and one can hope (and in many cases realize computationally) that the general questions of e.g. Gabor analysis can be numerically well approximated in the finite-dimensional context.

We will try to point out what the properties of the BGTR (S0,L2,SO') are which allow for such explanation resp. numerical approximations.