When it comes to the constructive realization of operators arising in
Fourier Analysis, be it the Fourier transform itself, or some
convolution operator, or more generally an (underspread)
pseudo-differential operator it is natural to make use of sampled version
of the ingredients.

The theory around the Banach Gelfand Triple (S0,L2,SO') which is based
on methods from Gabor and time-frequency analysis, combined with the
relevant function spaces (Wiener amalgams and modulation spaces) allows
to provide what we consider the appropriate setting and possibly the
starting point for qualitative as well as later on more quantitative
error estimates.



old draft to the title:
The Role of Function Spaces in Sampling Theory and Gabor Analysis


outlining that e.g. Wiener amalgams or modulation spaces appear to be as important in this theory as for example say Sobolev spaces in the theory PDE.