We want to show that we should move on from a separation of "abstract" versus "computation" or "applied" Harmonic Analysis towards the age of "conceptual harmonic analysis", where all the tools from both sides are integrated, including numerical and functional-analytic considerations.

In many cases the reduction from the view-point of LCA groups to the "trivial" setting of finite Abelian groups is what is needed in order to get the algebraic structures right, but also in order to do MATLAB implementations. On the other hand, once one moves to continuous variables unit vectors have to be replaced by Dirac measures, and one needs some form of generalized functions in order to give sense to the concept of the Fourier transform.

The setting of Banach Gelfand triples, based on the Segal Algebra S_0(G) (also called modulation space M^1 in Charly Groechenig's book) appears to provide the appropriate setting. This setting in turn allows to formulate truly "conceptual" statements that allow constructive approximative realization of operators such as the determination of the Fourier transform, the determination of the dual Gabor window (N.Kaiblinger), or the approximate calculation of a set of dual windows for a finite family determining a Riesz basis (by taking translates along some lattice) for the corresponding multi-window spline-type space (Feichtinger/Onchis).