The idea of ``Conceptual Harmonic Analysis'' grew out of the attempt to make objects arising in Fourier Analysis or Gabor Analysis (such as norms of functions, their Fourier transforms, dual Gabor atoms, etc.) computable. Using suitable function spaces such as the Segal algebra $S_0(R^d)$ it should be possible to find concrete algorithms which allow to compute approximations to the desired on real hardware in finite time, up to (at least potentially) arbitrary requested precision.

Going beyond the ideas of Abstract Harmonic Analysis, which only allows to identify the analogies between objects on different LCA (locally compact Abelian) groups, the idea of Conceptual Harmonic Analysis wants to see the connection between these settings to be used, for example, in order to use methods from discrete, periodic Gabor analysis (which are computationally realizable using MATLAB).

The Banach Gelfand Triple $(S0,L2,S0')(R^d)$, which can also be seen as a ``rigged Hilbert space'', allows to describe and justify a number of such procedures providing also a tool to deal with periodic, but also of continuous or discrete signals in a unified way. As it turns out the so-called $w*$-convergence is crucial for the description, and fine partitions of unity as well as regularizing operators play a crucial role in this context.