**ABSTRACT:**

The idea of this talks is the promote an idea which grew out of the experiences of a person educated in Abstract Harmonic Analysis and has turned into an application oriented mathematicians who likes to carry out experimental work, develop algorithms, and connect concepts from functional analysis with concrete models: CONCEPTUAL HARMONIC ANALYSIS goes beyond the view-point of AHA (Abstract Harmonic Analysis). CHA (computational harmonic analyis or Fourier analysis) is concerned with the question of finding efficient implementations of e.g. the DFT (discrete Fourier transform), in the form of FFT or even more recently FFTW (as built into MATLAB). AHA allows to explain the analogy between the discrete, the periodic and the general case, but it does not help to explain the approximation of one by the other.

The talk will raise some fundamental questions which arise in this context, often not even perceived as problems by the engineering community (working completely in the finite dimensional setting) or the pure mathematicians or physicists who have no problem to juggle with integrals.

The buzzword is ``structure preserving approximation'', which is a blend of numerical analysis, numerical linear algebra, function spaces, functional analysis, AHA, and so-called time-frequency analysis and Gabor analysis.

As time permits the lecture will state not only questions and problems but will also show a few first good examples where suitable function spaces allow to quantify the level of approximation. The corresponding modulation spaces have been introduced by the speaker around 1983 and allow an approach to distribution theory, starting from the Riemann integral only, using basic functional analysis.