Conceptual Harmonic Analysis:
Convolutions, Fourier Transforms and all that

Since almost 14 years the speaker tries to promote the idea of ``CONCEPTUAL HARMONIC ANALYIS'' as a way to combine or rather reconcile Abstract Harmonic Analysis (AHA) with Computational Harmonic Analysis (CHA) and much more. In particular, the long history Fourier Analysis (by now 200 years!) has contributed to a diversification of methods and standards. This has led to
the unpleasant situation that mathematicians, engineers or physicists have their own notations, their own settings and habits, and numerical work is often only seen as a way to illustrate the continuous theory, or to simulate a problem in order to improve the heuristic basis for the proper development of a mathematical theory.

Going back to Andre Weil and Hans Reiter one can say that the natural domain for Fourier Analysis are LCA groups. The same is true for time-frequency analysis and Gabor Analysis. But in the world of AHA we can discuss the analogy between different groups G. Once the dual group G^ has been identified we can define the forward and inverse Fourier transform, define time-frequency shifts and the STFT and discuss the reconstruction from samples (for band-limited functions or from the STFT).

Obviously one expects that the FFT should be useful in computing at least approximately the Fourier transform of a nice function, or the convolution of two functions, or perhaps even measures. We should motivate the approaches and ideally provide a guarantee (in the spirit of numerical integration methods) for computations to deliver good quantitative results. Ideally to approach should avoid unnecessary technicalities (such as Lebesgue integration or Frechet spaces such as S(R)), at least for the problems relevant for
digital signal processing. Of course, suitable function spaces are required in order to express properly that computations deliver a good approximation of a given signal.

In the talk the speaker will report on attempts to rebuild Fourier Analysis over LCA groups (including R^d) from scratch. First convolution of bounded measures is introduced via translation invariant systems and then the Fourier Stieltjes transform is introduced, up to the convolution theorem.
BUPUs (bounded uniform partitions play an important role here).
As an intermediate goal the space S_0(G) is introduced, and finally
the Banach Gelfand Triple (S_0,L_2,S_0*). Most spaces relevant for classical Fourier Analysis are then sandwiched between S_0 and S_0* and are isometrically invariant under the time-frequency shifts.

Overall, the focus of the talk will be on alternative ways to provide a proper foundation for AHA, it will talk about non-standard function spaces (avoiding Lebesgue spaces as a starting point) and suggest an interpretation of signals as ``mild distributions'' (members of S_0*), having a bounded STFT. On the other hand we need computational tools plus quantitative and constructive approximations of guaranteed quality.