Visions for the Third Century of Fourier Analysis

Hans G. Feichtinger (Vienna)

We are all aware of the fact that soon Fourier Analysis will celebrate its 200th birthday (the fundamental paper by J.B.~Fourier was published in 1822). Hence this talk will give a short panoramic view on the developments of the field, pointing out its importance for many branches of Mathematical Analysis.

The main part of the talk will be concerned with speculations and suggestions for future tasks in the field, for the coming years. The main goals concern three different directions:
\begin{enumerate}
\item Conceptual Harmonic Analysis, meaning an integration of ideas from Abstract and Computational Harmonic Analysis; making use of suitable function spaces in order to approximate and execute numerically efficient various tasks arising in the continuous domain;
\item Reinforce the connections to the applied fields, such as physics, chemistry, communication theory and other natural sciences;
\item Make the available results {\it more user-friendly}, i.e.\ ensure that existing algorithms or theoretical results are not only accessible to the expert who can tune the parameters her/himself, e.g.\ by providing examples of best practice, verifications of optimality or self-tuning of parameters.
\end{enumerate}

Overall the spirit should be more that of combining scientific knowledge already accumulated or coming up due to the efforts of a large community of mathematicians in the coming decades in a way that changes from the perspective: Instead of asking from the view-point of producers, asking what can we produce, to that of customers,
telling us what they look for. Thus providing ``consumer reports'', ask about customer satisfaction, find tools to measure costumer satisfaction and so on could be discussed (after the talk).