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:
- 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;
- Reinforce the connections to the applied fields, such as
physics, chemistry, communication theory and other natural sciences;
- Make the available results 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.
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).