File:    D:\conf\conf16\aveiro16\Av16FeiCourse.htm 

Draft for the course entitled 21th European Intensive Course on Complex Analysis, its Generalizations and Applications

COURSE:   A MODERN APPROACH TO FOURIER ANALYSIS: Foundations of Gabor Analysis 

Motivation: Although Fourier Analysis can look back on a history of almost 200 years now and is thus a mature field, the last decades have seen an enormous development of pratical and theoretical Fourier analysis, which so far has not found its way into the standard course, given to mathematicians or engineers. In fact, the way how Fourier Analysis, or on the applied side often ``Signal Analysis and Systems Theory'' is taught in engineering schools is using quite different methods, making it hard, to move between the theoretical side and the more practical side of the field. Given the fact that modern technology (mobile communication, data compression, digital imaging, medical or geophysical data processing) is making heavy use of different variants of Fourier analysis many of our students and colleagues are not aware of the new perspectives and teach the subject in a traditional way.

The aim of the course is to help bridging this gap. It is based on 40 years of studies of abstract harmonic analysis and function spaces, and meanwhile also more than 25 years of practical work, based on MATLAB (which will be only used as a demonstration tool in the course, but LTFAT is certainly a useful tool for those who want to go deeper on the applied side.

The key topics of the course are:

  1. A short review of traditional Fourier Analysis (just for comparison)
  2. The concept of a Short-Time Fourier Transform (quasi ``musical score"), phase space analysis of signals or distributions;
  3. A new approach to Systems Theory (BIBOS systems are convolution operators with bounded measures, convolution theorem);
  4. Frames and Riesz bases in Hilbert spaces, Gabor families, Foundations of Gabor analysis
  5. The Segal algebra SO(R^d)  as a space of  nice functions, well suited for Fourier analysis
  6. Wiener amalgam spaces and decomposition techniques, atomic representations
  7. The Banach Gelfand triple (SO,L2,SO') and ramifications for Fourier and Gabor analysis.
  8. Unitary Banach Gelfand triple isomorphisms and retracts everywhere (e.g. kernel theorem, Fourier transform,...)
  9. Banach frames and Riesz projection bases;
  10. Approximation by discrete measures or signals, computational aspects
  11. The idea of CONCEPTUAL HARMONIC ANALYSIS (unification of abstract and numerical HA);

The PREREQUESTES for the course are relatively minor (aside from an expected openness to go non-traditional ways):

Standard results from linear algebra and basic functional analysis (norms, operators, Hilbert spaces, weak convergence) will be used, but can be explained also during the course. Readiness to do some numerical experiments (e.g. in OCTAVE or MATLAB) will enhance the learning effect, but is not required. Material will be provided before, during and after the course.

A number of talks