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Course at the University of Wroclaw, May 17th to 23rd, 2016   held by Hans G. Feichtinger, Univ. Vienna, NuHAG

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 practical 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. Well presented YOUTUBE presentations at prestigious universities are no exception.

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) versus Fourier Analysis over finite groups (MATLAB);
  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. Spreading function and Kohn-Nirenberg symbol of operators;
  10. Banach frames and Riesz projection bases;
  11. Approximation by discrete measures or signals, computational aspects;
  12. 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 on related subjects can be downloaded from the NuHAG Talk Server  (you can select alternative speakers or titles there).

All the relevant papers are available from the NuHAG BIBTEX site:  http://www.univie.ac.at/nuhag-php/bibtex/

A repository of MATLAB files is found at http://www.univie.ac.at/nuhag-php/mmodule/

The ARI (Acoustic Research Institute, Austrian Academy of Science) offers free access to the LTFAT toolbox as well as the STX signal analyzer: LINK 

A good part of the early part of the course is well documented in the LECTURE NOTES (material used over the years at the University of Vienna) by the author.

For those who want to learn about the use of MATLAB in the context of Time-Frequency Analysis the MACHA11-Notes could be useful.

Hint: Our upcoming conference (June 6th to 11th) in Strobl, www.nuhag.eu/strobl16 will provide a number of talks/PDFs on related subjects.