"Tools and Ideas for Time-Frequency Analysis"Feichtinger, HansIt is the purpose of this presentation to share - as a senior person - some thoughts about questions related to Fourier Analysis and in particular related to Gabor Analysis and Time-Frequency Analysis which I found and still find interesting. In particular, I would like to describe in general terms the way how I explore the field and let the listeners look behind the scenes, by explaining some of the motivations and long-term goals. Obviously, over the years a whole battery of tools have been acquired, from WIENER AMALGAM SPACES to MODULATION SPACES, with the Segal algebra $S_0(R^d)$ and its dual, the space of mild distributions as a special case. Together with the Hilbert spaces $L^2(R^d)$ they form the so-called BANACH GELFAND TRIPLE, which plays a universal role (for applications, for classical Fourier Analysis, and of course for Gabor Analysis), and appears to provide the correct setting for the discussion of the ``discrete to continuous'' transition problem. Of course, MATLAB-based experiments allow obtaining insight concerning possible theoretical statements, but still much has to be done in order to ensure that this transition can be used for the quantitative evaluation of estimates in the continuous domain. As such the exploration of the field, using numerical methods and efficient implementations, but also suitable function spaces (not just complicated spaces for fun) combined with functional analytic arguments help a lot to make progress on the key problems of time-frequency analysis, and are still playing an important role in my current research work. In addition, this setting allows laying the ground for a new approach to Fourier Analysis (as done in my ETH course given in 2020), avoiding the heavy use of measure theory and $L^p$-spaces. Following Andre Weil and Hans Reiter, the correct setting is that of functions or distributions over LCA (locally compact Abelian) groups, and all of the mentioned tools have their natural interpretation even at this level of generality. In this way, I hope to promote the idea of CONCEPTUAL HARMONIC ANALYSIS, which is more than just the synthesis of Numerical/Computational and Abstract Harmonic Analysis. On the side, I also will indicate how pictures and illustrations help to make abstract objects ``more concrete'' and get a feeling concerning e.g. inclusion results for function spaces. |
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