A modern approach to the Fourier AnalysisHans G. Feichtinger (NuHAG, Univ. Vienna, Faculty Mathematics) given at (06.12.21 11:00) id: 3703 length: 60min status: type: LINK-Presentation: https://nuhagphp.univie.ac.at/dateien/talks/3703_FeiIndia21A.pdf ABSTRACT: Assuming the many of the participants of this conference may at some point also have to teach a course in Fourier Analysis I consider it eminently important that such a course should satisfy the needs of a modern society, with an increasing role for digital signal processing, or the simulation of linear systems using the computer. While Laurent Schwartz' theory of tempered distributions (almost as old as the speaker) is well established it is highly demanding in terms of mathematical background, and mostly relevant for the solution of PDEs. This talks should point out that a theory of ``{\it mild distributions}'' can be based on quite simple ideas, and not much more than the Riemann integral. The key to this modern (and {\it simple}!) approach to a theory of {\it generalized functions} provides a solid mathematical basis for objects such as Dirac measures or Dirac combs. Mild distributions have a Fourier transform, which is also a mild distributions, and they have a bounded spectrogram. Such spectrograms can nowadays be produced on real-time, and in this sense audio-signals are typical examples of objects which should undergo a ``localized frequency analysis''. The analogy to the number systems, with the natural embedding $\QRC$ is used as a motivation and orientation for a sequential approach. Actual computations are done in the field of rational numbers $\Qst$, and extended to $\Rst$ by taking limits (e.g. in order to define $\pi \cdot \sqrt{5}$). Finally a ``trick'' allows to extend the usual calculations to the even larger (and more flexible) field of complex numbers $\Cst$. By its genesis the function spaces used, namely the Segal algebra $\SORdN$ (also called {\it Feichtinger's algebra}), together with $\LtRdN$, the Hilbert space of square integrable functions, and the dual space $\SOPRdN$ (of mild distributions) are very well suited to deal with questions of time-frequency analysis and discuss the stability of so-called Gabor expansions of signals. In the audio case this can be viewed as a kind of graphical score computed from the recorded sound signal. |