25.03.2024 Vortrag von Hans Georg Feichtinger am 10.04.2024
Am Mittwoch, den 10.04.2024, hält Hans Georg Feichtinger im Rahmen seines Besuchs in Marburg einen Vortrag mit dem Titel "A PANORAMIC view on CONCEPTUAL HARMONIC ANALYSIS". Alle Interessierten sind dazu herzlich eingeladen.
Programm
- 16:00 Uhr: Tee/Kaffee
- 16:30-17:30 Uhr: Vortrag
- Anschließend findet eine Nachsitzung statt.
Der Vortrag findet auf den Lahnbergen im Raum SR XV (04C37) statt.
Abstrakt
The original intention of Conceptual Harmonic Analysis was the integration of Abstract and Computational Harmonic Analysis. More recently Time-Frequency Analysis (Analysis over Phase Space) appears as a common setting for the discussions of Gabor Analysis or the theory of pseudo-differential operators.
Classical Fourier Analysis, as it is also represented in many modern engineering books, deals with functions on different LCA groups, or in engineering terminology discrete vs. continuous signals, or periodic and non-periodic (decaying, say square integrable) functions.
Numerical Harmonic Analysis deals with the efficient development and implementation of algorithms (such as FFTW). NuHAG has done such things with respect to the irregular sampling problem, Gabor frames or Gabor multipliers.
Functional or approximation theoretical considerations help to analyze these mathematical methods. The goal can be to adapt a computational method so that the solution of a problem can be obtained up to some given error, usually described by suitable function spaces.
This is one of the reasons why it is important to dispose of a large repertoire of function spaces (of tempered distributions) which allow to formulate useful mapping properties for relevant linear operators, or describe the concentration of functions (e.g. by the decay of their STFT) in a phase space picture.
It is meanwhile well-recognized that Wiener Amalgam as well as Modulation spaces are a fairly useful tool, with Feichtinger's algebra $S_0(G) = M^1(G)$ as the smallest and its dual (the space of mild distributions) as the largest one among the unweighted ones. They constitute THE Banach Gelfand Triple $(S_0,L^2,S*_0)$ and allow the analysis of so-called Fourier Standard Spaces. BGTr-morphism and isomorphisms appear in abundance in Harmonic Analysis, such as the Kernel Theorem or the Kohn-Nirenberg description of operators.
As the last aspect let us mention that there is great interest in approximating the continuous (whether it is periodic, decaying or ongoing) by the discrete. Here a cascade of models (providing better and better approximation, typically in the sense of mild distributions) and allowing a nested approximation of functions from regular samples (say) is the starting point. This is where MATLAB experiments and simulations come in and play an important role for CHA.