Hans Feichtinger (Univeristy of Vienna): A function space defined by the Wigner transform and is Applications

Friday 2 Mar 2018, 14:00 → 16:00 Europe/Budapest

Tanacsterem

The so-called Feichtinger algebra S0, resp.\ the modulation space M1 on Rd can be defined by the integrability of the Wigner transform. Despite the quadratic character of the transform this defines a linear space and even a FOURIER invariant Banach space containing the Schwartz space of rapidly decreasing functions.
Based on such a Banach space of test functions one can develop a quite general but still comparatively convenient theory of Fourier transforms, covering the continuous and discrete, the periodic and the non-periodic case.
Using the Banach Gelfand triple, consisting of S0, contained in the Hilbert space L2 and this in turn embedded into the space SO′ of linear functionals one has a good way to describe questions from applied fields, such as slowly varying systems, audio signals (using spectrograms) and much more. Moreover it a key element in a variant of time-frequency analysis called Gabor Analysis, going back to the work of Dennes Gabor (published in 1946).
This Banach Gelfand Triple is a suitable tool for theoretical physics (rigged Hilbert space) and time-frequency analysis.
The spaces can also be characterized via so-called Wilson bases, which also played a significant role in the signal processing part of the discovery of gravitational waves by the LIGO team (Phyiscs Nobel-Prize 2017).