Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Gabor Analysis: Background, Concepts and Computational Issues

Vortrag von Prof. Dr. Hans Georg Feichtinger

**Datum:** 24.03.21 **Zeit:** 16.15 - 17.45 **Raum:** Online ZHACM

Gabor Analysis goes back to the fundamental paper of D.Gabor from 1946, who expressed (resp. conjectured) that every function can be expanded into a series of time-frequency shifted version of the standard Gaussian, meaning by building blocks of the form $$g_{k,n}(x) = exp(2 \pi i b n} g_0(x - ak), $$ with $g_0(t) = exp(-\pi t^2)$. By choosing $a=1=b$ he was hoping to expand ``every signal in a unique way'', thus having a natural interpretation of the energy at position $(k,n)$ in phase-space through the (expected) unique coefficients for such an expansion. Only since 35 years mathematicians have taken care of this problem, which provides a number of interesting challenges, going far beyond the original problem. From the speaker's point of view there is not only the computational problem arising (since Gabor Analysis can also be realized in the context of finite, Abelian groups), but also a variety of functional analytic concepts, including the idea of Banach frames and Banach Gelfand Triples. The methods developed in the last 25 years are also relevant for the teaching of classical Fourier Analysis (such a course has been held at ETH last semester by the speaker), but also provides a good platform for the formulation of ``Conceptual Harmonic Analysis'', where the main question concerns the relationship between the continuous formulation of a problem and the corresponding discrete version and its computational realization. PS: I would like to use my private ZOOM licence and record it (as I have done it with the course last semester, which was then uploaded to YouTube). This has been agreed with Prof. Helmut Boelckei, so there should not be problem. Participants will not be visible, but this also implies that questions can be only asked after the end of the talk (ie. in the offline part, following the actual presentation). I hope that this is also applicable in your case, as it takes place at the same institution.