University of Heidelberg, **Applied Analysis Seminar**

Dr. Markus Faulhuber (University of Vienna)

The Strohmer and Beaver Conjecture for Gaussian Gabor Frames

*INF 205, SR1*

The Strohmer and Beaver Conjecture for Gaussian Gabor Frames

The talk will start with an introduction to time-frequency and Gabor

analysis and most of the presented concepts can as well be found in

quantum mechanics. Actually, many methods of time-frequency analysis

were already described by John von Neumann in his textbook on quantum

mechanics. However, it is Dennis Gabor's name, due to his seminal

article "Theory of communication", which is eponymous for the special

function systems in time-frequency analysis which we are going to

discuss. The key idea of Gabor systems is to have a joint representation

of a one-dimensional, time-dependent signal (a square-integrable

function of one real variable) in the time-domain and in the frequency-

(or Fourier-) domain. These systems are, for example, used for wireless

communication.

A usual property we expect from a Gabor system is that it actually forms

a frame, which is a generalization of a basis, because otherwise we are

not able to stably reconstruct a signal from the measurements with

respect to the Gabor system. Already under very mild conditions on the

so-called window function (the function used to localize the signal), a

Gabor system has to be redundant in order form a frame.

In particular, we cannot get an orthonormal basis with a Gaussian Gabor

system (the window being a Gauss function) and need over-complete

systems in order to guarantee stable reconstruction. Furthermore, we are

going to study the problem on how to choose the time-bands and

frequency-bands (channels) in a most efficient way. This leads to the

Strohmer and Beaver conjecture for Gaussian Gabor frames which, in full

generality, is still open and there are obvious connections to other

(open) problems in many branches of analysis, such as lattice energy

minimization problems.

<< Back