Abstract:
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.