Abstract for the seminar talk <br>
delivered at the Wavelet Seminar of the MIT (Prof. G. Strang)
in April 1994 <br> 

<H2> <strong> Gabor expansions and coherent frames </strong> 
</H2> <br> 

In this talk some of the group theoretic structure behind
Gabor expansions as well as the so-called irregular sampling
theorem for band-limited functions is explained, allowing to 
pin-point some of the common structural features of those two
problems. We are talking about so-called <strong> coherent </strong>
frames, not just arbitrary un-structured frames, i.e. the 
frame is generated from one single building block ("atom")
by the action of a discrete subset of a (continuous) transformation
group. In the "regular" case we even have the situation, that
the dual frame is of the same form, with just another "window"
(transformed in exactly the same way as the atom). <br>

Based on this observation we point out efficient numerical
algorithms for the determination of the <strong> dual Gabor atom </strong>
(for the discrete and periodic case). These methods are based on
the conjugate gradient method and do not require to build the
(huge) system matrix for the corresponding linear equation
(the positive definite so-called frame operator). Whereas this
matrix  would be much too large (even from small-sized 1D problems, 
i.e. for signals of relatively small length) the explicit information
about the sparsity of the Gabor frame matrix can be used for
efficient algorithms. Cf. also the joint work with Sigang Qiu
in the NUHAG publication list for details.  
<hr> 

See our list of   <a href = "/papers/gabor.html">  Gabor publications </a> 
for details.