## Double Preconditioning for Gabor Frames

### Peter Balazs

(Acoustics Research Institute,
Austrian Academy of Science)

**Abstract:**
We present an application of the general idea of preconditioning
in the context of Gabor frames. While most (iterative) algorithms go for a
more or less costly exact numerical calculation of the dual Gabor atom, we
suggest here to look for ``very cheap methods'' to find an approximation of
the inverse Gabor frame matrix based on (double) preconditioning. We thereby
obtain very good approximations of the true dual Gabor atom at very low
computational costs.
Since the Gabor frame matrix commutes with certain time-frequency shifts it
is natural to make use of diagonal and circulant preconditioners sharing this
property.
Part of the efficiency of the proposed scheme results from the fact that all
the matrices involved share a well-known block matrix structure.
At least for the smooth Gabor atoms used typically the combination of these
two preconditioners leads consistently to very good results.
These claims are supported by numerical experiments
in the second part of the paper.