**ABSTRACT**
### A space of test functions for Gabor analysis

#### Feichtinger, H.G. and Zimmermann, G.

##### Department of Mathematics, University of Vienna, Vienna, Austria

We introduce the Banach space S0 in L2 which has a variety of properties making it a useful tool in Gabor
analysis. S0 can be characterized as the smallest time-frequency homogeneous Banach space of
(continuous) functions. We also present other characterizations of S0 turning it into a very flexible tool for
Gabor analysis and allowing for simplifications of various proofs.

A careful analysis of both the coefficient and the synthesis mapping in Gabor theory shows that an
arbitrary window in S0 not only is a Bessel atom with respect to arbitrary time-frequency lattices, but also
yields boundedness between S0 and l1. On the other hand, we can study properties of general L2-atoms since
they induce mappings from S0 to S0'. This enables us to introduce a new, very natural concept of weak
duality of Gabor atoms, applying also to the classical pair of the Gauss-function and its dual function
determined by Bastiaans.

Using the established results, we show a variety of properties that are desirable in applications, like the
continuous dependence of the canonical dual window on the given Gabor window and on the lattice;
continuity of thresholding and masking operators from signal processing; and an algorithm for the
reconstruction of bandlimited functions from samples of the Gabor transform in a corresponding horizontal
strip in the time-frequency plane. We also present an approximate Balian-Low Theorem stating that for
close-to-critical lattices, the dual Gabor atoms progressively lose their time-frequency localization.