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.