3 A Banach space of test functions for Gabor analysis Hans G. Feichtinger Georg Zimmermann Acknowledgments: This work was supported by the Austrian Science Foun- dation FWF (grant S7001-MAT.) ABSTRACT We introduce the Banach space S 0 ⊆ L 2 which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest time-frequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 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 Ga- bor theory shows that an arbitrary window in S 0 not only is a Bessel atom with respect to arbitrary time-frequency lattices, but also yields bound- edness between S0 and ℓ 1 . On the other hand, we can study properties of general L 2 -atoms since they induce mappings from S 0 to S ′ 0 . This en- ables 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 de- sirable 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 algo- rithm 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. 3.1 Introduction This preprint is in final form. It appeared in: Gabor Analysis and Algorithms: Theory and Applications H.G. Feichtinger and T. Strohmer (eds.) Applied and Numerical Harmonic Analysis Birkh¨ auser, Basel 1998, pp. 123–170. One of the main difficulties in Gabor theory is the choice of the appropriate window for the short time Fourier transformation, to ensure that the STFT