Abstract. A Gabor or Weyl-Heisenberg frame for L 2 (R d ) is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost.In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space M 1 (R d ), which is a dense subspace of L 2 (R d ). Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.

The key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their Weyl symbols. This observation has two implications in time-frequency analysis. First, it implies the stability of general Gabor frames under small dilations of the time-frequency set, previously known only for the case where the time-frequency set is a lattice. Secondly, it allows us to derive a new Balian-Low theorem (BLT) for Gabor systems with window in the standard window class and with general time-frequency families. In contrast to the classical versions of BLT the new BLT does not only exclude orthonormal bases and Riesz bases at critical density, but indeed it even excludes irregular Gabor frames at critical density.

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H (G), defined for a locally compact abelian group G. For G = R d , H is the usual Heisenberg group. For the case when G is the finite cyclic group Z n , only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C n . We also include applications to Gabor frames.

Many results and problems in Fourier and Gabor analysis are formulated in the continuous variable case, i.e., for functions on R. In contrast, a suitable setting for practical computations is the finite case, dealing with vectors of finite length. We establish fundamental results for the approximation of the continuous case by finite models, namely, the approximation of the Fourier transform and the approximation of the dual Gabor window of a Gabor frame. The appropriate function space for our approach is the Feichtinger space S 0 . It is dense in L 2 , much larger than the Schwartz space, and it is a Banach space.