  We consider three problems for Gabor frames that have recently received much
attention. The first problem concerns the approximation of dual Gabor frames in
$L_2(R)$ by finite-dimensional methods. Utilizing Wexler-Raz type duality
relations we derive a method to approximate the dual Gabor frame, that is much
simpler than previously proposed techniques. Furthermore it enables us to give
estimates for the approximation rate when the dimension of the finite model
approaches infinity. The second problem concerns the relation between the decay
of the window function $g$ and its dual $\gamma$. Based on results on
commutative Banach algebras and Laurent operators we derive a general condition
under which the dual $\gamma$ inherits the decay properties of $g$. The third
problem concerns the design of pulse shapes for orthogonal frequency division
multiplex (OFDM) systems for time- and frequency dispersive channels. In
particular, we provide a theoretical foundation for a recently proposed
algorithm to construct orthogonal transmission functions that are well
localized in the time-frequency plane.
