Some Problems Related to the Distributional Zak Transform.

We define the distributional Zak transform and study some of its properties. We show how the distributional Zak transform can be used as an effective tool in the theory of Gabor systems where the window function belongs to the Schwartz class S(ℝ) and where the product of the parameters defining the Gabor system is rational. In particular, we obtain a necessary and sufficient condition for the linear span of such a Gabor system to be dense in S(ℝ) in the topology of S(ℝ) and, if this is the case, we show that a dual window in the Schwartz class can be constructed. We also characterize when such a Gabor system satisfies the Riesz property. [ABSTRACT FROM AUTHOR]