Approximation of the Fourier Transform and the Dual Gabor Window

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 S0. It is dense in L 2, much larger than the Schwartz space, and it is a Banach space.