Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in L2(R), we determine a non-countable generalized frame for the non-separable space AP2(R) of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of AP2(R) are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.