Gabor systems and almost periodic functions

We give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $A{P}_2(\mathbb{R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame for the whole space $A{P}_2(\mathbb{R})\mathrm{.}$ 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 sequencies.