For each globally subanalytic set $X$, we construct a certain ring $\mathcal{C}^{\exp}(X)$ of complex-valued functions on $X$. We show that $\mathcal{C}^{\exp} = \{\mathcal{C}^{\exp}(X)\}_{X}$ is the smallest family of rings of functions on the subanalytic sets that is stable under integration and which contains all functions of the form $f$ and $\exp(if)$, where $f$ is globally subanalytic. We also show that $\mathcal{C}^{\exp}$ is closed under taking Fourier transforms, both in the sense of $L^1$ and of $L^2$. The proofs of these theorems involve extensive use of the subanalytic preparation theorem and also properties of almost periodic functions. This work is joint with R. Cluckers, G. Comte, J.-P. Rolin, and T. Servi.