In this paper, we are concerned with functions defined on the cube $Q^m=[-\pi, \pi]^m$ and functions defined on the torus $\T^m$. Especially, the harmonic analysis of Sobolev-type spaces is carefully studied. We analyze in particular periodic distributions and distributions on the torus. We introduce a space similar to $H^1_0$, for which we prove a Poincaré-Wirtinger inequality. We prove that the usual Rellich-Kondrachov result does not hold for these last space because of the lack of compactness. A result of absolute continuity and density of regular functions is then established and a theorem of traces is obtained.