Periodic distributions and periodic elements in modulation spaces

We characterize periodic elements in Gevrey classes, Gelfand-Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If $q\in [1,\infty )$, $\omega$ is a suitable weight and $(\maclE _0^E)'$ is the set of all $E$-periodic elements, then we prove that the dual of $M^{\infty ,q}_{(\omega )}\cap (\maclE _0^E)'$ equals $M^{\infty ,q'}_{(1/\omega )}\cap (\maclE _0^E)'$ by suitable extensions of Bessel's identity.