Periodic distributions and periodic elements in modulation spaces

Abstract: 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 $(\maclE <em>0<sup>E)&#39;$ is the set of all $E$-periodic elements, then we prove that the dual of $M^{\infty ,q}</em>{(\omega )}\cap (\maclE <em>0<sup>E)&#39;$ equals $M^{\infty ,q&#39;}</em>{(1/\omega )}\cap (\maclE _0<sup>E)&#39;$ by suitable extensions of Bessel's identity.