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∈[1,∞), ω is a suitable weight and (E0E)′ is the set of all E-periodic elements, then we prove that the dual of M(ω)∞,q∩(E0E)′ equals M(1/ω)∞,q′∩(E0E)′ by suitable extensions of Bessel's identity.