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 (E E 0 ) ′ is the set of all E-periodic elements, then we prove that the dual of M ∞,q(

A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.

We deduce Paley-Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipović spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space H ♭1 = S C , deduced in [12].

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We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.