Time-periodic Gelfand-Shilov spaces and global hypoellipticity on T×R^n

We introduce a class of time-periodic Gelfand-Shilov spaces of functions on $mathbb{T} imes mathbb{R}^n$, where mathbb{T} sim mathbb{R}/2pi mathbb{Z}$ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on $mathbb{R}^n$. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on $mathbb{T} imes mathbb{R}^n$.