  This article presents a convenient approach to Fourier analysis for the
investigation of functions and distributions defined in $\mathbb{T}^m \times
\mathbb{R}^n$. Our approach involves the utilization of a mixed Fourier
transform, incorporating both partial Fourier series on the torus for the
initial variables and partial Fourier transform in Euclidean space for the
remaining variables. By examining the behaviour of the mixed Fourier
coefficients, we achieve a comprehensive characterization of the spaces of fast
decaying smooth functions and distributions in this context. Additionally, we
apply our results to derive necessary and sufficient conditions for the
Schwartz global hypoellipticity of a class of differential operators defined on
$\mathbb{T} \times \mathbb{R}$, including all constant coefficient first order
differential operators.
