This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions on \(\mathbb {T}^m \times \mathbb {R}^n\) . Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus on the first m variables and partial Fourier transform in Euclidean space on the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of rapidly decreasing smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the global regularity of a few classes of first-order differential operators defined on \(\mathbb {T}^1 \times \mathbb {R}\) , including all constant coefficient first-order differential operators, as well as complex vector fields with variable coefficients depending on the first variable.
