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This article presents a convenient approach to Fourier analysis for theinvestigation of functions and distributions defined in $\mathbb{T}^m \times\mathbb{R}^n$. Our approach involves the utilization of a mixed Fouriertransform, incorporating both partial Fourier series on the torus for theinitial variables and partial Fourier transform in Euclidean space for theremaining variables. By examining the behaviour of the mixed Fouriercoefficients, we achieve a comprehensive characterization of the spaces of fastdecaying smooth functions and distributions in this context. Additionally, weapply our results to derive necessary and sufficient conditions for theSchwartz global hypoellipticity of a class first order differential operatorsdefined on $\mathbb{T} \times \mathbb{R}$, including all constant coefficientfirst order differential operators.

Fourier Analysis on $\mathbb{T}^m\times\mathbb{R}^n$ and Applications to Global Hypoellipticity