as we know, in $R^n$, for a function $f$, we can define its Fourier transform as $$\hat f(\xi)=\int_{R^3}f(x)e^{-ix\cdot \xi}d x,$$ with this, the Laplacian of $f$ can be elegently defined by $$\widehat{-\triangle f}(\xi)=|\xi|^2\hat f(\xi).$$ Also, the fractional Laplacian is defined by $$\widehat{(-\triangle)^\alpha f}(\xi)=|\xi|^{2\alpha}\hat f(\xi).$$
What I want to know is the corresponding result for functions defined in a torus $T^n=R^n/Z^n$.
Would you help me? Thank you.