We define a Fourier transform \(\mathcal{F}\) for a tempered distribution \(u\in \mathcal{S}'(\mathbb{R}^d)\) as follows,
\[ \mathcal{F}(u)(\varphi)=\hat{u}(\varphi)=u(\hat{\varphi}). \]It is not difficult to check, that \(\hat{u}\) is again a tempered distribution.
Remark