The Fourier transform on distributions coincide with the one on Schwartz spaces because with Fubini follows
\[ \hat{u}(\varphi) = \int \hat{u} \, \varphi = \int u \, \hat{\varphi} = u(\hat{\varphi}). \]for \( \varphi \in \mathcal{S}(\mathbb{R}^n) \).