Proposition 22

Let $f$ be a tempered distribution belonging to $\mathscr{M}^1(\mathbf{R}^n)$ (resp. to $L^2(\mathbf{R}^n)$). The Fourier transform of $f$ in $\mathscr{S}'(\mathbf{R}^n)$ is the tempered distribution associated with the Fourier transform of $f$ in $\mathscr{C}_0(\mathbf{R}^n)$ (resp. in $L^2(\mathbf{R}^n)$). Analogously for the inverse Fourier transform.