Definition 7

The transpose of the Fourier transform on $\mathscr{S}(\mathbf{R}^n)$ (resp. of the inverse Fourier transform) is called the Fourier transform on $\mathscr{S}'(\mathbf{R}^n)$, and is denoted by $\mathscr{F}$ (resp. is called the inverse Fourier transform, and is denoted by $\overline{\mathscr{F}}$).

For $f\in \mathscr{S}'(\mathbf{R}^n)$, the tempered distribution $\mathscr{F}(f)$ (resp. $\overline{\mathscr{F}}(f)$) is defined by $\varphi \mapsto \langle f,\mathscr{F}(\varphi )\rangle$ for $\varphi \in \mathscr{S}(\mathbf{R}^n)$ (resp. by $\varphi \mapsto$ $\langle f,\overline{\mathscr{F}}(\varphi )\rangle$ ).

The Fourier transform on $\mathscr{S}'(\mathbf{R}^n)$ is an automorphism of topological vector spaces whose inverse is the inverse Fourier transform $\overline{\mathscr{F}}$.