Definition 6

The transpose of the Fourier transform on $\mathscr{S}(\mathbf{R}^n)$ (resp. of the Fourier cotransform) is called the Fourier transform (resp. Fourier cotransform) on $\mathscr{S}'(\mathbf{R}^n)$.

The notation $\mathscr{F}$ (resp. $\overline{\mathscr{F}}$) is again used for the Fourier transform (resp. the Fourier cotransform) on $\mathscr{S}'(\mathbf{R}^n)$. The Fourier transform on $\mathscr{S}'(\mathbf{R}^n)$ is therefore an automorphism of topological vector spaces whose inverse is the Fourier cotransform. For every $f\in \mathscr{S}'(\mathbf{R}^n)$, the tempered distribution $\mathscr{F}(f)$ (resp. $\overline{\mathscr{F}}(f)$) is defined by the formula

$\langle \mathscr{F}(f), \varphi \rangle =\langle f,\mathscr{F}(\varphi )\rangle ($resp. $\langle \overline{\mathscr{F}}(f), \varphi \rangle =\langle f,\overline{\mathscr{F}}(\varphi )\rangle )$

for every $\varphi \in \mathscr{S}(\mathbf{R}^n)$.