Proposition 19

Let $f$ be a tempered distribution associated with a bounded measure $\nu \in \mathscr{M}^1(\mathbf{R}^n) ($resp. with $g\in L^2(\mathbf{R}^n))$. The Fourier transform of $f$ in $\mathscr{S}'(\mathbf{R}^n)$ is the tempered distribution associated with the Fourier transform of the measure $\nu ($resp. with the Fourier transform of g).

Let $\nu$ be a bounded measure on $\mathbf{R}^n$ and $f$ the tempered distribution associated with $\nu$. The Fourier transform $\mathscr{F}(\nu )$ is a continuous and bounded function on $\mathbf{R}^n$ (prop. 3 of II, p. 207). The tempered distribution associated with this function satisfies

for every $\varphi \in \mathscr{S}(\mathbf{R}^n)$, where the second equality is prop. 13 of II, p. 221, which is applicable here since the measure $\varphi \cdot \mu$ is bounded. The tempered distribution associated with $\mathscr{F}(\nu )$ is therefore $\mathscr{F}(f)$.

When $f$ is the tempered distribution associated with $g\in L^2(\mathbf{R}^n)$, one follows an entirely analogous procedure using formula (29) of II, p. 221.

A similar statement is also valid for the Fourier cotransform.