What does it mean for a distribution to have support on the real axis? I'm ask this because I just saw the following:
If $d$ is a distribution with support on the real axis, the Laplace-Borel transform coincides with the Fourier transform.
My attempt: Let $\mathcal{L}(d)(\xi)\equiv \langle d(z),\mathrm{e}^{z\xi}\rangle$ the Laplace-Borel of the distribution $d$. Here, $z$ and $\xi$ are in $\mathbb{C}$ and $d(z)$ indicates the variable where the distributions acts.
\begin{align} \mathcal{L}(d)(-i\xi)&\equiv\langle d(z),\mathrm{e}^{-iz\xi}\rangle\\ &=\int_{-\infty}^{\infty}d(x)\mathrm{e}^{-ix \xi} dx\text{ ? }\\ &=\mathcal{F}(d(x))(\xi)\text{ ?} \end{align} where $d(x)=d(z=x+i0)$?