Let the function $f$ be defined on $\mathbb{R}$ by $f(x)=e^{iax}$ with $a \in \mathbb{R}$. We have that $f \in S'(\mathbb{R})$, and $F \delta_a= e^{-iax}$ and $\overline{F} \delta_a=f$. where $F f $ is a Fourier transformation of $f$.
How we can deduce $Ff$? I try with inverse Fourier transformation but I don't understand how we can use it to obtain $Ff$.
Fourier transform is almost involutive, in the sense that $F^2=P$, where $P$ is the reflection operator defined by $(Pf)(x)=f(-x)$. From $F\delta_a=e^{-iax}$, we get $F\delta_{-a}=f$, and so $$ Ff = F^2\delta_{-a} = P\delta_{-a} = \delta_{-a} . $$