Let $X$ be a real random variable, $\phi(x)=E[e^{ixX}],x \in \mathbb{R},$ its characteristic function.
Prove that $\phi \in L^2$ if and only if $X$ is absolutely continuous and having a density $f \in L^2.$ In this case $$\int_{\mathbb{R}}(f(x))^2dx=\frac{1}{2\pi}\int_{\mathbb{R}}|\phi(x)|^2dx.$$ Is there an inversion formula for $f?$
The above equality seems to be Plancherel identity, any suggestions are welcomed.
Suppose $\mu$ is a positive Borel measure on $\mathbb R$ with $\mu(\mathbb R)=1.$ Define the "Fourier transform" of $\mu$ to be
$$\phi(x)=\hat {\mu}(x) = \int e^{ixt}\,d\mu(t).$$
(All integrals are over $\mathbb R .$) We want to show $\phi\in L^2$ iff $\mu$ is absolutely continuous with $d\mu = \psi\,dx$ and $\psi\in L^2.$ Let's concentrate on the $\implies$ part of the proof, since $\impliedby$ is pretty obvious.
Suppose $\phi\in L^2.$ Then by Plancherel, $\phi=\hat{\psi}$ for some $\psi\in L^2.$ Let $g$ be in the Schwartz space $\mathcal S.$ Using standard identities, we have
$$ \int g\phi = \int g \hat {\mu} = \int \hat{g}\,d\mu.$$
Similarly,
$$ \int g\phi = \int g\hat {\psi} = \int \hat {g}\psi.$$
So we have $\int \hat{g}\,d\mu = \int \hat {g}\psi$ for all $g\in\mathcal S.$ That implies $\int g\,d\mu = \int g\psi$ for all $g\in\mathcal S.$ Since $\mathcal S$ is dense in $C_0,$ the Riesz Representation theorem shows that $d\mu= \psi(x)\,dx,$ which is the desired result.