I am looking for a proof of Mathias theorem, as described on wikipedia and possible applications of it. (Polya's theorem has a nice consequence per example, it allows to show that if $X_1+X=^dX_2+X$ we may not have $X_1=^dX_2$).
The theorem states that:
A real-valued, even, continuous, absolutely integrable function φ, with φ(0) = 1, is a characteristic function if and only if :
$(-1)^n \left ( \int_{\mathbf{R}} \varphi(pt)e^{-\frac{t^2}{2}}H_{2n}(t)dt \right ) \geq 0$ for $n = 0,1,2,…$, and all $p > 0$. Here $H_{2n}$ denotes the Hermite polynomial of degree $2n$.