$f(x)=\frac{1}{2}|x|$ is a fundamental solution for $\frac{d^2}{dx^2}$ on $\mathbb R$

Question

How to prove that $f(x)=\frac{1}{2}|x|$ is a fundamental solution for $\frac{d^2}{dx^2}$ on $\mathbb R$.I think it suffice to prove $\hat{(L(F))}=(-4\pi^2x^2)\hat F(x)=1$,where $F$ is a distribution of $\frac{1}{2}|x|$.But I'm stuck in calculating $\hat F(x)$.I guess I should find a function $g$ satisfying $\int \frac{1}{2}|x|\hat \phi(x)dx=\int g(x)\phi(x)dx$ which follows $\hat F=g$.

Answer 1

$\frac{d}{dx}\left(\frac{1}{2}|x|\right)=\frac{1}{2}\operatorname{sign}(x)$, so $$\frac{d^2}{dx^2}\left(\frac{1}{2}|x|\right)=\frac{d}{dx}\left(\frac{1}{2}\operatorname{sign}(x)\right)= \delta_0.$$