I need to prove that the following fundamental solution of the wave equation: $$ E(x,t)= \mathfrak{F}^{-1}\Bigg(\frac{\sin ( t|.|)}{|.|}\Bigg)\frac{\theta(t)}{(2\pi)^{n/2}}(x) \in \mathcal{S}'(\mathbb{R}^{1,n})$$ satisfies: $$ supp(E) \subset \{ (t,x) \in \mathbb{R}^{1,n}: |x| \leq t\}$$ (where $\mathfrak{F}^{-1}$ - inverse Fourie transform wrt. "spatial" coordinates).
It is also hinted that Paley-Wiener-Schwartz theorem is used in this.
I have found a theorem that states the following:
Let $u \in \mathcal{D}'(\mathbb{R}^n)$ and $f \in C^\infty(\mathbb{R}^n)$. If $fu=0$ then $$ supp(u) \subset \{ x\in \mathbb{R}^n | f(x)=0\} $$
Is this theorem somehow applicable here?