Commutativity of distributional convolution

Prove that if one of the generalized functions $F$ and $G$ is an element of $\mathcal{E}^{\prime }$, and the other an element of $\mathcal{D}^{\prime }$, then the convolution $F * G$ is defined, is an element of the space $\mathcal{D}^{\prime }$, and the relation $F * G = G * F$ holds.

Prove that if one of the generalized functions $F$ and $G$ is an element of $\mathcal{E}^{\prime }$, and the other an element of $\mathcal{S}^{\prime }$, then the convolution $F * G$ is defined, is an element of the space $\mathcal{S}^{\prime }$, and the relation $F * G = G * F$ holds.

Let $\check f(x)=f(-x)$. Prove the identity $(f_1*f_2)=\check f_1*\check f_2$, where $f_1$ and $f_2$ are generalized functions, one of which has compact support.