Convolution of $\mathcal{C}^\infty$ is analytic

Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as:

$$(f*g)(x) = \int_\mathbb{R}f(x-z)g(z)dz$$

an analytic function ?

Otherwise, under which conditions the convolution is analytic ?

Any references or solutions will be highly appreciated.

Thank you!