Analycity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$

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Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$analytic on $\mathbb{R}\backslash\{0\}$ but non-analytic at $x=0$. 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 ?

Examples of functions $f$ an $g$ can be $x\mapsto x^{-(1+\alpha)}e^{-1/x}1_{x>0}$ with $\alpha>0$.

Any references or solutions will be highly appreciated.

Thank you!