I am looking for a reference to the following question, which I am sure has been studied. Let $f: \mathbb{R}^n \to \mathbb{R}$ and define \begin{align} g(x)= f(x) \star e^{-\|x\|^2}, \end{align} where $\star$ denots convolution operation.
What are some minimum or mild assumptions that we can make on $f(x)$ such that $g(x)$ is a real-analytic function?
I am looking for a reference or book that has such a theorem. Note, that I am especially interested in $n>1$.
By Morera's theorem if $$F(z)=\int_{-\infty}^\infty f(x)e^{-(z-x)^2}dx$$ converges locally uniformly then $F$ is entire.
From there we get that if for all $c > 0$ as $r\to \infty$ $$\int_{-r}^r|f(x)|dx = O(e^{r^2-cr})$$ then $F$ is entire.
This condition is of course not necessary (try with $f(x)=(\sin(e^{e^x}))'$) and I doubt there is any minimal condition.