Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual $\psi^*(x)=\overline{\psi(x^{-1})}$. Another (perhaps easier) question is whether the above convolutions span the vector space $C^\infty(G)$ of smooth functions on $G$.
Both questions have puzzled me for a while, and I wonder if they are well known to the experts. (I have assumed that $G$ is compact for simplicity. Of course similar questions can be asked about compactly supported functions of any order over $G$.)
The only thing that comes to my mind about this type of functions is as follows:
Given $f\in L^2(G)$, the function $f\ast \tilde{f}$, where $\tilde{f}(g)=\overline{f(g^{-1})}$, is a function of positive type associated with the left regular representation of $G$. For the involved terminology and definitions see Appendix C of the following book:
Kazhdan's Property (T)., by B. Bekka, P. de la Harpe, A. Valette.
It is available here.