Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on the sphere $f$ and $g$ that both have such expansions, then their product $h(\theta,\varphi)=f(\theta,\varphi)\,g(\theta,\varphi)$ also has an expansion. Now, my question is: how do the coefficients $f_\ell^m$, $g_\ell^m$ and $h_\ell^m$ relate?
Since the spherical harmonics expansion is quite similar to a Fourier series, I would expect some sort of convolution theorem to apply, but I can only really find one for the other way around. That is, that spherical convolution corresponds to multiplication of the coefficients in the expansion. I am interested in multiplication of spherical functions, and what that does to the coefficients.
I did find this question, but it does not really seem to be answered. In particular, the accepted answer refers to spherical convolution (which is the wrong way around).