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).
Best Answer
I think I found what you were looking for (I've been looking all day!). See equations A.33-A.38 in http://geo.mff.cuni.cz/~lh/phd/cha.pdf : $$ h_{nm} = \sum_{n_1m_1} \sum_{n_2m_2}f_{n_1m_1}g_{n_2m_2}Q^{nm}_{n_1m_1n_2m_2} $$ where $$ Q^{nm}_{n_1m_1n_2m_2} = \sqrt{\frac{(2n_1+1)(2n_2+2)}{4\pi(2n+1)}} C^{n0}_{n_10n_20} C^{nm}_{n_1m_1n_2m_2} $$ and $C^{nm}_{n_1m_1n_2m_2}$ are the Clebsch-Gordan coefficients as mentioned by user "Start wearing purple" above