How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

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).

• Such relation involves Clebsch-Gordan coefficients. See, e.g. here. Jun 13 '13 at 13:32
• Thanks! That's pretty much what I was looking for (except I'd prefer it in a form that gives me $h_\ell^m$ based on the coefficients of $f$ and $g$, but I'll try to figure that out myself). You may want to consider posting it as an answer, so I can mark the question as answered. Jun 13 '13 at 14:08

1 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