Recently, I've been interested in expanding an angular function over the spherical harmonics, and particularly in retrieving the amplitude of the part corresponding to a given degree \(\ell\). More precisely, let \(F(\Omega) = F(\theta,\phi) =\sum_{\ell} \sum_{m} Y_{\ell m} (\Omega)\). The projection of \(F\) onto the subspace spanned by the harmonics with a given degree \(\ell\) (I believe this space is generally denoted by \(\mathcal{H}_{\ell}\)) is:

\begin{equation}

\label{eq:proj1}

\operatorname{Proj}_{\ell} \left [F \right ] (\Omega) = \sum_{m= - \ell}^{\ell} c_{\ell m} Y_{\ell m} (\Omega)

\end{equation}

which can be rewritten using \(c_{\ell m} = \int \text{d}\Omega Y^*_{\ell m} (\Omega) F(\Omega)\) and the addition theorem of spherical harmonics as:

\begin{equation}

\label{eq:proj2}

\operatorname{Proj}_{\ell} \left [F \right ] (\Omega) = \frac{2 \ell +1}{4 \pi} \int \text{d}\Omega ' F(\Omega ') P_{\ell} \left [\cos \left ( \widehat{\Omega , \Omega '} \right ) \right ]

\end{equation}

and I want to determine the coefficient

\begin{equation}

\label{eq:cl1}

c_{\ell} = \sum_{m= - \ell}^{\ell} \left | c_{\ell m} \right |^2 = \int \text{d}\Omega \, \left | \operatorname{Proj}_{\ell} \left [F \right ] (\Omega) \right |^2

\end{equation}

At this point I can either compute all the individual \(c_{\ell m}\)'s and sum them up (which is probably the faster option) or develop \eqref{eq:cl1} using \eqref{eq:proj1} and the closure relation to yield:

\begin{equation}

\label{eq:cl2}

c_{\ell} = \int \text{d}\Omega \int \text{d}\Omega ' \, F^*(\Omega) F(\Omega ') P_{\ell} \left [\cos \left ( \widehat{\Omega , \Omega '} \right ) \right ]

\end{equation}

I'm pretty sure \eqref{eq:cl2} is a standard result (with a more elegant proof than the pedestrian strategy I used) but I couldn't find it anywhere.

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