23 April 2015

Projection onto the subspace of spherical harmonics with the same degree

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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