0. 
\label{eq:17}$$ These relations ensure that our estimates of the power spectra, $$\hat{C}_l^E = \sum_{l'} (\tilde{P}^{-1}_{ll'} \tilde{C}_{l'}^E +
\tilde{M}^{-1}_{ll'} \tilde{C}_{l'}^B), \quad
\hat{C}_l^B = \sum_{l'} (\tilde{P}^{-1}_{ll'} \tilde{C}_{l'}^B +
\tilde{M}^{-1}_{ll'} \tilde{C}_{l'}^E),
\label{eq:estimates}$$ satisfy $\langle \hat{C}_l^E \rangle = \sum_{l'} {}_{-2}\bar{K}_{ll'} C_{l'}^E$ and the equivalent relation for $C_l^B$. The normalised window function ${}_{-2}\bar{K}_{ll'}={}_{-2} K_{ll'} / \sum_L {}_{-2} K_{lL}$, where ${}_{-2}K_{ll'}$ is given by [@chon04] $${}_{-2}K_{ll'} \equiv \frac{2l'+1}{2} \int f(\beta) d^l_{2\,-2}(\beta)
d^{l'}_{2\,-2}(\beta) \, {{\rmn{d}}}\cos\beta.$$ Here, $d^l_{mn}(\beta)$ are the reduced Wigner rotation matrices and $f(\beta)$ is the apodizing function that is applied to the correlation functions. An example of the pseudo-inverses obtained via this route is shown in Fig. \[fig:power\_and\_window\], along with the mean of the recovered $C_l$s. The correlation functions have been apodized with a Gaussian with HWHM of $4\degr$. The apodizing function was deliberately chosen to be narrow to ensure that the resulting pseudo-inverses $\tilde{P}^{-1}_{ll'}$ and $\tilde{M}^{-1}_{ll'}$ are well localised. Given that the pseudo-inverses are significantly broader than $P_{ll'}$ and $M_{ll'}$, the window function ${}_{-2}K_{ll'}$ (not shown in the figure) inherits the width of the pseudo-inverses. For this reason the spectral resolution of the recovered $C_l$s is rather poor compared to what should ultimately be achievable for a $15\degr$-radius survey. The effect of this 