these with the appropriate Legendre transforms [@chon04]. For observations over smaller parts of the sky, $P_{ll'}$ and $M_{ll'}$ will not be invertible. In this case we do not have the spectral resolution to obtain unbiased estimates of the $C_l$s at every $l$. Various approaches have been suggested to regularise the inversion including the use of band-powers [@hivon02; @hansen02], or apodized Legendre transforms of the (incomplete) correlation functions [@szapudi01; @chon04]. The latter method provides estimates of the $C_l$s convolved with a window function that depends only on the choice of apodizing function that is applied to the correlation function. @chon04, building on earlier work by @crittenden02, showed how to generalise the correlation function method to ensure that the estimated polarization power spectra did not mix $E$ and $B$ power in the mean. This is particularly useful for visualisation of the $B$-mode power spectrum which can easily be swamped by the dominant $E$ modes. We do not reproduce the method of @chon04 here, but note only that it provides a convenient way of constructing pseudo-inverses $\tilde{P}^{-1}_{ll'}$ and $\tilde{M}^{-1}_{ll'}$ to $P_{ll'}$ and $M_{ll'}$ respectively with the properties that $$\sum_{l'} (\tilde{P}^{-1}_{ll'} P_{l'l''} +\tilde{M}^{-1}_{ll'} M_{l'l''})
= {}_{-2}\bar{K}_{ll''} , \quad
\sum_{l'} (\tilde{P}^{-1}_{ll'} M_{l'l''} +\tilde{M}^{-1}_{ll'} P_{l'l''})
= 