but typically it is desirable to excise a larger region to remove further regions of high foreground emission. One can attempt to choose $w({\hat{\bmath{n}}})$ to improve the accuracy of subsequent power spectrum estimates. For example, on small scales, where the instrument noise dominates the signal, the optimal weighting is essentially inverse-noise-variance weighting [@efstathiou04], while on large scales uniform weighting is preferable. In this paper we shall assume that the weight is chosen to be real and to go smoothly to zero as excised regions are approached. The former restriction ensures that the weighting does not alter the polarization direction, while the latter makes $w({\hat{\bmath{n}}})$ approximately band-limited to some $l_{\rmn{max}}$. Such edge apodization allows us to deal with the effects of the resulting $E$-$B$ mixing in the pseudo-$C_l$ covariance matrices in a perturbative manner; see Section \[sec:approx-cov\]. The problem of $E$-$B$ mixing without edge apodization, i.e. taking $w({\hat{\bmath{n}}})$ everywhere one or zero, is discussed extensively by @lewis02 and @lewis03 in the context of isolating $B$ modes at the map level.

Following @hansen02b, we define pseudo-multipoles $\tilde{E}_{lm}$ and $\tilde{B}_{lm}$ to be the $E$ and $B$-mode multipoles of the weighted polarization map, so that $$\tilde{E}_{lm} \pm i \tilde{B}_{lm}
= \sum_{(lm)'} {}_{\pm2}I_{(lm)(lm)'} (E_{(lm)'}\pm iB_{(lm)'}),
