the key issues that are specific to the analysis of polarization, i.e. the geometric mixing of $E$ and $B$ modes. We then move on to develop approximations for arbitrary (smooth) weight functions, and test these against the exact covariances in cases where we can easily compute the latter (in particular, for azimuthally-symmetric weight functions). Simple rules-of-thumb for band-power variances are also derived in Section \[subsec:rot\] as limiting cases of the more general expressions; some of the manipulations of the $3j$ symbols required for these derivations are relegated to Appendix \[app:useful\_sums\]. Finally, in Section \[sec:implications\] we consider the implications of our results for the detection of gravitational waves with $B$-mode polarization estimated with pseudo-$C_l$ methods.

Throughout, we illustrate our results with the best-fitting power-law concordance $\Lambda$CDM model to the *Wilkinson Microwave Anisotropy Probe* (*WMAP*) and 2dF Galaxy Redshift Survey data (@spergel03; Table 7), but with the tensor-to-scalar ratio $r=0.15$ corresponding to $\phi^2$ inflation. We include the effects of gravitational lensing in the $B$-mode power spectrum, but ignore effects due to its non-Gaussianity. Theoretical power spectra, pseudo-$C_l$s and estimated power spectra in all figures include the smoothing effect of a Gaussian beam of ten-arcmin FWHM.

Properties of polarization pseudo-[$C_l$]{}s {#sec:properties}
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We 