and $M_{ll'}=0$ since then only $w_0$ is non-zero and the $3j$ symbol forces $l=l'$ and hence $K$ to be even. The sum of $P_{ll'}$ and $M_{ll'}$ is approximately equal to the matrix that relates the temperature $C_l$s to the mean pseudo-$C_l$s [@hivon02] at high $l$. Their sum differs from the temperature case only by the presence of non-zero azimuthal quantum numbers $\pm 2$ in the $3j$ symbol. The mean of $\tilde{C}_l^{EB}$ evaluates to $$\langle \tilde{C}_l^{EB} \rangle = \sum_{l'} (P_{ll'} - M_{ll'}) C_{l'}^{EB},
\label{eq:13}$$ and so vanishes if parity is preserved in the mean ($C_l^{EB}=0$).

An example of the mean pseudo-$C_l$s for observations over a circular region of radius $15\degr$ is shown in Fig. \[fig:power\_and\_window\]. The weight function is uniform inside a circle of radius $10\degr$, but the remaining $5\degr$ is apodized as $\cos^2 [36(\theta-10\degr)]$ so that $w({\hat{\bmath{n}}})$ falls smoothly to zero at the edge of the observed region. For this weight function the matrix $P_{ll'}$ is roughly two orders of magnitude larger than $M_{ll'}$, and both are narrow compared to features in $C_l^E$ and $C_l^B$ except for the reionization signal on the largest scales. On small scales the mean pseudo-$C_l$s can be approximated by removing $C_{l}^E$ and $C_{l}^B$ from the 