We could also form $\tilde{C}_l^{EB} \equiv \sum_m \tilde{E}_{lm}
\tilde{B}_{lm}^*/(2l+1)$, but, as we show shortly, this vanishes in the mean in the absence of parity violations. However, it may provide a useful test on parity-violating physics or unaccounted-for systematics, and so future data should, of course, be validated with $\tilde{C}_l^{EB}$.

In the absence of instrument noise, the mean pseudo-$C_l$s (i.e.averaged over CMB realisations) are related to the true power spectra by convolutions. We find [@hansen02b] $$\langle \tilde{C}_l^E \rangle = \sum_{l'} (P_{ll'} C_{l'}^E +
M_{ll'} C_{l'}^B),
\quad
\langle \tilde{C}_l^B \rangle = \sum_{l'} (M_{ll'} C_{l'}^E +
P_{ll'} C_{l'}^B),
\label{eq:10}$$ where $$\begin{aligned}
P_{ll'} &\equiv& \frac{1}{2l+1} \sum_{mm'} |{}_+ I_{(lm)(lm)'}|^2
        = \frac{2l'+1}{8\pi} \sum_L (2L+1) w_L [1+(-1)^K]
\left( \begin{array}{ccc} l & l' & L \\ -2 & 2 &0 \end{array} \right)^2 ,
\label{eq:11} \\
M_{ll'} &\equiv& \frac{1}{2l+1} \sum_{mm'} |{}_- I_{(lm)(lm)'}|^2
        = \frac{2l'+1}{8\pi} \sum_L (2L+1) w_L [1-(-1)^K]
\left( \begin{array}{ccc} l & l' & L \\ -2 & 2 &0 \end{array} \right)^2 .
\label{eq:12}\end{aligned}$$ Here $K \equiv l+l'+L$ and we have introduced the power spectrum of the weight function $w_l \equiv \sum_m |w_{lm}|^2 / (2l+1)$. A non-zero matrix $M_{ll'}$ biases the pseudo-$C_l$s by mixing e.g. $E$-mode power into $\tilde{C}_l^B$. For observations with uniform weight over the full sky, $w({\hat{\bmath{n}}})=\rmn{constant}$ 