convolutions in equation (\[eq:10\]) to give $$\langle \tilde{C}_l^E \rangle \approx C_l^E \sum_{l'} P_{ll'} + C_l^B
\sum_{l'} M_{ll'} , \quad
\langle \tilde{C}_l^B \rangle \approx C_l^B \sum_{l'} P_{ll'} + C_l^E
\sum_{l'} M_{ll'}. 
\label{eq:5-36}$$ For $\tilde{C}_l^E$ the mixing term described by $M_{ll'}$ is essentially negligible, and the mean of $\tilde{C}_l^E$ is approximately a scaled version of the underlying power spectrum. The relevant scaling is discussed below. For $\tilde{C}_l^B$ mixing is important, despite $P_{ll'}$ dominating over $M_{ll'}$, since $C_l^E \gg C_l^B$. This additional contribution to $\langle
\tilde{C}_l^B \rangle$ from geometric mixing of $E$ and $B$ modes can be clearly seen in Fig. \[fig:power\_and\_window\], and, as expected, looks like a scaled version of the $E$-mode power spectrum.

The normalisations $\sum_{l'} P_{ll'}$ and $\sum_{l'}M_{ll'}$ evaluate to $$\begin{aligned}
\sum_{l'} P_{ll'} + M_{ll'} &=& \sum_L \frac{2L+1}{4\pi} w_L 
= w^{(2)} f_{\rmn{sky}},
\label{eq:5-35a}
\\
\sum_{l'} P_{ll'} - M_{ll'} &=& \sum_{L \leq l}
\frac{2L+1}{4\pi} w_L
\left( 1-4\frac{L(L+1)}{l(l+1)} + 3\frac{(L+2)!}{(L-2)!}
\frac{(l-2)!}{(l+2)!} \right),
\label{eq:5-35}\end{aligned}$$ where $w^{(2)}f_{\rmn{sky}}$ is defined by $4\pi w^{(i)}
f_{\rmn{sky}} \equiv \int w^i({\hat{\bmath{n}}}) \, {{\rmn{d}}}{\hat{\bmath{n}}}$, and $f_{\rmn{sky}}$ is the fraction of sky with non-zero weighting. Equation (\[eq:5-35a\]) follows from the expansions of $P_{ll'}$ and $M_{ll'}$ in equations (\[eq:11\]) and (\[eq:12\]), and the orthogonality of the $3j$ symbols. It is the same 