normalisation as for the temperature anisotropies. Equation (\[eq:5-35\]) can be established following the method used to derive equation (66) in @chon04; see also Appendix \[app:useful\_sums\]. Note that for $l$ much greater than the assumed band-limit of $w({\hat{\bmath{n}}})$, the normalisation $\sum_{l'} M_{ll'}$ is suppressed by a factor $O(l_{\rmn{max}}^2/l^2)$ compared to $\sum_{l'}
P_{ll'}$. This is consistent with the expectation that $E$-$B$ mixing is suppressed at high $l$ [@lewis02; @bunn03]. We can gain further insight into this result by noting that \[for smooth $w({\hat{\bmath{n}}})$\] $$\begin{aligned}
\sum_{L \leq l} (2L+1) L(L+1) w_L &=& \int |\nabla
w_{\rmn{s},l}|^2 \, {{\rmn{d}}}{\hat{\bmath{n}}}, \label{eq:14a} \\
\sum_{L \leq l} (2L+1) \frac{(L+2)!}{(L-2)!}
w_L &=& \int w_{\rmn{s},l} \nabla^2(\nabla^2+2) w_{\rmn{s},l} \, {{\rmn{d}}}{\hat{\bmath{n}}},
\label{eq:14b}\end{aligned}$$ where $w_{\rmn{s},l}({\hat{\bmath{n}}}) \equiv \sum_{L\leq l, M}w_{LM} Y_{LM}({\hat{\bmath{n}}})$ is $w({\hat{\bmath{n}}})$ smoothed with a top-hat ($l$-space) filter. For band-limited weight functions, at $l \gg l_{\rmn{max}}$ the smoothing has little effect and so to leading order $$\sum_{l'} M_{ll'} \sim \frac{1}{2\pi} \int \frac{|\nabla w|^2}{l(l+1)}
\, {{\rmn{d}}}{\hat{\bmath{n}}}\quad (l \gg l_{\rmn{max}}) .
\label{eq:15}$$ At a given scale $l$, the relative importance of mixing of $E$- and $B$-mode power in the pseudo-$C_l$s is thus controlled by $1/l^2$ times the ratio of the average squared gradient of the weight function to the average square weight function.

Inversion of 