relative separation in comoving units between $(x,\theta,\phi)$ and $(x',\theta',\phi')$. The equation (\[eq3-24\]) is inverted as $$W(kR) = 4\pi \int {S_K}^2(x) dx W_R(x) X_0(k,x).
\label{eq3-25}$$ Since $R$ is much less than the curvature scale, $R \ll |K|^{-1/2}$, the above equation accurately approximates to $$W(kR) = 4\pi \int x^2 dx W_R(x) j_0(kx) =
  \int d^3x W_R(x) e^{-i{{\mbox{\scriptsize\boldmath $k$}}}\cdot{{\mbox{\scriptsize\boldmath $x$}}}}
\label{eq3-26}$$ Thus, the function $W(kR)$ is the Fourier transform of the kernel, or the spherical window function of the kernel. The popular choices are the top-hat window function $W(kR) = 3 j_1(kR)/(kR) = 3(\sin kR - kR
\cos kR)/(kR)^3$, and the Gaussian window function $W(kR) = \exp(-k^2
R^2/2)$. Useful window functions are given in Appendix \[app3\], including the $m$-weight Epanechnikov kernel, which might also be useful.

There is a useful addition theorem for $X_l$ [e.g., @rat95; @mat00], $$X_0(k,x_{\rm rel}) =
  \sum_{l=0}^\infty (2l+1) X_l(k,x) X_l(k,x') P_l(\cos\theta_{\rm rel}),
\label{eq3-27}$$ where $X_l$ is given by equation (\[eqa2-4\]), $P_l(x)$ are the Legendre polynomials, $\theta_{\rm rel}$ is the relative angle between the directions $(\theta,\phi)$ and $(\theta',\phi')$. The sum rule (\[eq3-27\]) holds even for a non-flat case $K \neq 0$ as well as a flat case $K=0$. The Legendre polynomials are also expanded by the spherical harmonics $Y_l^m$, $$P_l(\cos\theta_{\rm 