k_1) \delta^D (k^\prime - k_2)   \nonumber \\
&& = \frac{\delta^D(k - k^\prime) }{k^2} \delta_{\ell \ell^\prime} \delta_{m m^\prime} P(k),\end{aligned}$$ where the last equality follows from Eq. and some algebraic simplifications. From this, we see that forming the power spectrum from $\overline{T}_{\ell m} (k)$ modes is remarkably similar to forming it from the rectilinear Fourier modes. Comparing Eqs. and , we see that if (roughly speaking) one can form $P(k)$ by squaring $\widetilde{T} (\mathbf{k})$ and normalizing appropriately, one can equally well form $P(k)$ by squaring $\overline{T}_{\ell m} (k)$ and normalizing (albeit with a different—and $k$ dependent—normalization that we will derive more explicitly in Section \[sec:FiniteVolume\]).

To understand why the squaring of $\overline{T}_{\ell m} (k)$ produces such a similar result to squaring $\widetilde{T} (k)$ (with both giving a result proportional to the power spectrum), notice that Eq. can be simplified to give $$\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int d\Omega_k Y^*_{\ell m} ({\hat{\mathbf{k}}})  \widetilde{T} (\mathbf{k})\bigg{|}_{|\mathbf{k}| = k},$$ where $ \widetilde{T} (\mathbf{k})$ is restricted to the shell where $|\mathbf{k}| = k$. In this form, one sees that an alternate way to understand our spherical harmonic Bessel modes is to view them as a spherical harmonic decomposition of $ \widetilde{T} (\mathbf{k})$ in Fourier space. In other 