Bessel functions rather than some other set of basis functions is arbitrary. However, we will now show that spherical Bessel functions are a particularly good choice for describing temperature fields that are statistically translation invariant. Translation-invariant fields admit a representation in terms of their power spectrum $P(k)$, which we define implicitly via the equation[^6] $$\label{eq:RectilinearPspecDef}
\langle \widetilde{T} (\mathbf{k}) \widetilde{T}^* (\mathbf{k^\prime}) \rangle = (2 \pi)^3 \delta^D (\mathbf{k} - \mathbf{k}^\prime) P(k),$$ where the angled brackets $\langle \cdots \rangle$ signify an ensemble average over random realizations of the cosmological temperature field $T(\mathbf{r})$, whose Fourier transform $\widetilde{T} (\mathbf{k})$ we define by the convention $$\label{eq:forwardNormal}
\widetilde{T} (\mathbf{k}) = \int \! d^3 r \,e^{-i \mathbf{k} \cdot \mathbf{r}} T(\mathbf{r})$$ with the inverse transform given by $$\label{eq:inverseNormal}
T(\mathbf{r}) = \int \! \frac{d^3 k}{(2 \pi)^3} e^{i \mathbf{k} \cdot \mathbf{r}} \widetilde{T} (\mathbf{k}).$$ Unless otherwise stated, this Fourier convention for the temperature field will be the one used for all Fourier transforms in this paper. Ideally, our spherical Fourier-Bessel description should be directly relatable to $P(k)$, for it would be pointless if an estimation of the power spectrum required first returning to position space. We will now show that this requirement is met by our $\overline{T}_{\ell m} (k)$ modes.

To relate 