words, going from the rectilinear Fourier modes to spherical harmonic Bessel modes is simply a change of basis—to spherical harmonics—in angular Fourier coordinates. Now, suppose one were to form an estimate of $P(k)$ in by squaring $ \widetilde{T} (\mathbf{k})$ and then averaging over a shell of constant $|\mathbf{k}| = k$. Parseval’s theorem ensures that such a squaring and averaging operation is basis-independent. Thus, it does not matter whether the Fourier amplitudes on the shell of constant $|\mathbf{k}| = k$ are expressed in a spherical harmonic basis. Squaring and averaging $\overline{T}_{\ell m} (k)$ must therefore also yield the power spectrum, up to some $k$-dependent conversion factors to account for the radius of shells in Fourier space. Note that Eq. also cements the interpretation (suggested by our notation) that the quantity $k$ of our Fourier-Bessel basis is the total magnitude of the wavevector $\mathbf{k}$, rather than some wavenumber that only pertains to radial fluctuations.

Rotationally invariant fields in the spherical Fourier-Bessel formalism {#sec:RotationalInvarianceOnly}
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While the cosmological temperature field is expected to possess translationally invariant statistics, contaminants in an intensity mapping survey (such as foreground emission) will in general not possess such symmetry. This difference in symmetry will result in different signatures 