Description of the problem
Suppose we have a function which can be expressed in terms of spherical harmonics
\[\begin{equation} \label{eq:radial_fun} f(r, \theta, \phi) = \sum_{l=0}^\infty \sum_{m=-l}^l g_{l,m}(r) \, Y_l^m(\hat{\mathbf{r}}) \end{equation}\]What is the Fourier transform (FT) of $f(r, \theta, \phi)$?
\[\begin{equation} \label{eq:radial_ft} {\cal F}[f](\mathbf{k}) = (2\pi)^{-3/2} \int f(r, \theta, \phi) \, e^{-i \mathbf{k} \cdot \mathbf{r}} \, \mathrm{d}\mathbf{r}\, \end{equation}\]Expansion of plane-wave in spherical waves
To do this, we will need the expansion of a plane wave in spherical waves1:
\[\begin{align} e^{i \mathbf{k} \cdot \mathbf{r}} &= 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l i^l \, j_l(kr) \, Y_l^m(\hat{\mathbf{k}}) \, Y_l^{m*}(\hat{\mathbf{r}}) \label{eq:pw_sph_expan} \\ &= 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l i^l \, j_l(kr) \, Y_l^{m*}(\hat{\mathbf{k}}) \, Y_l^{m}(\hat{\mathbf{r}}) \label{eq:pw_sph_expan_2} \end{align}\]where
- $j_l(kr)$ is the spherical Bessel function, 2
- $Y_l^m(\theta, \phi)$ is the complex spherical harmonics, 3
- The superscript * denotes complex conjugation,
- The hat ^ denotes unit vector,
- $Y_l^{m*}(\theta, \phi) = (-1)^m Y_l^{-m}(\theta, \phi)$ is used from Eq.\eqref{eq:pw_sph_expan} to Eq.\eqref{eq:pw_sph_expan_2}.
Real spherical harmoics
As shown in the previous post, complex and real spherical harmonics can be inter-transformed by a unitary matrix $U^l$, i.e.
\[\begin{equation} \label{eq:c2r_umatrix} Y_l^m(\theta, \phi) = \sum_{m'=-l}^l U^l_{mm'} \,Y_{l,m'}(\theta, \phi) \end{equation}\]Inserting Eq.\eqref{eq:c2r_umatrix} into Eq.\eqref{eq:pw_sph_expan_2}, one get
\[\begin{align} e^{i \mathbf{k} \cdot \mathbf{r}} &= 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l i^l \, j_l(kr) \, \sum_{m_1=-l}^l U_{mm_1}^{l*} Y_{lm_1}(\hat{\mathbf{k}}) \, \sum_{m_2=-l}^l U_{mm_2}^{l} Y_{lm_2}(\hat{\mathbf{r}}) \\ &= 4\pi \sum_{l=0}^\infty \sum_{m_1=-l}^l \sum_{m_2=-l}^l i^l \, j_l(kr) \, Y_{lm_1}(\hat{\mathbf{k}}) Y_{lm_2}(\hat{\mathbf{r}}) \sum_{m=-l}^l \, U_{mm_1}^{l*} U_{mm_2}^{l} \\ &= 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l i^l \, j_l(kr) \, Y_{lm}(\hat{\mathbf{k}}) \, Y_{lm}(\hat{\mathbf{r}}) \label{eq:pw_real-sph_expan} \end{align}\]where we have utilized the fact that $U$ is a unitary matrix in the last line, i.e.
\[\begin{equation} U^{\dagger}\cdot U = \mathbb{1} \qquad\Longleftrightarrow\qquad \sum_{m} U_{mj}^*\cdot U_{mk} = \delta_{jk} \end{equation}\]Final results
Substituting Eq.\eqref{eq:pw_sph_expan} into Eq.\eqref{eq:radial_ft}, we have
\[\begin{equation} \label{eq:rad_ft_1} {\cal F}[f](\mathbf{k}) = (2\pi)^{-3/2} \int\!\mathrm{d}\mathbf{r}\, \left[ 4\pi \sum_{l'=0}^\infty \sum_{m'=-l'}^{l'} i^{-l'} \, j_{l'}(kr) \, Y_{l'}^{m'}(\hat{\mathbf{k}}) \, Y_{l'}^{m'*}(\hat{\mathbf{r}}) \right] \left[ \sum_{l=0}^\infty \sum_{m=-l}^l g_{l,m}(r) \, Y_l^m(\hat{\mathbf{r}}) \right] \end{equation}\]Utilizing the orthogonality of spherical harmonics4
\[\begin{equation} \label{eq:sph_ortho} \int_{\theta=0}^\pi \int_{\phi=0}^{2\pi} Y_l^m(\theta, \phi) Y_{l'}^{m'*}(\theta, \phi) \mathrm{d}\Omega = \delta_{ll'} \, \delta_{mm'} \end{equation}\]where $\mathrm{d}\Omega = \sin\theta\, \mathrm{d}\theta \mathrm{d}\phi$. Then, Eq.\eqref{eq:rad_ft_1} reduces to
\[\begin{equation} \label{eq:rad_ft_2} {\cal F}[f](\mathbf{k}) = \sqrt{\frac{2}{\pi}} \sum_{l,m} i^{-l} \, Y_l^m(\hat{\mathbf{k}}) \cdot \int_0^\infty j_l(kr) g_{lm}(r) r^2\mathrm{d}r \end{equation}\]Comparing Eq.\eqref{eq:rad_ft_2} and Eq.\eqref{eq:radial_fun}, one can see that $f$ and ${\cal F}[f]$ are of similar forms.
Note that the integral over $r$ in Eq.\eqref{eq:rad_ft_2} is the spherical Bessel transform.5