Spherical Bessel Transforms

Our algorithm for evaluating the scaled spherical Bessel transform equations \ref{eqn:sbtr} and \ref{eqn:sbtk}¹ follows earlier work in references \cite{Talman_2009,Hamilton_2000}.

With the change to dimensionless variables \[\kappa = \log (k/k_0) \quad , \quad \rho = \log (r/r_0) \; ,\] we can write equation \ref{eqn:sbtr} as a convolution suitable for FFT evaluation \[(r/r_0)^\alpha\cdot f_{\ell m}(r) = \int_{-\infty}^{+\infty} G(\rho + \kappa) F(\kappa) d\kappa \label{eqn:conv}\] with dimensionless functions \[G(s) = e^{\alpha s}\, j_{\ell}(k_0 r_0 e^s) \quad , \quad F(s) = c\, e^{(3-\alpha)s}\,k_0^3\,\tilde{f_{\ell m}}(k_0 e^s) \; ,\] where the \(r\)-weighting parameter \(\alpha\) is arbitrary at this point. The function \(F\) depends on \(f\), but \(G(s)\) depends only on the values of \(k_0 r_0\) and \(\alpha\) (see Fig. \ref{fig:fplot}), and has asymptotes \[G_{+}(s) = \frac{1}{k_0 r_0}\, e^{(\alpha-1)s} \quad , \quad G_{-}(s) = \frac{2^{-(\ell+1)}\sqrt{\pi}}{\Gamma(\ell+3/2)}\,(k_0 r_0)^\ell\,e^{(\alpha+\ell)s} \; ,\] where \(G_+\) is the envelope of oscillations with a rapidly decreasing period \(2\pi e^{-s}/(k_0 r_0)\).

In order to simplify the calculations, we use our freedom to chose \(k_0 r_0\) and \(\alpha\) to symmetrize \(G(s)\) via \[\alpha = \frac{1-\ell}{2} \quad , \quad \left( \frac{k_0 r_0}{2}\right)^{\ell+1} = \frac{\Gamma(\ell+3/2)}{\sqrt{\pi}} \;, \label{eqn:symm}\] which yields \[G_+(s) = G_-(-s) = \frac{1}{2}\left[ \frac{\sqrt{\pi}}{\Gamma(\ell+3/2)}\right]^{1/(\ell+1)}\cdot \exp\left(-(\ell+1)s/2\right) \; .\] We can write this using \[s_0 = \frac{2}{\ell + 1} \quad , \quad G_0 = \frac{1}{k_0 r_0} \;,\] as \[G_+(s) = G_-(-s) = G_0 e^{-s/s_0} \;.\]