Multipole Transforms


Our goal is efficient and robust numerical evaluation of Fourier integrals of the form \[f(\vec{r}) = N_n(a,b) \int\, d^n k\, e^{+b i (\vec{k}\cdot\vec{r})}\, \tilde{f}(\vec{k}) \quad , \quad \tilde{f}(\vec{k}) = \tilde{N}_n(a,b) \int\, d^n r\, e^{-b i (\vec{k}\cdot\vec{r})}\,f(\vec{r}) \; , \label{eqn:fourier}\] with normalization factors \[N_n(a,b) = |b|^{n/2} (2\pi)^{-n(1+a)/2} \quad , \quad \tilde{N}_n(a,b) = |b|^{n/2} (2\pi)^{-n(1-a)/2} \; ,\] where the constants \(a\) and \(b\) establish our choice of Fourier convention1. We focus on two- and three-dimensional (\(n=2,3\)) transforms of functions that can be adequately represented with a small number of (not necessarily low order) multipoles. Specifically, for \(n=2\), we expand \[f(r,\varphi_r) = \sum_{m=-\infty}^{+\infty} f_m(r)\,\Phi_m(\varphi_r) \quad , \quad \tilde{f}(k,\varphi_k) = \sum_{m=-\infty}^{+\infty} \tilde{f_m}(k)\,\Phi_m(\varphi_k) \; , \label{eqn:multipole2}\] using the polar basis functions \[\Phi_m(\varphi) \equiv \frac{1}{\sqrt{2\pi}}\, e^{i m \varphi}\] with orthonormality2 (\(\delta_D\) and \(\delta\) are the Dirac and Kronecker delta functions, respectively): \[\sum_{m=-\infty}^{+\infty} \Phi_m(\varphi)\Phi^\ast_m(\varphi') = \delta_D(\varphi-\varphi') \label{eqn:dirac2}\] and \[\int_0^{2\pi} d\varphi\, \Phi_m(\varphi) \Phi_{m'}^\ast(\varphi) = \delta_{m m'} \; . \label{eqn:kronecker2}\] Similarly, for \(n = 3\), we expand \[f(r,\theta_r,\varphi_r) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{+\ell} f_{\ell m}(r)\, Y_{\ell m}(\theta_r,\varphi_r) \quad, \quad \tilde{f}(k,\theta_k,\varphi_k) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{+\ell} \tilde{f_{\ell m}}(k)\, Y_{\ell m}(\theta_k,\varphi_k) \; , \label{eqn:multipole3}\] using the spherical-harmonic basis functions3 (with associated Legendre polynomials \(P_{\ell}^m\)) \[Y_{\ell m}(\theta,\varphi) \equiv \sqrt{\frac{2\ell+1}{2}\frac{(\ell-m)!}{(\ell+m)!}}\, P_{\ell}^m(\cos\theta) \Phi_m(\varphi)\] with orthonormality4 \[\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{+\ell} Y_{\ell m}(\theta,\varphi)Y_{\ell m}^\ast(\theta',\varphi') = \delta_D(\cos\theta-\cos\theta') \delta_D(\varphi-\varphi') \label{eqn:dirac3}\] and5 \[\int d\Omega\, Y_{\ell m}(\theta,\varphi) Y_{\ell' m'}^\ast(\theta,\varphi) = \delta_{\ell\ell'}\delta_{m m'} \; . \label{eqn:kronecker3}\] In the special case of a three-dimensional \(f(\vec{r})\) that is cylindrically symmetric, i.e. has no \(\phi_r\) dependence, only \(m = 0\) terms contribute to the multipole expansion equation \ref{eqn:multipole3}. Since \[Y_{\ell 0}(\theta,\varphi) = \sqrt{\frac{2\ell+1}{4\pi}}\,L_{\ell}(\mu) \label{eqn:sphericalmu}\] with \(L_{\ell}\) the Legendre polynomial and \(\mu \equiv \cos\theta\), it is then convenient to replace equation \ref{eqn:multipole3} with the equivalent expansion \[f(r,\mu_r) = \sum_{\ell=0}^\infty f^{(\mu)}_{\ell}(r) L_{\ell}(\mu_r) \quad , \quad \tilde{f}(k,\mu_k) = \sum_{\ell=0}^\infty \tilde{f}^{(\mu)}_{\ell}(k) L_{\ell}(\mu_k) \; , \label{eqn:multipole3mu}\] in which the coefficient functions are simply rescaled \[f^{(\mu)}_{\ell}(r) = \sqrt{\frac{2\ell+1}{4\pi}}\,f_{\ell 0}(r) \quad , \quad \tilde{f}^{(\mu)}_{\ell}(k) = \sqrt{\frac{2\ell+1}{4\pi}}\,\tilde{f_{\ell 0}}(k) \; . \label{eqn:coefsmu}\]

  1. Use \(a = 1\) and \(b = 1\) for the convention of references (Dodelson 2003, Weinberg 2013).