INTRODUCTION Our goal is efficient and robust numerical evaluation of Fourier integrals of the form f() = N_n(a,b) \int\, d^n k\, e^{+b i (\cdot)}\, () \quad , \quad () = _n(a,b) \int\, d^n r\, e^{-b i (\cdot)}\,f() \; , with normalization factors N_n(a,b) = |b|^{n/2} (2\pi)^{-n(1+a)/2} \quad , \quad _n(a,b) = |b|^{n/2} (2\pi)^{-n(1-a)/2} \; , where the constants a and b establish our choice of Fourier convention[1]. 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) = ^{+\infty} f_m(r)\,\Phi_m(\varphi_r) \quad , \quad (k,\varphi_k) = ^{+\infty} (k)\,\Phi_m(\varphi_k) \; , using the polar basis functions \Phi_m(\varphi) \equiv {}\, e^{i m \varphi} with orthonormality[2] (δD and δ are the Dirac and Kronecker delta functions, respectively): ^{+\infty} \Phi_m(\varphi)\Phi^\ast_m(\varphi') = \delta_D(\varphi-\varphi') and \int_0^{2\pi} d\varphi\, \Phi_m(\varphi) ^\ast(\varphi) = \; . Similarly, for n = 3, we expand f(r,\theta_r,\varphi_r) = ^{\infty}^{+\ell} f_{\ell m}(r)\, Y_{\ell m}(\theta_r,\varphi_r) \quad, \quad (k,\theta_k,\varphi_k) = ^{\infty}^{+\ell} }(k)\, Y_{\ell m}(\theta_k,\varphi_k) \; , using the spherical-harmonic basis functions[3] (with associated Legendre polynomials Pℓm) Y_{\ell m}(\theta,\varphi) \equiv {2}{(\ell+m)!}}\, P_{\ell}^m(\cos\theta) \Phi_m(\varphi) with orthonormality[4] ^{\infty}^{+\ell} Y_{\ell m}(\theta,\varphi)Y_{\ell m}^\ast(\theta',\varphi') = \delta_D(\cos\theta-\cos\theta') \delta_D(\varphi-\varphi') and[5] \int d\Omega\, Y_{\ell m}(\theta,\varphi) Y_{\ell' m'}^\ast(\theta,\varphi) = \; . In the special case of a three-dimensional $f()$ that is cylindrically symmetric, _i.e._ has no ϕr dependence, only m = 0 terms contribute to the multipole expansion equation [eqn:multipole3]. Since Y_{\ell 0}(\theta,\varphi) = {4\pi}}\,L_{\ell}(\mu) with Lℓ the Legendre polynomial and μ ≡ cosθ, it is then convenient to replace equation [eqn:multipole3] with the equivalent expansion f(r,\mu_r) = ^\infty f^{(\mu)}_{\ell}(r) L_{\ell}(\mu_r) \quad , \quad (k,\mu_k) = ^\infty ^{(\mu)}_{\ell}(k) L_{\ell}(\mu_k) \; , in which the coefficient functions are simply rescaled f^{(\mu)}_{\ell}(r) = {4\pi}}\,f_{\ell 0}(r) \quad , \quad ^{(\mu)}_{\ell}(k) = {4\pi}}\,}(k) \; . [1] Use a = 1 and b = 1 for the convention of references . [2] http://dlmf.nist.gov/1.17#E12 [3] http://dlmf.nist.gov/14.30#E1 [4] http://dlmf.nist.gov/1.17#E25 [5] http://dlmf.nist.gov/14.30#E8