Fast SGL Fourier transforms for scattered data

Abstract Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L n − l − 1 ( l + 1 / 2 ) ( r 2 ) r l Y l m ( ϑ , φ ) , | m | ≤ l n ∈ N , L n − l − 1 ( l + 1 / 2 ) being a generalized Laguerre polynomial, Y l m a spherical harmonic, constitute an orthonormal polynomial basis of the space L 2 on R 3 with radial Gaussian (multivariate Hermite) weight exp ⁡ ( − r 2 ) . We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in R 3 . In this paper, we present fast SGL Fourier transforms for scattered data. The idea is to employ well-known basal fast algorithms to determine a three-dimensional trigonometric polynomial that coincides with the bandlimited function of interest where the latter is to be evaluated. This trigonometric polynomial can then be evaluated efficiently using the well-known non-equispaced FFT (NFFT). We prove an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.