Papers › Fast SGL Fourier transforms for scattered data
Fast SGL Fourier transforms for scattered data
Christian Wülker
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Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type Lₙ₋ₗ₋₁^((l + 1/2))(r²) rˡ Yₗₘ(ϑ,φ), |m| ≤l < n ∈ℕ, Lₙ₋ₗ₋₁^((l + 1/2)) being a generalized Laguerre polynomial, Yₗₘ a spherical harmonic, constitute an orthonormal polynomial basis of the space L² on ℝ³ with radial Gaussian (multivariate Hermite) weight exp(-r²). We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in ℝ³. 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 proof an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.
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