Fast SGL Fourier transforms for scattered data

TL;DR

This paper introduces fast algorithms for SGL Fourier transforms applicable to scattered data, leveraging non-equispaced FFTs and providing error estimates validated through numerical experiments.

Contribution

It presents novel fast SGL Fourier transform algorithms for scattered data using NFFT, improving computational efficiency and accuracy.

Findings

Algorithms achieve high accuracy in numerical experiments.

Error estimates are rigorously established.

Significant speedup over previous methods.

Abstract

Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^l Y_{lm}(\vartheta,\varphi)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal polynomial basis of the space $L^2$ on $\mathbb{R}^3$ 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 $\mathbb{R}^3$. 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 proof an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.