A fast analysis-based discrete Hankel transform using asymptotic expansions

TL;DR

This paper introduces a fast, stable algorithm for computing the discrete Hankel transform and related expansions using asymptotic Bessel function expansions, FFT, and error-based parameter selection, achieving near-optimal efficiency.

Contribution

It presents a novel, asymptotic expansion-based algorithm that significantly improves computational speed and stability for Hankel transforms and related expansions.

Findings

Algorithm operates in O(N(log N)^2 / log log N) time

Numerical results confirm high efficiency and stability

Parameters are chosen based on error bounds for optimal performance

Abstract

A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The algorithm is based on an asymptotic expansion for Bessel functions of large arguments, the fast Fourier transform, and the Neumann addition formula. All the algorithmic parameters are selected from error bounds to achieve a near-optimal computational cost for any accuracy goal. Numerical results demonstrate the efficiency of the resulting algorithm.