A fast analysis-based discrete Hankel transform using asymptotic expansions
This work provides a more efficient algorithm for a fundamental transform used in various scientific computing domains, offering improved speed over existing methods.
The paper presents a fast and numerically stable algorithm for computing the discrete Hankel transform of order 0 and evaluating Schlömilch and Fourier-Bessel expansions in O(N(log N)^2 / log log N) operations, achieving near-optimal computational cost for any accuracy goal.
A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schlömilch 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.