A Nonuniform Fast Hankel Transform

TL;DR

This paper introduces a fast, accurate algorithm for computing discrete Hankel transforms of moderate order on nonuniform points, combining Bessel expansions and nonuniform FFTs for efficient performance.

Contribution

It presents a novel algorithm that efficiently computes nonuniform discrete Hankel transforms with adjustable precision, improving speed and accuracy over previous methods.

Findings

Algorithm achieves $O((m+n) \, \log\min(n,m))$ complexity.

Demonstrates high accuracy and speed in various numerical tests.

Applicable to multiple regimes and practical applications.

Abstract

We describe a fast algorithm for computing discrete Hankel transforms of moderate orders from $n$ nonuniform points to $m$ nonuniform frequencies in $O((m+n)\log\min(n,m))$ operations. Our approach combines local and asymptotic Bessel function expansions with nonuniform fast Fourier transforms. The order of each expansion is adjusted automatically according to error analysis to obtain any desired precision $\varepsilon$. Several numerical examples are provided which demonstrate the speed and accuracy of the algorithm in multiple regimes and applications.