A Nonuniform Fast Hankel Transform

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.