FAST HANKEL TRANSFORMS*

Abstract

Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type:

Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*.

If the input function f(λ exp (iω)) is known to be analytic in the region o < λ < ∞, |ω|≤ω₀ of the complex plane, we can show that the absolute error on the output function is less than (K(ω₀)/r) · exp (−ρω₀/Δ), Δ being the logarthmic sampling distance.

Due to the explicit expansions of H* the tails of the infinite summation

((m−n)Δ) can be handled analytically.

Since the only restriction on the order is ν > − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).