##### Fast Hankel transforms

Authors:
H. K. Johansen and K. Sørensen

Journal name: Geophysical Prospecting

Issue: Vol 27, No 4, December 1979 pp. 876 - 901

DOI: 10.1111/j.1365-2478.1979.tb01005.x

Organisations:
Wiley

Language: English

Info: Article, PDF ( 1.33Mb )

Summary:

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 (ρ*x*)/sinh (*a*ρ*x*), 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|≤ω_{0} of the complex plane, we can show that the absolute error on the output function is less than (*K*(ω_{0})/*r*) · exp (−ρω_{0}/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).

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