Fast Hankel transforms
H. K. Johansen and K. Sørensen
Journal name: Geophysical Prospecting
Issue: Vol 27, No 4, December 1979 pp. 876 - 901
Info: Article, PDF ( 1.33Mb )
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).