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FAST HANKEL TRANSFORMS *
Author(s) -
JOHANSEN H. K.,
SØRENSEN K.
Publication year - 1979
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.1979.tb01005.x
Subject(s) - hankel transform , sine and cosine transforms , mathematics , smoothness , sine , mathematical analysis , fast fourier transform , fourier transform , trigonometric functions , function (biology) , series (stratigraphy) , filter (signal processing) , discrete fourier transform (general) , discrete cosine transform , discrete sine transform , fourier analysis , algorithm , image (mathematics) , fractional fourier transform , geometry , computer science , paleontology , evolutionary biology , computer vision , biology , artificial intelligence
A bstract 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).