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Zbl 0903.65101
Barakat, R.; Parshall, E.
Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. (English)
[J] Appl. Math. Lett. 9, No.5, 21-26 (1996). ISSN 0893-9659

Summary: The purpose of this communication is to present an algorithm for evaluating zero-order Hankel transforms using ideas first put forward by {\it L. N. G. Filon} [|On a quadrature formula for trigonometric integrals, Proc. R. Soc. Edinb. 49, 38-47 (1928/29; JFM 55.0946.02)] in the context of finite range Fourier integrals. In Filon quadrature philosophy, the integrand is separated into the product of an (assumed) slowly varying component and a rapidly oscillating one (in our problem, the former is $h(p)$ and the latter is $J_0(rp)p)$. Here only $h(p)$ is approximated by a quadratic over the basic subinterval instead of the entire integrand $h(p)J_0(rp)p$ being approximated. Since only $h(p)$ has to be approximated, only a relatively small number of subintervals is required. In addition, the error incurred is relatively independent of the magnitude of $r$. There is a profound difference between the finite range Fourier integral and the zero-order Hankel transform in that $\exp(irp)$ is periodic and translationally invariant, whereas $J_0(rp)$ is an almost periodic decaying function.

MSC 2000:: *65R10 Integral transforms (numerical methods)
65D20 Computation of special functions
44A15 Special transforms
44A20 Integral transforms of special functions

Keywords: Bessel functions; algorithm; zero-order Hankel transforms; Filon quadrature; finite range Fourier integral

Citations: JFM 55.0946.02

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