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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. (English) Zbl 0903.65101
Summary: The purpose of this communication is to present an algorithm for evaluating zero-order Hankel transforms using ideas first put forward by 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 $h\left(p\right)$ and the latter is ${J}_{0}\left(rp\right)p\right)$. Here only $h\left(p\right)$ is approximated by a quadratic over the basic subinterval instead of the entire integrand $h\left(p\right){J}_{0}\left(rp\right)p$ being approximated. Since only $h\left(p\right)$ 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 $exp\left(irp\right)$ is periodic and translationally invariant, whereas ${J}_{0}\left(rp\right)$ is an almost periodic decaying function.
##### MSC:
 65R10 Integral transforms (numerical methods) 65D20 Computation of special functions, construction of tables 44A15 Special transforms (Legendre, Hilbert, etc.) 44A20 Integral transforms of special functions