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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 {\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.

65R10Integral transforms (numerical methods)
65D20Computation of special functions, construction of tables
44A15Special transforms (Legendre, Hilbert, etc.)
44A20Integral transforms of special functions
Full Text: DOI
[1] Filon, L.: On a quadrature formula for trigonometric integrals. Proc. roy. Soc. Edinburgh 49, 38-47 (1928--1929) · Zbl 55.0946.02
[2] Trantner, J.: Integral transforms in mathematical physics. (1966)
[3] Havie, T.: Remarks on an expansion for integrals of rapidly oscillating functions. Bit, 16-29 (1973) · Zbl 0273.65019
[4] Gaskell, J.: Linear systems, Fourier transforms, and optics. (1978)
[5] Mason, J.: Cylindrical Bessel functions for a large range of complex arguments. Comput. phys. Commun. 30, 1-11 (1983)
[6] Abramowitz, M.; Stegun, I.: Handbook of mathematical functions. (1968) · Zbl 0171.38503
[7] Sneddon, I.: Fourier transforms. (1951) · Zbl 0038.26801
[8] R. Barakat, E. Parshall and B. Sandler, Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy for optical diffraction theory and beam propagation, (in preparation).