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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(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:
65R10Integral transforms (numerical methods)
65D20Computation of special functions, construction of tables
44A15Special transforms (Legendre, Hilbert, etc.)
44A20Integral transforms of special functions