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Bessel transforms and rational extrapolation. (English) Zbl 0554.65010
A numerical method is developed which handles the Bessel transform of functions having slow rates of decrease, i.e. $f\left(u\right)=O\left({u}^{-\alpha }\right)$, $u\to +\infty$ $\left(\alpha >0\right)$ in the Bessel transform ${H}_{v}\left(\lambda \right)={\int }_{0}^{\infty }f\left(u\right){J}_{v}\left(\lambda u\right)du,v>-1/2·$ The method replaces ${H}_{v}$ by a related damped transform for which the sinc quadrature rule provides an efficient and accurate approximation. It is then shown that the value of ${H}_{v}\left(\lambda \right)$ can be obtained from the damped transform by extrapolation with the Thiele algorithm.
MSC:
 65D20 Computation of special functions, construction of tables 65R10 Integral transforms (numerical methods) 44A15 Special transforms (Legendre, Hilbert, etc.) 44A20 Integral transforms of special functions 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
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