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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. $$f(u)=O(u^{-\alpha})$$, $$u\to +\infty$$ $$(\alpha >0)$$ in the Bessel transform $$H_ v(\lambda)=\int^{\infty}_{0}f(u)J_ v(\lambda u)du,v>-1/2.$$ 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 $$H_ v(\lambda)$$ can be obtained from the damped transform by extrapolation with the Thiele algorithm.

##### MSC:
 65D20 Computation of special functions and constants, construction of tables 65R10 Numerical methods for integral transforms 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A20 Integral transforms of special functions 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$
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##### References:
 [1] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. NBS Appl. Math. Ser.55, 375-417. New York: Dover 1964 [2] Crump, K.S.: Numerical inversion of Laplace Transforms using a Fourier series approximation. J. Assoc. Comput. Mach.23, 89-96 (1976) · Zbl 0315.65074 [3] de Balaine, G., Franklin, J.N.: The calculation of Fourier integrals. Math. Comput.20, 570-89 (1966) · Zbl 0196.49102 [4] Erd?lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 1. New York: McGraw-Hill 1954 · Zbl 0055.36401 [5] Erd?lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 2. New York: McGraw-Hill 1954 · Zbl 0055.36401 [6] Henrici, P.: Applied and Computational Complex Analysis, vol. 2. New York: John Wiley 1977 · Zbl 0363.30001 [7] Hildebrand, F.B.: Introduction to Numerical Analysis. (2nd ed.) New York: McGraw-Hill 1974 · Zbl 0279.65001 [8] Longman, I.M.: Note on a method for computing infinite integrals of oscillatory functions. Proc. Camb. Philos.52, 764-68 (1956) · Zbl 0072.33803 [9] Olver, F.W.J.: Asymptotics and Special Functions. New York: Academic Press 1974 · Zbl 0303.41035 [10] Sidi, A.: Extrapolation methods for oscillatory infinite integrals. J. Inst. Math. App.26, 1-20 (1980) · Zbl 0464.65002 [11] Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev.23, 165-224 (1981) · Zbl 0461.65007 [12] Stenger, F.: Explicit, nearly optimal, linear rational approximation with preassigned poles. (in preparation) · Zbl 0592.41019 [13] Stenger, F.: Optimal convergence of minimum norm approximations inH p . Numer. Math.29, 342-62 (1978) · Zbl 0437.41030 [14] Widder, D.V.: The Laplace Transform. Princeton: University Press 1941 · Zbl 0063.08245 [15] Wuytack, L.: A new technique for rational extrapolation to the limit. Numer. Math.17, 215-221 (1971) · Zbl 0225.65007 [16] Wynn, P.: On a procrustean technique for the numerical transformation of slowly convergent sequences and series. Proc. Camb. Philos.52, 663-71 (1960) · Zbl 0072.33802
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