Foundations of Hankel transform algorithms

Suter, Bruce W.

doi:10.1090/qam/1106392

Author: Bruce W. Suter
Journal: Quart. Appl. Math. 49 (1991), 267-279
MSC: Primary 65R10; Secondary 44A15
DOI: https://doi.org/10.1090/qam/1106392
MathSciNet review: MR1106392
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Abstract: A brief survey of existing Hankel (Fourier-Bessel) transform algorithms is presented along with a natural way to classify these algorithms. In several cases these algorithms were derived originally by methods that were unnecessarily complicated and not sufficiently general. By using operator notation and Radon transform methods, derivations and generalizations are straightforward. These improvements and generalizations are given at the appropriate places in the discussion.

W. L. Anderson, Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering, Geophysics 44, 1287–1305 (1979)

W. L. Anderson, Fast Hankel transforms using related and lagged convolutions, ACM Trans. Math. Software 8, 344–368 (1982)

W. L. Anderson, Algorithm 558 : Fast Hankel transforms using related and lagged convolutions, ACM Trans. Math. Software 8, 369–370 (1982)

W. L. Anderson, Computation of Green’s tensor integral for three-dimensional electromagnetic problems using fast Hankel transforms, Geophysics 49, 1754–1759 (1984)

George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. MR 0205512

H. H. Barrett, The Radon transform and its applications, E. Wolf (editor), Progress in Optics, North Holland, Amsterdam, Holland, 21 219–286 (1984)

Ronald N. Bracewell, The Fourier transform and its applications, 2nd ed., McGraw-Hill Book Co., New York, 1978. MR 549484
S. M. Candel, An algorithm for the Fourier-Bessel transform, Comput. Phys. Comm. 23 (1981), no. 4, 343–353. MR 628681, DOI https://doi.org/10.1016/0010-4655%2881%2990175-2
Sébastien M. Candel, Dual algorithms for fast calculation of the Fourier-Bessel transform, IEEE Trans. Acoust. Speech Signal Process. 29 (1981), no. 5, 963–972. MR 629032, DOI https://doi.org/10.1109/TASSP.1981.1163658
Sébastien M. Candel, Simultaneous calculation of Fourier-Bessel transforms up to order $N$, J. Comput. Phys. 44 (1981), no. 2, 243–261. MR 645838, DOI https://doi.org/10.1016/0021-9991%2881%2990050-4

E. Cavanaugh and B. D. Cook, Numerical evaluation of Hankel transforms via Gaussian-Laguerre polynomial expansions, IEEE Trans. Acoust. Speech Signal Process. 27, 361–366 (1979)

A. D. Chave, Numerical integration of related Hankel transforms by quadrature and continued fraction expansion, Geophysics 48, 1671–1686 (1983)

A. D. Chave and C. S. Cox, Controlled electromagnetic sources for measuring electrical conductivity beneath the oceans, 1. Forward problem and model study, J. Geophysical Research 87, 5327–5338 (1982)

P. Cornille, Computation of Hankel transforms, SIAM Rev. 14 (1972), 278–285. MR 311080, DOI https://doi.org/10.1137/1014032
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
Stanley R. Deans, The Radon transform and some of its applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 709591
F. R. DiNapoli and R. L. Deavenport, Theoretical and numerical Green’s function field solution in a plane multilayered medium, J. Acoust. Soc. Amer. 67 (1980), no. 1, 92–105. MR 553860, DOI https://doi.org/10.1121/1.383794
Alan V. Oppenheim, George V. Frisk, and David R. Martinez, Computation of the Hankel transform using projections, J. Acoust. Soc. Amer. 68 (1980), no. 2, 523–529. MR 579187, DOI https://doi.org/10.1121/1.384765
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831

D. P. Ghosh, The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements, Geophysical Prospecting 19, 192–217 (1971)

K. Gopalan, Fournier-Bessel expansion: Numerical evaluation and application in the representation and feature extraction of speech signals, Ph. D. Dissertation, University of Akron, Akron, Oh., 1984

K. Gopalan and C. S. Chen, Fast computation of zero order Hankel transform, J. Franklin Inst. 316 (1983), no. 4, 317–326. MR 718212, DOI https://doi.org/10.1016/0016-0032%2883%2990098-4

E. W. Hansen, Fast Hankel transform algorithm, IEEE Trans. Acoust. Speech Signal Process. 33, 666–671 (1985), Errata 34, 623–624 (1986)

Sigurdur Helgason, The Radon transform, Progress in Mathematics, vol. 5, Birkhäuser, Boston, Mass., 1980. MR 573446

W. E. Higgins and D. C. Munson, An algorithm for computing general integer order Hankel transforms, IEEE Trans. Acoust. Speech Signal Process. 35, 86–97 (1987)

John David Jackson, Mathematics for quantum mechanics. An introductory survey of operators, eigenvalues, and linear vector spaces, W. A. Benjamin, Inc., New York, 1962. MR 0144622

E. Kausel and G. Bouchovalas, Computation of Hankel transforms using the fast Fourier transform algorithm, Massachusetts Institute of Technology, Civil Engineering Department, Research Report R79–12, 1979

E. Kausel and J. M. Roesset, Stiffness matrices for layered soils, Bull. Seismol. Soc. Amer. 71, 1743–1761 (1981)

M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge University Press, New York, 1960. MR 0115085
I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Soc. 52 (1956), 764–768. MR 82193
I. M. Longman, Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, Math. Tables Aids Comput. 11 (1957), 166–180. MR 91538, DOI https://doi.org/10.1090/S0025-5718-1957-0091538-0

D. R. Mook, An algorithm for the numerical evaluation of the Hankel and Abel transforms, IEEE Trans. Acoust. Speech Signal Process. 31, 979–985 (1983)

Patricia K. Murphy and Neal C. Gallagher, Fast algorithm for the computation of the zero-order Hankel transform, J. Opt. Soc. Amer. 73 (1983), no. 9, 1130–1137. MR 717882, DOI https://doi.org/10.1364/JOSA.73.001130

A. V. Oppenheim, G. V. Frisk, and D. R. Martinez, An algorithm for the numerical evaluation of the Hankel transform, Proc. IEEE 66, 264–265 (1978)

Alan V. Oppenheim, George V. Frisk, and David R. Martinez, Computation of the Hankel transform using projections, J. Acoust. Soc. Amer. 68 (1980), no. 2, 523–529. MR 579187, DOI https://doi.org/10.1121/1.384765
T. N. L. Patterson, Table errata: “The optimum addition of points to quadrature formulae” (Math. Comp. 22 (1968), 847–856; addendum, ibid. 22 (1968), no. 104, loose microfiche suppl. C1-C11), Math. Comp. 23 (1969), no. 108, 892. MR 400633, DOI https://doi.org/10.1090/S0025-5718-1969-0400633-3
Johann Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Computed tomography (Cincinnati, Ohio, 1982) Proc. Sympos. Appl. Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1982, pp. 71–86 (German). MR 692055

S. Sheng and A. E. Siegman, Nonlinear-optical calculations using fast-transform methods: Second harmonic generation with depletion and diffracton, Phys. Rev. 21, 599–606

A. E. Siegman, Quasi fast Hankel transform, Optics Letters 1, 13–15 (1977)

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972

B. W. Suter, Some Numerical Applications of the Radon Transform, Ph. D. Dissertation, University of South Florida, Tampa, Fl., 1988

B. W. Suter, Fast nth-order Hankel transform algorithm, IEEE Trans. Signal Process. 39, 532–536 (1991)

James D. Talman, Numerical Fourier and Bessel transforms in logarithmic variables, J. Comput. Phys. 29 (1978), no. 1, 35–48. MR 510459, DOI https://doi.org/10.1016/0021-9991%2878%2990107-9

R. K. Verma, Detectability by electromagnetic sounding systems, IEEE Trans. on Geoscience Electronics 15, 232–251 (1977)

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746