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Computing the Hilbert transform on the real line

Author: J. A. C. Weideman
Journal: Math. Comp. 64 (1995), 745-762
MSC: Primary 65R10; Secondary 44A15, 65D30
MathSciNet review: 1277773
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Abstract: We introduce a new method for computing the Hilbert transform on the real line. It is a collocation method, based on an expansion in rational eigenfunctions of the Hilbert transform operator, and implemented through the Fast Fourier Transform. An error analysis is given, and convergence rates for some simple classes of functions are established. Numerical tests indicate that the method compares favorably with existing methods.

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  • [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
  • [2] N.K. Bary, A treatise on trigonometric series, Vol. I, Macmillan, New York, 1964. MR 0171116 (30:1347)
  • [3] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid. Mech. 25 (1967), 559-592.
  • [4] C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, Wiley-Interscience, New York, 1983.
  • [5] J. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69 (1987), 112-142. MR 892255 (88e:65093)
  • [6] -, The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials, J. Approx. Theory 61 (1990), 98-105. MR 1047151 (91c:41039)
  • [7] P. L. Butzer and R. J. Nessel, Fourier analysis and approximation, Vol. I, Academic Press, New York, 1971. MR 0510857 (58:23312)
  • [8] C. I. Christov, A complete orthonormal system of functions in $ {L^2}( - \infty ,\infty )$ space, SIAM J. Appl. Math. 42 (1982), 1337-1344. MR 678221 (84b:42018)
  • [9] P. J. Davis and P. Rabinowitz, Methods of numerical integration, 2nd ed., Academic Press, San Diego, 1984. MR 760629 (86d:65004)
  • [10] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms, Vols. I and II, McGraw-Hill, New York, 1954. MR 0061695 (15:868a)
  • [11] P. Henrici, Applied and computational complex analysis, Vol. III, Wiley-Interscience, New York, 1986. MR 0372162 (51:8378)
  • [12] J. R. Higgins, Completeness and basis functions of sets of special functions, Cambridge Univ. Press, Cambridge, 1977. MR 0499341 (58:17240)
  • [13] E. Hille, Analytic function theory, Vol. II, Ginn, Boston, 1962. MR 0201608 (34:1490)
  • [14] R. James, Pseudospectral methods for the Benjamin-Ono equation, Master's Paper, Oregon State Unversity (unpublished).
  • [15] R. James and J. A. C. Weideman, Pseudospectral methods for the Benjamin-Ono equation, Advances in Computer Methods for Partial Differential Equations--VII (R. Vichnevetsky, D. Knight, and G. Richter, eds.), IMACS, New Brunswick, 1992, pp. 371-377.
  • [16] R. Kress and E. Martensen, Anwendung der Rechteckregel auf die reelle Hilberttransformation mit unendlichem Intervall, Z. Angew. Math. Mech. 50 (1970), T61-T64. MR 0282529 (43:8240)
  • [17] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), 1082-1091. MR 0398275 (53:2129)
  • [18] A. C. Pipkin, A course on integral equations, Springer-Verlag, New York, 1991. MR 1125074 (92g:45001)
  • [19] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series, Vol. I: Elementary functions, Gordon and Breach, New York, 1986. MR 874986 (88f:00013)
  • [20] F. Stenger, Approximations via Whittaker's cardinal function, J. Approx. Theory 17 (1976), 222-240. MR 0481786 (58:1885)
  • [21] H. Weber, Numerical computation of the Fourier transform using Laguerre functions and the fast Fourier transform, Numer. Math. 36 (1981), 197-209. MR 611492 (82c:65095)
  • [22] W. T. Weeks, Numerical inversion of Laplace transforms using Laguerre functions, J. Assoc. Comput. Mach. 13 (1966), 419-429. MR 0195241 (33:3444)
  • [23] J. A. C. Weideman, The eigenvalues of Hermite and rational spectral differentiation matrices, Numer. Math. 61 (1992), 409-432. MR 1151779 (92k:65071)
  • [24] -, Computation of the complex error function, SIAM J. Numer. Anal. 31 (1994), 1497-1518. MR 1293526 (95h:65012a)
  • [25] -, Computing integrals of the complex error function, Mathematics of Computation 1943-1993: A half-century of computational mathematics (W. Gautschi, ed.), Proc. Sympos. Appl. Math., vol. 48, 1994, pp. 403-407. MR 1314879 (96a:65026)
  • [26] N. Wiener, Extrapolation, interpolation, and smoothing of stationary time series, M.I.T. Press, Cambridge, 1949.

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Keywords: Hilbert transform, orthogonal rational eigenfunctions, Fast Fourier Transform
Article copyright: © Copyright 1995 American Mathematical Society

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