Computing the Hilbert transform on the real line

Weideman, J. A. C.

doi:10.1090/S0025-5718-1995-1277773-8

Computing the Hilbert transform on the real line
HTML articles powered by AMS MathViewer

by J. A. C. Weideman PDF

Math. Comp. 64 (1995), 745-762 Request permission

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.

References

Handbook of mathematical functions

N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116

Internal waves of permanent form in fluids of great depth

25

Absorption and scattering of light by small particles

John P. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69 (1987), no. 1, 112–142. MR 892255, DOI 10.1016/0021-9991(87)90158-6
John P. Boyd, The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials, J. Approx. Theory 61 (1990), no. 1, 98–105. MR 1047151, DOI 10.1016/0021-9045(90)90026-M
Paul L. Butzer and Rolf J. Nessel, Fourier analysis and approximation, Pure and Applied Mathematics, Vol. 40, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory. MR 0510857
C. I. Christov, A complete orthonormal system of functions in $L^{2}(-\infty ,\,\infty )$ space, SIAM J. Appl. Math. 42 (1982), no. 6, 1337–1344. MR 678221, DOI 10.1137/0142093
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
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
John Rowland Higgins, Completeness and basis properties of sets of special functions, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Cambridge Tracts in Mathematics, No. 72. MR 0499341
Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608

Pseudospectral methods for the Benjamin-Ono equation

R. Kress and E. Martensen, Anwendung der Rechteckregel auf die reelle Hilberttransformation mit unendlichem Intervall, Z. Angew. Math. Mech. 50 (1970), T61–T64 (German). MR 282529, DOI 10.1002/zamm.19700500125
Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 398275, DOI 10.1143/JPSJ.39.1082
Allen C. Pipkin, A course on integral equations, Texts in Applied Mathematics, vol. 9, Springer-Verlag, New York, 1991. MR 1125074, DOI 10.1007/978-1-4612-4446-2
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986
Frank Stenger, Approximations via Whittaker’s cardinal function, J. Approximation Theory 17 (1976), no. 3, 222–240. MR 481786, DOI 10.1016/0021-9045(76)90086-1
H. Weber, Numerical computation of the Fourier transform using Laguerre functions and the fast Fourier transform, Numer. Math. 36 (1980/81), no. 2, 197–209. MR 611492, DOI 10.1007/BF01396758
William T. Weeks, Numerical inversion of Laplace transforms using Laguerre functions, J. Assoc. Comput. Mach. 13 (1966), 419–429. MR 195241, DOI 10.1145/321341.321351
J. A. C. Weideman, The eigenvalues of Hermite and rational spectral differentiation matrices, Numer. Math. 61 (1992), no. 3, 409–432. MR 1151779, DOI 10.1007/BF01385518
J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31 (1994), no. 5, 1497–1518. MR 1293526, DOI 10.1137/0731077
J. A. C. Weideman, Computing integrals of the complex error function, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 403–407. MR 1314879, DOI 10.1090/psapm/048/1314879

Extrapolation, interpolation, and smoothing of stationary time series