Abstract
In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form
where the bar indicates the Cauchy principal value and \(f\) is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When \(x=0\), the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of \(\omega \) are derived for each fixed \(x\ge 0\), which clarify the large \(\omega \) behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of \(x\), we classify our discussion into three regimes, namely, \(x=\mathcal O (1)\) or \(x\gg 1\), \(0<x\ll 1\) and \(x=0\). Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency \(\omega \) increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.
Similar content being viewed by others
Notes
The use of increased precision is just to show the convergence rates of our methods.
This can be done with a minor adaptation of Oliver’s method for the LU decomposition of a tridiagonal matrix [28]. We omit the details.
References
Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications. Cambridge University Press, Cambridge (2003)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington D.C. (1964)
Asheim, A., Huybrechs, D.: Complex Gaussian quadrature for oscillatory integral transforms. Report TW 594, (2011)
Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Capobianco, M.R., Criscuolo, G.: On quadrature for Cauchy principal value integrals of oscillatory functions. J. Comput. Appl. Math. 156, 471–486 (2003)
Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)
Chung, K.C., Evans, G.A., Webster, J.R.: A method to generate generalized quadrature rules for oscillatory integrals. Appl. Numer. Math. 34, 85–93 (2000)
Deaño, A., Huybrechs, D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197–219 (2009)
Domínguez, V., Graham, I.G., Smyshlyaev, V.P.: Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal. 31, 1253–1280 (2011)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
Gentleman, W.M.: Implementing Clenshaw–Curtis quadrature. II. Comm. ACM 15, 343–346 (1972)
Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM, Philadelphia (2007)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)
Hasegawa, T., Torii, T.: An automatic quadrature for Cauchy principal value integrals. Math. Comput. 56, 741–754 (1991)
Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006)
Iserles, A., Nørsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. A 461, 1383–1399 (2005)
Iserles, A., Nørsett, S.P.: On quadrature methods for highly oscillatory integrals ans their implementation. BIT Numer. Math. 44, 755–772 (2004)
King, F.W.: Hilbert Transforms: Volume 1. Cambridge University Press, Cambridge (2009)
King, F.W., Smethells, G.J., Helleloid, G.T., Pelzl, P.J.: Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach. Comput. Phys. Commun. 145, 256–266 (2002)
Krommer, A.R., Ueberhuber, C.W.: Computational Integration. SIAM, Philadelphia (1998)
Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67, 95–101 (1996)
Levin, D.: Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 38, 531–538 (1982)
Lyness, J.N.: The Euler Maclaurin expansion for the Cauchy principal value integral. Numer. Math. 46, 611–622 (1985)
Milovanović, G.V.: Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures. Comput. Math. Appl. 36, 19–39 (1998)
Martin, P.A.: On the null-field equations for water-wave radiation problems. J. Fluid Mech. 113, 315–332 (1981)
Monegato, G., Lyness, J.N.: The Euler–Maclaurin expansion and finite-part integrals. Numer. Math. 81, 273–291 (1998)
Okecha, G.E.: Quadrature formulae for Cauchy principal value integrals of oscillatory kind. Math. Comput. 49, 259–268 (1987)
Oliver, J.: Relative error propagation in the recursive solution of linear recurrence relations. Numer. Math. 9, 323–340 (1967)
Olver, F.W.J.: Numerical solution of second-order linear difference equation. J. Res. Nat. Bur. Standards Sect. B 71B, 111–129 (1967)
Olver, S.: Computing the Hilbert transform and its inverse. Math. Comput. 80, 1745–1767 (2011)
Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA. J. Numer. Anal. 26, 213–227 (2006)
Olver, S.: GMRES for the differentiation operator. SIAM J. Numer. Anal. 47, 3359–3373 (2009)
Piessens, R., Branders, M.: On the computation of Fourier transforms of singular functions. J. Comput. Appl. Math. 43, 159–169 (1992)
Ursell, F.: Integrals with a large parameter: Hilbert transforms. Math. Proc. Camb. Soc. 93, 141–149 (1983)
Wang, H., Xiang, S.: Uniform approximations to Cauchy principal value integrals of oscillatory functions. Appl. Math. Comput. 215, 1886–1894 (2009)
Wang, H., Xiang, S.: On the evaluation of Cauchy principal value integrals of oscillatory functions. J. Comput. Appl. Math. 234, 95–100 (2010)
Wong, R.: Asymptotic expansion of the Hilbert transform. SIAM J. Math. Anal. 11, 92–99 (1980)
Wong, R.: Quadrature formulas for oscillatory integral transforms. Numer. Math. 39, 351–360 (1982)
Wong, R.: Asymptotic Approximations of Integrals. SIAM, Philadelphia (2001)
Xiang, S.: Efficient Filon-type methods for \(\int _a^bf(x)e^{i\omega g(x)}dx\). Numer. Math. 105, 633–658 (2007)
Xiang, S., Chen, X., Wang, H.: Error bounds for approximation in Chebyshev points. Numer. Math. 116, 463–491 (2010)
Acknowledgments
We thank Andreas Asheim for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
H. Wang and D. Huybrechs were supported by the Fund for Scientific Research—Flanders through Research Project G.0617.10. Lun Zhang is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (FWO), Belgium.
Rights and permissions
About this article
Cite this article
Wang, H., Zhang, L. & Huybrechs, D. Asymptotic expansions and fast computation of oscillatory Hilbert transforms. Numer. Math. 123, 709–743 (2013). https://doi.org/10.1007/s00211-012-0501-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-012-0501-9