Numerical Algorithms

, Volume 26, Issue 1, pp 77–92 | Cite as

A Padé-based algorithm for overcoming the Gibbs phenomenon

  • Tobin A. Driscoll
  • Bengt Fornberg

Abstract

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques.

Fourier series Padé approximation interpolation Gibbs phenomenon 

References

  1. [1]
    G.A. Baker and P. Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and Its Applications, Vol. 59, 2nd ed. (Cambridge Univ. Press, Cambridge, 1996).Google Scholar
  2. [2]
    N.S. Banerjee and J.F. Geer, Exponential approximations using Fourier series partial sums, Technical Report 97-56, ICASE (1997).Google Scholar
  3. [3]
    C. Brezinski, The asymptotic behavior of sequences and new series transformations based on the Cauchy product, Rocky Mountain J. Math. 21 (1991) 71–84.Google Scholar
  4. [4]
    R.G. Brookes, The local behavior of the quadratic Hermite-Padé approximation, in: Computational Techniques and Applications, eds. W.L. Hogarth and B.J. Noye (Hemisphere Publ. Corp., New York, 1990) pp. 569–575.Google Scholar
  5. [5]
    J.S.R. Chisholm and A.K. Common, Generalisations of Padé approximation for Chebyshev and Fourier series, in: E.B. Christoffel: The Influence of His Work on Mathematics and the Physical Sciences, eds. P.L. Butzer and F. Fehér (Birkhäuser, Basel, 1981) pp. 212–231.Google Scholar
  6. [6]
    K.S. Eckhoff, On a high order numerical method for functions with singularities, Math. Comp. 67 (1998) 1063–1087.Google Scholar
  7. [7]
    B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, Cambridge, 1996).Google Scholar
  8. [8]
    J.F. Geer, Rational trigonometric approximations using Fourier series partial sums, J. Sci. Comput. 10 (1995) 325–356.Google Scholar
  9. [9]
    J. Geer and N.S. Banerjee, Exponentially accurate approximations to piecewise smooth periodic functions, J. Sci. Comput. 12 (1997) 253–287.Google Scholar
  10. [10]
    A. Gelb and E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999) 101–135.Google Scholar
  11. [11]
    D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997) 644–668.Google Scholar
  12. [12]
    W.B. Gragg, Laurent, Fourier, and Chebyshev-Padétables, in: Padé and Rational Approximation, Theory and Applications, eds. E.B. Saff and R.S. Varga (Academic Press, New York, 1977).Google Scholar
  13. [13]
    G. Kvernadze, T. Hagstrom and H. Shapiro, Locating discontinuities of a bounded function by the partial sums of its Fourier series, J. Sci. Comput. 14 (1999) 301–327.Google Scholar
  14. [14]
    G. Kvernadze, Locating discontinuities of a bounded function by spectral methods, Ph.D. thesis, University of New Mexico (1998).Google Scholar
  15. [15]
    R.E. Shafer, On quadratic approximation, SIAM J. Numer. Anal. 11 (1974) 447–460.Google Scholar
  16. [16]
    R.D. Small and R.J. Charron, Continuous and discrete nonlinear approximation based on Fourier series, IMA J. Num. Anal. 8 (1988) 281–293.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Tobin A. Driscoll
    • 1
  • Bengt Fornberg
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

Personalised recommendations