BIT Numerical Mathematics

, Volume 51, Issue 1, pp 7–41 | Cite as

Gibbs phenomenon and its removal for a class of orthogonal expansions

  • Ben AdcockEmail author


We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.

Next, we demonstrate how the Gibbs phenomenon can be both mitigated and completely removed from such expansions using a number of different techniques. As a by-product, we introduce a generalisation of the classical Lidstone polynomials.


Polyharmonic expansions Gibbs phenomenon Accelerating convergence 

Mathematics Subject Classification (2000)

41A58 41A25 65B10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1974) Google Scholar
  2. 2.
    Adcock, B.: Univariate modified Fourier methods for second order boundary value problems. BIT Numer. Math. 49(2), 249–280 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Adcock, B.: Convergence acceleration of modified Fourier series in one or more dimensions. Math. Comp. 80(273), 225–261 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Adcock, B.: Multivariate modified Fourier series and application to boundary value problems. Numer. Math. 115(4), 511–552 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Adcock, B.: On the convergence of expansions in polyharmonic eigenfunctions. Technical report NA2010/06, DAMTP, University of Cambridge (2010) Google Scholar
  6. 6.
    Adcock, B., Iserles, A., Nørsett, S.P.: From high oscillation to rapid approximation II: expansions in Birkhoff series. IMA J. Numer. Anal. (to appear) (2010) Google Scholar
  7. 7.
    Agarwal, R., Wong, P.: Error Inequalities in Polynomial Interpolation and Their Applications. Springer, Berlin (1993) zbMATHGoogle Scholar
  8. 8.
    Baszenski, G., Delvos, F.J.: Accelerating the rate of convergence of bivariate Fourier expansions. In: Approximation Theory IV, pp. 335–340 (1983) Google Scholar
  9. 9.
    Birkhoff, G.D.: Boundary value and expansion problems of ordinary linear differential equations. Trans. Am. Math. Soc. 9(4), 373–395 (1908) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boyd, J.P.: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178, 118–160 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Boyd, J.P.: Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of 1/n. J. Comput. Phys. 228, 1404–1411 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Brunner, H., Iserles, A., Nørsett, S.P.: The computation of the spectra of highly oscillatory Fredholm integral operators. J. Integral Equ. Appl. (2010, to appear) Google Scholar
  13. 13.
    Driscoll, T.A., Fornberg, B.: A Padé-based algorithm for overcoming the Gibbs phenomenon. Numer. Algorithms 26, 77–92 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Eckhoff, K.S.: Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Math. Comput. 61(204), 745–763 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Eckhoff, K.S.: Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions. Math. Comput. 64(210), 671–690 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Eckhoff, K.S.: On a high order numerical method for functions with singularities. Math. Comput. 67(223), 1063–1087 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Fornberg, B., Flyer, N.: The Gibbs phenomenon for radial basis functions. In: Jerri, A. (ed.) The Gibbs Phenomenon in Various Representations and Applications. Sampling Publishing, Potsdam (2007) Google Scholar
  18. 18.
    Gelb, A., Gottlieb, D.: The resolution of the Gibbs phenomenon for “spliced” functions in one and two dimensions. Comput. Math. Appl. 33(11), 35–58 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, 1st edn. Society for Industrial and Applied Mathematics, Philadelphia (1977) zbMATHGoogle Scholar
  20. 20.
    Gottlieb, D., Shu, C.W.: On the Gibbs’ phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gottlieb, D., Shu, C.W., Solomonoff, A., Vandeven, H.: On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math. 43(1–2), 91–98 (1992) MathSciNetGoogle Scholar
  22. 22.
    Huybrechs, D.: On the Fourier extension of non-periodic functions. SIAM J. Numer. Anal. 47(6), 4326–4355 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Iserles, A., Nørsett, S.P.: From high oscillation to rapid approximation I: modified Fourier expansions. IMA J. Numer. Anal. 28, 862–887 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jerri, A.: The Gibbs Phenomenon in Fourier Analysis, Splines, and Wavelet Approximations. Springer, Berlin (1998) zbMATHGoogle Scholar
  25. 25.
    Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis, 3rd edn. Interscience, New York (1958) zbMATHGoogle Scholar
  26. 26.
    Lorentz, G.G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation. Addison–Wesley, London (1983) zbMATHGoogle Scholar
  27. 27.
    Lyness, J.N.: Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comput. 25, 87–104 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Lyness, J.N.: Computational techniques based on the Lanczos representation. Math. Comput. 28(125), 81–123 (1974) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Lyness, J.N.: The calculation of trigonometric Fourier coefficients. J. Comput. Phys. 54, 57–73 (1984) CrossRefzbMATHGoogle Scholar
  30. 30.
    Minkin, A.M.: Equiconvergence theorems for differential operators. J. Math. Sci. 96, 3631–3715 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Naimark, M.A.: Linear Differential Operators. Harrap, Bromley (1968) zbMATHGoogle Scholar
  32. 32.
    Olver, S.: On the convergence rate of a modified Fourier series. Math. Comput. 78, 1629–1645 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Platte, R., Trefethen, L.N., Kuijlaars, A.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev. (2010, to appear) Google Scholar
  34. 34.
    Shaw, J.K., Johnson, L.W., Riess, R.D.: Accelerating convergence of eigenfunction expansions. Math. Comput. 30(135), 469–477 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Sidi, A.: Practical Extrapolation Methods. Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, vol. 10. Cambridge University Press, Cambridge (2003) CrossRefzbMATHGoogle Scholar
  36. 36.
    Smitheman, S.A., Spence, E.A., Fokas, A.S.: A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon. IMA J. Numer. Anal. 30(4), 1184–1205 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Srivastava, H., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht (2001) zbMATHGoogle Scholar
  38. 38.
    Tadmor, E.: Filters, mollifiers and the computation of the Gibbs’ phenomenon. Acta Numer. 16, 305–378 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Young, R.M.: An Introduction to Nonharmonic Fourier Series, 1st edn. Academic Press, San Diego (2001) zbMATHGoogle Scholar
  40. 40.
    Zygmund, A.: Trigonometric Series, vol. 1. Cambridge University Press, Cambridge (1959) zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations