Numerische Mathematik

, Volume 71, Issue 4, pp 511–526

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

  • David Gottlieb
  • Chi-Wang Shu

DOI: 10.1007/s002110050155

Cite this article as:
Gottlieb, D. & Shu, CW. Numer Math (1995) 71: 511. doi:10.1007/s002110050155

Summary.

This paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function \(f(x)\) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods. A numerical example is also provided.

Mathematics Subject Classification (1991): 42A15, 41A05, 41A25

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • David Gottlieb
    • 1
  • Chi-Wang Shu
    • 1
  1. 1.Division of Applied Mathematics, Brown University, Providence, RI 02912, USA US