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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
Article

Abstract

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.

Keywords

Polyharmonic expansions Gibbs phenomenon Accelerating convergence 

Mathematics Subject Classification (2000)

41A58 41A25 65B10 

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Copyright information

© Springer Science + Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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