A Review of David Gottlieb's Work on the Resolution of the Gibbs Phenomenon (Communications in Computational Physics, 2011)
Given a piecewise smooth function, it is possible to construct a global expansionin some complete orthogonal basis, such as the Fourier basis. However, the localdiscontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic. The global expansionsare contaminated by the presence of a local discontinuity, and the result is that thepartial sums are oscillatory and feature non-uniform convergence. This characteristicbehavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-WangShu showed that these slowly and non-uniformly convergent global approximationsretain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and thestory of its resolution.