TY - JOUR
T1 - A Review of David Gottlieb's Work on the Resolution of the Gibbs Phenomenon
AU - Sigal Gottlieb, Jae-Hun Jung & Saeja Kim
JO - Communications in Computational Physics
VL - 3
SP - 497
EP - 519
PY - 2011
DA - 2011/03
SN - 9
DO - http://doi.org/10.4208/cicp.301109.170510s
UR - https://global-sci.org/intro/article_detail/cicp/7508.html
KW - 
AB - <p style="text-align: justify;">Given a piecewise smooth function, it is possible to construct a global expansion
in some complete orthogonal basis, such as the Fourier basis. However, the local
discontinuities of the function will destroy the convergence of global approximations,
even in regions for which the underlying function is analytic. The global expansions
are contaminated by the presence of a local discontinuity, and the result is that the
partial sums are oscillatory and feature non-uniform convergence. This characteristic
behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang
Shu showed that these slowly and non-uniformly convergent global approximations
retain within them high order information which can be recovered with suitable postprocessing.
In this paper we review the history of the Gibbs phenomenon and the
story of its resolution.</p>