A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Author:

Gottlieb Sigal,Jung Jae-Hun,Kim Saeja

Abstract

AbstractGiven 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.

Publisher

Global Science Press

Subject

Physics and Astronomy (miscellaneous)

Reference47 articles.

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4. Gottlieb D. , The effect of local features on global expansions, The SIAM 2008 John von Neumann Lecture, SIAM Annual Meeting, (2008), http://tinyurl.com/yce34h9.

5. Letter to the Editor;Michelson;Fourier’s Series, Nature,1898

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