What is it about?
The Gibbs phenomenon occurs when a function with jump discontinuities is expanded into a Fourier series. This phenomenon, which is normally undesirable, results in slow convergence and is accompanied by a characteristic oscillatory behavior. The phenomenon is relevant to many areas of science and engineering. Many methods are used as remedies for the Gibbs phenomenon. Perhaps the simplest is the so-called method of Fejer averaging. This paper examines the Gibbs phenomenon and the method of Fejer averaging through the lens of asymptotics.
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Why is it important?
Our methods help us comprehend the Gibbs phenomenon. They also enhance our understanding of methods that can overcome or alleviate the phenomenon.
Perspectives
The simple formulas developed here nicely illustrate that, when applicable, asymptotics can provide significant insight into physical and mathematical phenomena. Formulas like ours can: (1) be developed for further Gibbs-phenomenon remedies (other than the remedy of Fejer averaging); (2) lead to comprehensive comparisons of the various remedies between themselves; (3) be used to improve older remedies and to develop new ones.
George Fikioris
National Technical University of Athens
Read the Original
This page is a summary of: Asymptotic approximations elucidating the Gibbs phenomenon and Fejér averaging, Asymptotic Analysis, March 2017, IOS Press,
DOI: 10.3233/asy-171408.
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