Gibbs Phenomenon
Theorem
The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out.
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Proof
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Source of Name
This entry was named for Josiah Willard Gibbs.
Historical Note
The Gibbs phenomenon was first reported in print in $1848$ by Henry Wilbraham.
However, this went unnoticed at the time.
Josiah Willard Gibbs published a short note in $1899$ on the subject of the Fourier series of a square wave, but failed to report on the phenomenon at that time.
Later that year he published a correction to that note in which the overshoot was described accurately.
In $1906$ Maxime Bôcher gave a complete analysis of the mathematics behind the phenomenon, and called it the Gibbs Phenomenon.
Wilbraham's paper was later brought to light, but by that time it was generally attributed to Gibbs.
In the words of Horatio Scott Carslaw ($1925$):
- We may still call this property of Fourier's series (and certain other series) Gibbs's phenomenon; but we must no longer claim that the property was first discovered by Gibbs.
Sources
- 1848: On a certain periodic function (The Cambridge and Dublin Mathematical Journal Vol. 3: pp. 198 – 201)
- 1899: J. Willard Gibbs: Fourier Series (Nature Vol. 59: p. 200)
- 1899: J. Willard Gibbs: Fourier Series (Nature Vol. 59: p. 606)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,17897 97444 \ldots$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Gibbs phenomenon
- Weisstein, Eric W. "Gibbs Phenomenon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GibbsPhenomenon.html