Gibbs Phenomenon

From ProofWiki
Jump to navigation Jump to search

Theorem

The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out.

Gibbs-Phenomenon-9.png
Gibbs-Phenomenon-19.png



Proof


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