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The Gibbs phenomenon

Near a jump in a function, a Fourier series will always overshoot a little, no matter how many terms we include. This is called the Gibbs phenomenon, and the overshoot moves closer to the jump but does not shrink.

overshoot≈9% of the jump\text{overshoot} \approx 9\%\ \text{of the jump}the Gibbs phenomenon

Symbols

jump\text{jump}the discontinuity in f(x)

Example

A square wave always has an overshoot of about 9% right at the jumps, even with very many terms in the series.

At the jump point itself, the series converges to the average of the left and right limits.
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Part of Laplace, Fourier and PDEs: Fourier series.