Why does a Fourier series overshoot at sharp edges?
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Gibbs phenomenon
Why does a Fourier series overshoot at sharp edges?
Imagine you're trying to smooth out the edges of a jagged mountain range for a map. You want a clear, continuous line, but the rough edges make it difficult.
As you add more detail to your map, trying to smooth out the jagged edges, you notice that instead of getting a perfectly smooth line, you end up with peaks and valleys that don't quite match the actual sharpness of the mountain range's edges. This is what happens with the Gibbs phenomenon when using Fourier series to approximate discontinuous functions.
Example
When mapping a mountain range with sharp peaks, instead of a smooth curve, you get a series of peaks and valleys around those sharp points.
Remember this
The Gibbs phenomenon causes overshoots and undershoots around discontinuities when approximating functions with Fourier series.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
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