Imagine you are trying to draw a square wave using only sine waves. The first few sine waves will do a good job of approximating the square wave, but as you add more sine waves, the approximation will start to look more and more like a triangle wave. This is because the sine waves are not able to perfectly reproduce the sharp corners of the square wave.

The Gibbs phenomenon is the name given to this overshoot of the Fourier series at discontinuities. It can be reduced by using a technique called windowing.
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