Gibbs phenomenon

Summary

The Gibbs phenomenon is the overshoot of the partial sums of a Fourier series near a jump discontinuity of the function. As more terms are added, the overshoot moves closer to the jump but does not shrink. It tends to about $9\%$ of the size of the jump on each side. It shows that the convergence of a Fourier series at a jump cannot be uniform. It matters in signal processing, where truncating frequencies causes ringing near sharp edges.