Gibbs phenomenon

A Fourier partial sum's persistent overshoot near a jump, which never shrinks away as more terms are added.

Near a jump discontinuity of size $J$ , a Fourier partial sum $S_N$ overshoots the jump on the near side by an amount that does not shrink to $0$ as $N\to\infty$ -- instead it converges to a fixed fraction of $J$ (the Gibbs constant), while the ripple's location slides toward the jump and its width shrinks like $1/N$ . This is the Gibbs phenomenon: an exact, provable feature of every Fourier partial sum near every jump, not a numerical error.

Examples

A square wave's Fourier partial sums always overshoot each jump by about 9%, no matter how many terms are added.