Interactive experiment · Harmonic analysis
Move the slider to increase the number of odd-frequency sine waves in the approximation.
Consider the partial sum
As \(N\) increases, this approaches the square wave at its points of continuity. Near its jumps, the persistent overshoot is an example of the Gibbs phenomenon.
Solid line: partial sum · Dashed line: limiting square wave (away from the jump points).
The approximation rapidly becomes almost constant away from the discontinuities, but the overshoot near each jump does not simply disappear as more terms are added. Instead, it becomes localized into a smaller neighborhood.