Convergence and the Gibbs Phenomenon

We have been writing f(x) = \sum(\cdots) with an equals sign, but when does the series genuinely converge to f? For the smooth functions of physics the answer is reassuring, captured by Dirichlet's theorem.

If f is periodic and piecewise smooth (finitely many jumps and corners per period), then its Fourier series converges at every point x to

f(x) wherever f is continuous;
the midpoint of the jump, \tfrac12\big(f(x^-) + f(x^+)\big), at a discontinuity.

So at the steps of the square wave, the series doesn't pick a side — it settles dead centre, at 0. That is not a flaw; it is the only democratic choice.

The Gibbs overshoot

Near a jump something stubborn happens: the partial sums overshoot the true value, and the overshoot never shrinks. Adding harmonics makes the spike narrower and pushes it closer to the jump — but its height stays at about 9% of the jump forever. This is the Gibbs phenomenon. Convergence is pointwise (every fixed point away from the jump converges) but not uniform (the worst-case error near the jump refuses to vanish).

Watch the spike refuse to die

The square wave again, zoomed near its jump at x = 0. As N climbs, the ripple crowds toward the step and narrows — but the first peak stays stubbornly about 9\% above the line. More terms move the error around; they never quite kill it.