Convergence and the Gibbs Phenomenon

Does the series actually add up to the function?

Flip a light switch and the current jumps almost instantly from off to on — a genuine discontinuity, not a smooth ramp. Engineers describe such jumps with square-wave-like signals and happily write them as a sum of smooth sine waves. But that raises an honest question: at the exact instant of the jump, what is the sum of infinitely many perfectly smooth curves even supposed to equal? The function doesn't have a single value there — it has two, a "just before" and a "just after." Something has to give, and what actually happens is one of the more elegant surprises in all of mathematics.

We have been writing f(x) = \sum(\cdots) with a confident equals sign, but when does the series genuinely converge to f? Away from any jump the answer is completely reassuring; at a jump it is stranger, and wonderfully precise. Both are 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

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 fair, democratic choice a limit of continuous curves can possibly make.

Worked example: checking the average, exactly

Take the square wave f(x) = +1 for 0 < x < \pi and f(x) = -1 for -\pi < x < 0, with Fourier series

f(x) = \frac{4}{\pi}\Big(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\Big).

There is a jump at x = 0: the left-hand limit is f(0^-) = -1 and the right-hand limit is f(0^+) = +1. Dirichlet's theorem predicts the series converges there to the average, \tfrac12\big((-1) + (+1)\big) = 0.

Check it directly, with no approximation needed at all. Every single term in the series is a sine, and \sin(k \cdot 0) = 0 for every whole number k. So the partial sum at x=0 is

S_N(0) = \frac{4}{\pi}\Big(\sin 0 + \frac{\sin 0}{3} + \cdots\Big) = 0 \quad\text{for every } N.

Not just in the limit — S_N(0) equals the predicted average 0 exactly, for N=1 already, and stays there forever. The theorem's prediction and the direct calculation agree perfectly. (Away from the jump it's less immediate — e.g. at x = \pi/2, where f is continuous and equal to 1, the partial sums only creep toward 1 as N grows, rather than landing on it exactly.)

Contrast: convergence away from the jump is slow but honest

At the jump, S_N(0) hit the exact answer 0 immediately, for every N. Away from the jump the story is different — still true, just not instant. At x = \pi/2, where f is continuous and equals 1, plugging in \sin(k\pi/2) for successive odd k turns the series into the famous, slowly-converging Leibniz sum for \pi/4:

S_N\!\left(\tfrac{\pi}{2}\right) = \frac{4}{\pi}\Big(1 - \frac13 + \frac15 - \frac17 + \cdots\Big).

Adding terms one at a time gives S_1 \approx 1.273, S_2 \approx 0.849, S_3 \approx 1.104, S_4 \approx 0.922 — rocking back and forth, only gradually settling toward 1 as more terms pile on. That is ordinary pointwise convergence: no shortcuts, just patient improvement, term by term, exactly as Dirichlet's theorem promises for a point where the function is continuous.

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).

The 9% isn't a rough eyeball figure — it can be pinned down exactly. The overshoot fraction works out to \frac{2}{\pi}\operatorname{Si}(\pi) - 1 \approx 0.0895, using the sine-integral function \operatorname{Si} — a specific, unavoidable constant of nature (of this mathematical nature, anyway), often called the Wilbraham–Gibbs constant. It is the same number no matter which piecewise-smooth function you jump-discontinuity-test it on.

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.

Convergence at a point is not the same thing as the approximation becoming uniformly good everywhere near it — and the Gibbs overshoot is exactly where that distinction bites. Pick any fixed point x_0 a hair's breadth away from the jump (but not equal to it): as N \to \infty, the partial sums at that particular x_0 genuinely do converge to f(x_0). That part is completely true.

What does not happen is the overshoot going away as a whole. However large you make N, there is always some point, freshly squeezed even closer to the jump than before, where the partial sum still overshoots by about 9\%. More terms shrink the overshoot's width — cramming it nearer and nearer to the discontinuity — but never its height. It is a permanent feature of truncating the series, not a bug that patience or more computation eventually fixes.

The bump was first spotted mathematically by the English mathematician Henry Wilbraham in 1848 — and then almost entirely forgotten. Fifty years later the physicist Albert Michelson built a mechanical "harmonic analyser," a contraption of gears and springs that could add up dozens of sine waves and draw the result with a pen. When Michelson fed it a square wave, the pen persistently overshot the corners by a stubborn little spike. He assumed his machine was faulty — a mechanical imperfection, surely nothing a mathematician needed to worry about.

In 1899 the physicist Josiah Willard Gibbs proved otherwise: the spike was not a flaw in Michelson's gears at all, but an inescapable mathematical fact about truncating any Fourier series at a jump. It is a lovely example of a surprising truth being mistaken for a mistake — and today the effect carries Gibbs's name, even though Wilbraham quietly got there first.

This point-versus-everywhere distinction isn't a one-off curiosity of series — the exact same tension between "converges here" and "converges uniformly nearby" resurfaces once sums of discrete harmonics become integrals over a continuum of frequencies, in the Fourier transform. Ringing artefacts near sharp edges in a filtered signal or a reconstructed image are, at heart, the Gibbs phenomenon wearing different clothes.