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Concept

Gibbs' phenomenon

The overshoot near a jump in a Fourier partial sum, which narrows as terms are added and never shortens. It is why a truncated Fourier series of a square wave rings at the edges, and the overshoot settles at about nine per cent of the jump.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · Fourier series Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

analysis · Arc length xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

A limit that forgets to be continuous

Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.

analysis · Uniform convergence

Named alongside it

The objects these essays reach for when they reach for this one.

ContinuityConvergenceCounterexampleApproximationArc lengthDerivativeFunction spaceHarmonicsLimitLimit functionOrthogonalityPeriodicity

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