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Concept

Fourier analysis

The decomposition of a function into sines and cosines, whose amounts are found by projecting the function onto each one in turn. It turns convolution into multiplication, which is why it settles questions about sums of independent quantities and about differential equations alike.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

The target, multiplied by one harmonic at a time. Four panels, each showing the square wave multiplied by a single sine. The areas cancel exactly except against the harmonics the wave actually contains.

Where the coefficients come from

The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

analysis · Fourier series The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

analysis · Fourier series A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

analysis · Fourier series A lopsided distribution added to itself, and the shape that returns. On the left, the exact distribution of a sum of copies of one lopsided distribution, standardised, for several counts: the shapes converge. On the right, the bell curve convolved with itself, which is the bell curve again.

The shape that averaging leaves alone

Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.

probability · Central limit A quadratic in the exponent, completed. Two panels sharing an x-axis. Above, the parabola −x² + 2x with its top at x = 1 marked. Below, e raised to that parabola: a bell centred at the same x = 1, with peak height e^1, beside the faint unmoved bell e^(−x²).

One number under every bell

The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.

algebra · Completing the square A plucked string, released. A string plucked into a tent shape at 0.5, drawn at times 0, 0.1, 0.25, 0.5, 0.75, 1: the corner splits into two corners that run apart, reflect off the ends upside down and meet again.

A plucked string keeps its corners

Heat smooths a sharp profile at once, because each harmonic decays at a rate set by the square of its frequency. Change one time derivative into two and nothing decays at all: each harmonic swings for ever, the corner of a pluck splits in two and runs along the string, and after one period the shape comes back exactly. The same sines, the same coefficients — and a flow that loses nothing.

analysis · Fourier series The Dirichlet kernel, a spike with ripples that do not die. The Dirichlet kernel for N = 4, 12 on the interval from −π to π: a central spike of height 2N + 1 and side ripples whose total area in absolute value grows with N.

The ripples that make a series run away

Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.

analysis · Fourier series

Named alongside it

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

SpectrumConvergenceEigenfunctionHarmonicsHeat equationIrreversibilityNormal distributionPeriodicityScalingCesaro summationCompleting the squareComplex numbers

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