Fourier Analysis

Joseph Fourier gave a bold answer while studying something else. A prefect under Napoleon and a veteran of the Egyptian expedition, he was working out how heat spreads through solid bodies. In 1807 he claimed that any function on an interval, however jagged, can be written as a sum of sines and cosines, with coefficients given by integrals:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos nx + b_n \sin nx\right), \qquad a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx .$$

The examiners, among them Lagrange and Laplace, were unconvinced, and Lagrange in particular insisted it could not be true. Fourier won a prize in 1812 for the work, with the jury complaining about its rigour, and published his great book only in 1822. His method solved problems nothing else could, and its foundations were shaky.

Making sense of Fourier's claim occupied the rest of the century and forced analysis to grow up. In 1829 Dirichlet proved convergence for reasonably well-behaved functions. To show the limits, he offered a function equal to 1 at rational points and 0 at irrational ones, which fits no reasonable notion of a curve at all. Riemann defined the integral precisely in order to handle Fourier coefficients. Georg Cantor, asking where a Fourier series could fail and still determine its function, was led to infinite sets and the beginnings of set theory. The rigorous real analysis of the nineteenth century grew largely out of Fourier's problem.

The final answer came late. Lennart Carleson proved in 1966 that the Fourier series of every square-integrable function converges almost everywhere, a result so hard that many experts had expected the opposite.

Fourier's boldest claim was that even a function with jumps is a sum of smooth sine waves. Take the square wave that switches between $+1$ and $-1$ every half-period. Its Fourier series is

$$f(x) = \frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \frac{\sin 7x}{7} + \cdots\right).$$

One term gives a rounded hump. Adding the third harmonic flattens the top, and with each odd harmonic the sum squares off further, closer and closer to the corners. Every term is smooth, yet the infinite sum jumps. This is exactly what Euler and Lagrange had thought impossible.

Two surprises lie in the details. Put $x = \frac{\pi}{2}$, where the square wave equals 1. The series becomes

$$1 = \frac{4}{\pi}\left(1 - \frac13 + \frac15 - \frac17 + \cdots\right), \quad \text{so} \quad \frac{\pi}{4} = 1 - \frac13 + \frac15 - \frac17 + \cdots ,$$

the Leibniz series for $\pi$, falling out as a by-product. Second, near each jump the partial sums always overshoot, by about 9% of the jump, however many terms are added. The overshoot squeezes closer to the jump but never shrinks. This is the Gibbs phenomenon. Henry Wilbraham noticed it in 1848, and it is named after Josiah Willard Gibbs, who described it in 1898–99. It shows up today as "ringing" around sharp edges in compressed images.

That the series converges at every point except the jumps, and exactly what happens at the jumps (it converges to the midpoint, 0), was proved by Dirichlet in 1829. It was one of the first theorems of rigorous analysis.

Meanwhile the idea escaped into science. Light splits into a spectrum, sound into frequencies, and the X-ray pattern of a crystal is the Fourier transform of its atoms, which is how molecular biology read the structure of DNA. In 1965 James Cooley and John Tukey published the fast Fourier transform, and made the transform cheap enough to run on every phone, camera and modem. Gauss, it later emerged, had found the same algorithm around 1805.

At the research frontier, harmonic analysis now asks how waves travelling in many directions can pile up. That question is disguised as a puzzle about rotating a needle, the Kakeya problem, which was solved in three dimensions only in 2025.