A Fourier series expresses any periodic (repeating) function as an infinite sum of simple sine and cosine waves of different frequencies. Introduced by Joseph Fourier in 1807 to study heat flow — and initially met with serious skepticism from leading mathematicians of the era — this technique became one of the most practically consequential ideas in all of mathematics, underlying MP3 audio compression, JPEG image compression, and virtually all of modern digital signal processing.
A musical chord sounds complex, but it's really just several simple pure tones (single-frequency sine waves) played simultaneously. Remarkably, this idea generalises far beyond music: any periodic (repeating) pattern whatsoever — no matter how jagged, complicated, or seemingly irregular — can always be broken down into a sum of simple sine and cosine waves of different frequencies. This is the Fourier series.
MP3 audio compression works by converting a sound wave into its Fourier-style frequency components, then discarding the frequency components too subtle for human hearing to notice — dramatically shrinking file size while preserving the sound quality that actually matters perceptually. JPEG image compression uses a closely related two-dimensional version of the same fundamental idea (the discrete cosine transform) applied to blocks of pixels instead of sound waves over time.
Fourier's original motivation had nothing to do with sound or signal processing at all — he was studying how heat spreads through solid objects over time, and needed a way to represent arbitrary initial temperature distributions using functions his heat equation (see the Differential Equation entry) could handle. The technique's staggering later usefulness across sound, images, and virtually all forms of digital data was a much later, far-reaching development well beyond Fourier's original 19th-century purpose.
For a periodic function f(x) with period 2π, its Fourier series is:
where the sum runs over all positive integers n, and each coefficient aₙ, bₙ measures how strongly that particular frequency component contributes to the overall function.
The coefficients are computed using specific definite integrals:
(integrated over one full period). These formulas work because sine and cosine waves of different frequencies are, in a precise technical sense, "orthogonal" to each other — analogous to how perpendicular directions in ordinary space don't interfere with one another, allowing each frequency's individual contribution to be cleanly extracted.
A square wave (jumping abruptly between two fixed values) is about as far from a smooth sine wave as a periodic function can be — yet it has a genuine, well-defined Fourier series: a sum of sine waves at odd-numbered frequencies with rapidly decreasing amplitude. Adding more and more terms produces an increasingly accurate approximation to the sharp-cornered square wave, using nothing but smooth curves.
Right at a sharp jump discontinuity (like in the square wave example), the Fourier series approximation persistently "overshoots" the true jump by a fixed percentage (roughly 9%), no matter how many additional terms are included — this overshoot narrows to occupy an ever-smaller region right around the jump, but never fully disappears in amplitude. This surprising and initially puzzling behaviour, named the Gibbs phenomenon after Josiah Willard Gibbs (who explained it rigorously in 1899, though it had been observed slightly earlier by Henry Wilbraham in 1848), is an important practical consideration in signal processing and numerical analysis.
When Fourier first presented his heat-conduction work to the French Academy of Sciences in 1807, prominent mathematicians including Joseph-Louis Lagrange objected strongly, doubting that a discontinuous or sharply-cornered function could genuinely be represented by a sum of perfectly smooth trigonometric functions. Fourier's original paper was initially rejected for publication specifically due to concerns about this lack of full mathematical rigour — it was only published considerably later (1822, in his book Théorie analytique de la chaleur), by which time the underlying mathematical questions about convergence had been substantially, though still not completely, clarified.
Establishing precisely when, and in what specific technical sense, a Fourier series actually converges back to the original function turned out to be a genuinely deep and subtle mathematical question, occupying analysts for well over a century after Fourier's original work. Peter Gustav Lejeune Dirichlet (1829) gave the first rigorous sufficient conditions guaranteeing pointwise convergence for reasonably well-behaved functions. The much harder, more general question of convergence for arbitrary functions merely required to be square-integrable (a considerably weaker technical condition) was resolved only in 1966, by Lennart Carleson, in a landmark and highly technical result now called Carleson's theorem — for which Carleson was later awarded the Abel Prize (2006), partly in recognition of this achievement.
Fourier series apply specifically to periodic (repeating) functions. The Fourier transform generalises the same essential idea to non-periodic functions defined across the entire real line, decomposing them into a continuous (rather than discrete) spectrum of frequency components. The Fourier transform is, if anything, even more widely used in practice than the Fourier series itself, appearing throughout quantum mechanics (where it connects a particle's position and momentum representations), telecommunications engineering, and audio/image signal processing (see the Fourier Transform entry for the fuller treatment).
Computing a discrete Fourier-style decomposition directly, term by term, requires a number of arithmetic operations that grows proportionally to n² for n data points — computationally impractical for large, real-world datasets like audio files or images. The Fast Fourier Transform algorithm (popularised by James Cooley and John Tukey in 1965, though closely related ideas trace back to unpublished work by Carl Friedrich Gauss around 1805, predating Fourier's own famous 1807 work) reduces this to just O(n log n) operations (see the Big-O Notation entry) — a dramatic efficiency improvement that made real-time digital signal processing genuinely practical, and is now considered one of the most important and widely-used algorithms in all of computing.