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In mathematics, the Fourier theorem is a theorem stating that a periodic function f(x), which is reasonably continuous, may be expressed as the sum of a series of sine and cosine terms (called the Fourier series), each of which has specific amplitude and phase coefficients known as Fourier coefficients. The theorem was developed by the French mathematician J.B. Fourier around 1800.

A simple statement of the theorem follows: Any physical function that varies periodically with time with a frequency f can be expressed as a superposition of sinusoidal components of frequencies: f, 2f, 3f, 4f, ...

The application of this theorem to sound is known as Fourier analysis and Fourier synthesis. To understand the relation between a picture of ripples and a description in terms of tones, consider the generic behavior of a string stretched between two endpoints. It leads to one of the most useful techniques in mathematics, a way of using superposition to characterize waves.

The term Fourier theorem applies to any of a set of theorems stating that a function may be represented by a Fourier series provided that it meets certain, very general continuity and periodicity conditions.

Fourier's theorem has a far more general range of application than just to waves on strings. Any wave can be decomposed as a sum of some given collection of other waves. The theorem is particularly useful when you know how to describe a complete set of fundamental modes, and when the system obeys the superposition principle.

Stated another way, by Sir James Jeans, "Fourier's theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves - in brief, every curve can be built up by piling up waves."

• David Bohm
• Fourier transform
• Karl Pribram

• SFU.ca - 'Fourier Theorem'
• USYD.edu.au - 'The Fourier Theorem'
• PariCenter.com 'Brain and Mathematics', Karl Pribram, MD, PhD