Fourier theorem 1561206 220441954 2008-06-19T21:50:04Z Shanes 94147 Reverted edits by [[Special:Contributions/68.41.127.97|68.41.127.97]] ([[User talk:68.41.127.97|talk]]) to last version by 68.83.150.120 In [[mathematics]], the '''Fourier theorem''' is a [[theorem]] stating that a [[periodic function]] ''f''(''x''), which is reasonably [[continuous function|continuous]], may be expressed as the sum of a [[series (mathematics)|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 [[Jean Baptiste Joseph Fourier|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'', 2''f'', 3''f'', 4''f'', ... 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 (mathematics)|function]] may be represented by a Fourier series provided that it meets certain, very general [[continuous function|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." ==See also== * [[David Bohm]] * [[Fourier transform]] * [[Karl H. Pribram]] ==External links== * [http://www.sfu.ca/sonic-studio/handbook/Fourier_Theorem.html SFU.ca] - 'Fourier Theorem' * [http://oldsite.vislab.usyd.edu.au/CP3/Four1/node3.html USYD.edu.au] - 'The Fourier Theorem' * [http://www.paricenter.com/library/papers/pribram01.php PariCenter.com] 'Brain and Mathematics', Karl Pribram, MD, PhD [[Category:Mathematical theorems]]