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In mathematics, a Fourier series () is a way to represent a (wave-like) function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis. ==History== The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.〔These three did some important early work on the wave equation, especially D'Alembert. Euler's work in this area was mostly comtemporaneous/ in collaboration with Bernoulli, although the latter made some independent contributions to the theory of waves and vibrations ((see here, pg.s 209 & 210, )).〕 Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 ''Mémoire sur la propagation de la chaleur dans les corps solides'' (''Treatise on the propagation of heat in solid bodies''), and publishing his ''Théorie analytique de la chaleur'' (''Analytical theory of heat'') in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles. The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet〔Lejeune-Dirichlet, P. "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". (In French), transl. "On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits". Journal für die reine und angewandte Mathematik, Vol. 4 (1829) pp. 157–169.〕 and Bernhard Riemann〔D. Mascre, Bernhard Riemann: Posthumous Thesis on the Representation of Functions by Trigonometric Series (1867). (Landmark Writings in Western Mathematics 1640–1940 ), Ivor Grattan-Guinness (ed.); pg. 492. Elsevier, 20 May 2005.Accessed 7 Dec 2012.Theory of Complex Functions: Readings in Mathematics ), by Reinhold Remmert; pg 29. Springer, 1991. Accessed 7 Dec 2012.〕 expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fourier series」の詳細全文を読む スポンサード リンク
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