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A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
"Chapter 2: Development in Trigonometric Series". An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and ...
The Fourier series for the identity function suffers from the Gibbs phenomenon near the ends of the periodic interval. Every Fourier series gives an example of a trigonometric series. Let the function f ( x ) = x {\displaystyle f(x)=x} on [ − π , π ] {\displaystyle [-\pi ,\pi ]} be extended periodically (see sawtooth wave ).
A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation, [20] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series ...
Sergei Vladimirovich Konyagin, "On divergence of trigonometric Fourier series everywhere", C. R. Acad. Sci. Paris 329 (1999), 693–697. Jean-Pierre Kahane, Some random series of functions, second edition. Cambridge University Press, 1993. ISBN 0-521-45602-9
Pages in category "Fourier series" The following 31 pages are in this category, out of 31 total. ... Trigonometric series; Trigonometric Series; W. Weyl integral;
In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes).
A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions.The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion is applied to periodic functions.