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Fourier transforms. A Fourier series ( / ˈfʊrieɪ, - iər / [ 1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. [ 2] By expressing a function as a sum of sines and cosines, many problems involving ...
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 Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
In physics, engineering and mathematics, the Fourier transform ( FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex -valued function of frequency.
Fourier transforms. In mathematics, Fourier analysis ( / ˈfʊrieɪ, - iər /) [ 1] is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum ...
In general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. If f is of bounded variation, then its Fourier series converges everywhere. If f is continuous and its Fourier coefficients are absolutely summable, then the ...
Parseval's identity. 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 ...
Carleson's theorem. Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise ( Lebesgue) almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson ( 1966 ). The name is also often used to refer to the extension of the result by Richard Hunt ( 1968) to Lp functions for p ∈ ...
A Fourier series uses an orthonormal basis of trigonometric functions. The series expansion is applied to periodic functions. The trigonometric functions are solutions to a Laplacian eigenvalue problem on an interval with periodic boundary conditions. In contrast, a generalized Fourier series can apply to any functions.