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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 ...
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 ...
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 ...
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 ...
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.
Fourier inversion theorem. In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
Half range Fourier series. In mathematics, a half range Fourier series is a Fourier series defined on an interval instead of the more common , with the implication that the analyzed function should be extended to as either an even (f (-x)=f (x)) or odd function (f (-x)=-f (x)). This allows the expansion of the function in a series solely of ...
Discrete Fourier series. In digital signal processing, a discrete Fourier series (DFS) a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).