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The '''Fourier transform''', named for [[Jean Baptiste Joseph Fourier]], is an [[integral transform]] that re-expresses a function in terms of [[trigonometric function|sinusoidal]] [[basis function]]s, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). There are many closely-related variations of this transform, summarized below, depending upon the type of function being transformed. See also: [[List of Fourier-related transforms]].
Fourier transforms have many scientific applications — in [[physics]], [[number theory]], [[combinatorics]], [[signal processing]], [[probability theory]], [[statistics]], [[cryptography]], [[acoustics]], [[oceanography]], [[optics]], [[geometry]], and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component [[frequency|frequencies]] and their [[amplitude]]s.) This wide applicability stems from several useful properties of the transforms:
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Most often, the unqualified term "Fourier transform" refers to the [[continuous Fourier transform]], representing any square-[[Lebesgue integral|integrable]] function ''f''(''t'') as a sum of [[complex number|complex]] exponentials with angular frequencies ω and complex amplitudes ''F''(ω):
:<math>f(t) = \mathcal{F}^{-1}(F)(t)
= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(\omega) e^{i\omega t}\,d\omega.</math>
This is actually the ''inverse'' continuous Fourier transform, whereas the Fourier transform expresses ''F''(ω) in terms of ''f''(''t''); the original function and its transform are sometimes called a ''transform pair''. See [[continuous Fourier transform]] for more information, including a table of transforms, discussion of the transform properties, and the various conventions
The continuous transform is actually a generalization of an earlier concept, a [[Fourier series]], which was specific to periodic (or finite-___domain) functions ''f''(''x'') (with period 2π), and represents these functions as a [[series (mathematics)|series]] of sinusoids:
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:<math>x_k = \frac{1}{n} \sum_{j=0}^{n-1} f_j e^{2\pi ijk/n} \quad \quad k = 0,\dots,n-1</math>
where ''f''<sub>''j''</sub> are the Fourier amplitudes. Although applying this formula directly would require O(''n''<sup>2</sup>) operations, it can be computed in only O(''n'' log ''n'') operations using a [[fast Fourier transform]] (FFT) algorithm (see [[Big O notation]]), which makes Fourier transformation a practical and important operation on computers.
These Fourier variants can also be generalized to Fourier transforms on arbitrary [[locally compact]] [[abelian]] [[topological group]]s, which are studied in [[harmonic analysis]]; there, one transforms from a group to its [[dual group]]. This treatment also allows a general formulation of the [[convolution theorem]], which relates Fourier transforms and [[convolution]]s. See also the [[Pontryagin duality]] for the generalized underpinnings of the Fourier transform.
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