Editing Fourier transform (section)

{{Short description|Mathematical transform that expresses a function of time as a function of frequency}}
{{distinguish|text =[[Separation of variables|Fourier method]] or Fourier's original [[sine and cosine transforms]]}}
{{Fourier transforms}}
[[File:CQT-piano-chord.png|thumb|An example application of the Fourier transform is determining the constituent pitches in a [[music]]al [[waveform]]. This image is the result of applying a [[constant-Q transform]] (a [[Fourier-related transform]]) to the waveform of a [[C major]] [[piano]] [[chord (music)|chord]]. The first three peaks on the left correspond to the frequencies of the [[fundamental frequency]] of the chord (C, E, G). The remaining smaller peaks are higher-frequency [[overtone]]s of the fundamental pitches. A [[pitch detection algorithm]] could use the relative intensity of these peaks to infer which notes the pianist pressed.]]

In [[mathematics]], the '''Fourier transform''' ('''FT''') is an [[integral transform]] that takes a [[function (mathematics)|function]] as input then outputs another function that describes the extent to which various [[Frequency|frequencies]] are present in the original function. The output of the transform is a [[complex number|complex]]-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the [[Operation (mathematics)|mathematical operation]]. When a distinction needs to be made, the output of the operation is sometimes called the [[frequency domain]] representation of the original function. The Fourier transform is analogous to decomposing the [[sound]] of a musical [[Chord (music)|chord]] into the [[sound intensity|intensities]] of its constituent [[Pitch (music)|pitches]].

[[File:Fourier transform time and frequency domains (small).gif|class=skin-invert-image|thumb|right|The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.]]

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the [[#Uncertainty principle|uncertainty principle]]. The [[critical point (mathematics)|critical]] case for this principle is the [[Gaussian function]], of substantial importance in [[probability theory]] and [[statistics]] as well as in the study of physical phenomena exhibiting [[normal distribution]]  (e.g., [[diffusion]]). The Fourier transform of a Gaussian function is another Gaussian function. [[Joseph Fourier]] introduced [[sine and cosine transforms]] (which [[Sine and cosine transforms#Relation with complex exponentials|correspond to the imaginary and real components]] of the modern Fourier transform) in his study of [[heat transfer]], where Gaussian functions appear as solutions of the [[heat equation]].

The Fourier transform can be formally defined as an [[improper integral|improper]] [[Riemann integral]], making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.<ref group=note>Depending on the application a [[Lebesgue integral]], [[distribution (mathematics)|distributional]], or other approach may be most appropriate.</ref> For example, many relatively simple applications use the [[Dirac delta function]], which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.<ref group=note>{{harvtxt|Vretblad|2000}} provides solid justification for these formal procedures without going too deeply into [[functional analysis]] or the [[distribution (mathematics)|theory of distributions]].</ref>

The Fourier transform can also be generalized to functions of several variables on [[Euclidean space]], sending a function of {{nowrap|3-dimensional}} 'position space' to a function of {{nowrap|3-dimensional}} momentum (or a function of space and time to a function of [[4-momentum]]). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in [[quantum mechanics]], where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly [[vector-valued function|vector-valued]].<ref group=note>In [[relativistic quantum mechanics]] one encounters vector-valued Fourier transforms of multi-component wave functions. In [[quantum field theory]], operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for example {{harvtxt|Greiner|Reinhardt|1996}}.</ref> Still further generalization is possible to functions on [[group (mathematics)|groups]], which, besides the original Fourier transform on [[Real number#Arithmetic|{{math|'''R'''}}]] or {{math|'''R'''<sup>''n''</sup>}}, notably includes the [[discrete-time Fourier transform]] (DTFT, group = {{math|[[integers|'''Z''']]}}), the [[discrete Fourier transform]] (DFT, group = [[cyclic group|{{math|'''Z''' mod ''N''}}]]) and the [[Fourier series]] or circular Fourier transform (group = {{math|[[circle group|''S''<sup>1</sup>]]}}, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle [[periodic function]]s. The [[fast Fourier transform]] (FFT) is an algorithm for computing the DFT.