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Fourier transform

From Wikipedia, the free encyclopedia
(Redirected fromFourier uncertainty principle)
Mathematical transform that expresses a function of time as a function of frequency
Not to be confused withFourier method or Fourier's originalsine and cosine transforms.
Fourier transforms
An example application of the Fourier transform is determining the constituent pitches in amusicalwaveform. This image is the result of applying aconstant-Q transform (aFourier-related transform) to the waveform of aC majorpianochord. The first three peaks on the left correspond to the frequencies of thefundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequencyovertones of the fundamental pitches. Apitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.

Inmathematics, theFourier transform (FT) is anintegral transform that takes afunction as input then outputs another function that describes the extent to which variousfrequencies are present in the original function. The output of the transform is acomplex-valued function of frequency. The termFourier transform refers to both this complex-valued function and themathematical operation. When a distinction needs to be made, the output of the operation is sometimes called thefrequency domain representation of the original function. The Fourier transform is analogous to decomposing thesound of a musicalchord into theintensities of its constituentpitches.

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 theuncertainty principle. Thecritical case for this principle is theGaussian function, of substantial importance inprobability theory andstatistics as well as in the study of physical phenomena exhibitingnormal distribution (e.g.,diffusion). The Fourier transform of a Gaussian function is another Gaussian function.Joseph Fourier introducedsine and cosine transforms (whichcorrespond to the imaginary and real components of the modern Fourier transform) in his study ofheat transfer, where Gaussian functions appear as solutions of theheat equation.

The Fourier transform can be formally defined as animproperRiemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.[note 1] For example, many relatively simple applications use theDirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.[note 2]

The Fourier transform can also be generalized to functions of several variables onEuclidean space, sending a function of3-dimensional "position space" to a function of3-dimensional momentum (or a function of space and time to a function of4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as inquantum 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 possiblyvector-valued.[note 3] Still further generalization is possible to functions ongroups, which, besides the original Fourier transform onR orRn, notably includes thediscrete-time Fourier transform (DTFT, group =Z), thediscrete Fourier transform (DFT, group =Z modN) and theFourier series or circular Fourier transform (group =S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handleperiodic functions. Thefast Fourier transform (FFT) is an algorithm for computing the DFT.

Definition

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The Fourier transform of a complex-valued (Lebesgue) integrable functionf(x){\displaystyle f(x)} on the real line, is the complex valued functionf^(ξ){\displaystyle {\hat {f}}(\xi )}, defined by the integral[1]

Fourier transform

f^(ξ)=∫−∞∞f(x) e−i2πξxdx,∀ξ∈R.{\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \xi \in \mathbb {R} .}    

Eq.1

Evaluating the Fourier transform for all values ofξ{\displaystyle \xi } produces thefrequency-domain function, and it converges at all frequencies to a continuous function tending to zero at infinity. Iff(x){\displaystyle f(x)} decays with all derivatives, i.e.,lim|x|→∞f(n)(x)=0,∀n∈N,{\displaystyle \lim _{|x|\to \infty }f^{(n)}(x)=0,\quad \forall n\in \mathbb {N} ,} thenf^{\displaystyle {\widehat {f}}} converges for all frequencies and, by theRiemann–Lebesgue lemma,f^{\displaystyle {\widehat {f}}} also decays with all derivatives.

First introduced inFourier'sAnalytical Theory of Heat.,[2][3][4][5] the corresponding inversion formula for "sufficiently nice" functions is given by theFourier inversion theorem, i.e.,

Inverse transform

f(x)=∫−∞∞f^(ξ) ei2πξxdξ,∀ x∈R.{\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} .}    

Eq.2

The functionsf{\displaystyle f} andf^{\displaystyle {\widehat {f}}} are referred to as aFourier transform pair.[6]  A common notation for designating transform pairs is:[7]f(x) ⟷F f^(ξ),{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ),}   for example  rect⁡(x) ⟷F sinc⁡(ξ).{\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi ).}

By analogy, theFourier series can be regarded as an abstract Fourier transform on the groupZ{\displaystyle \mathbb {Z} } ofintegers. That is, thesynthesis of a sequence of complex numberscn{\displaystyle c_{n}} is defined by the Fourier transformf(x)=∑n=−∞∞cnei2πnPx,{\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},}such thatcn{\displaystyle c_{n}} are given by the inversion formula, i.e., theanalysiscn=1P∫−P/2P/2f(x)e−i2πnPxdx,{\displaystyle c_{n}={\frac {1}{P}}\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx,}for some complex-valued,P{\displaystyle P}-periodic functionf(x){\displaystyle f(x)} defined on a bounded interval[−P/2,P/2]∈R{\displaystyle [-P/2,P/2]\in \mathbb {R} }. WhenP→∞,{\displaystyle P\to \infty ,} the constituentfrequencies are a continuum:nP→ξ∈R,{\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R} ,}[8][9][10] andcn→f^(ξ)∈C{\displaystyle c_{n}\to {\hat {f}}(\xi )\in \mathbb {C} }.[11]

In other words, on the finite interval[−P/2,P/2]{\displaystyle [-P/2,P/2]} the functionf(x){\displaystyle f(x)} has a discrete decomposition in the periodic functionsei2πxn/P{\displaystyle e^{i2\pi xn/P}}. On the infinite interval(−∞,∞){\displaystyle (-\infty ,\infty )} the functionf(x){\displaystyle f(x)} has a continuous decomposition in periodic functionsei2πxξ{\displaystyle e^{i2\pi x\xi }}.

Lebesgue integrable functions

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See also:Lp space § Lp spaces and Lebesgue integrals

Ameasurable functionf:R→C{\displaystyle f:\mathbb {R} \to \mathbb {C} } is called (Lebesgue) integrable if theLebesgue integral of its absolute value is finite:‖f‖1=∫R|f(x)|dx<∞.{\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty .}Iff{\displaystyle f} is Lebesgue integrable then the Fourier transform, given byEq.1, is well-defined for allξ∈R{\displaystyle \xi \in \mathbb {R} }.[12] Furthermore,f^∈L∞∩C(R){\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R} )} is bounded,uniformly continuous and (by theRiemann–Lebesgue lemma) zero at infinity.

The spaceL1(R){\displaystyle L^{1}(\mathbb {R} )} is the space of measurable functions for which the norm‖f‖1{\displaystyle \|f\|_{1}} is finite, modulo theequivalence relation of equalityalmost everywhere. The Fourier transform onL1(R){\displaystyle L^{1}(\mathbb {R} )} isone-to-one. However, there is no easy characterization of the image, and thus no easy characterization of the inverse transform. In particular,Eq.2 is no longer valid, as it was stated only under the hypothesis thatf(x){\displaystyle f(x)} decayed with all derivatives.

WhileEq.1 defines the Fourier transform for (complex-valued) functions inL1(R){\displaystyle L^{1}(\mathbb {R} )}, it is not well-defined for other integrability classes, most importantly the space ofsquare-integrable functionsL2(R){\displaystyle L^{2}(\mathbb {R} )}. For example, the functionf(x)=(1+x2)−1/2{\displaystyle f(x)=(1+x^{2})^{-1/2}} is inL2{\displaystyle L^{2}} but notL1{\displaystyle L^{1}} and therefore the Lebesgue integralEq.1 does not exist. However, the Fourier transform on the dense subspaceL1∩L2(R)⊂L2(R){\displaystyle L^{1}\cap L^{2}(\mathbb {R} )\subset L^{2}(\mathbb {R} )} admits a unique continuous extension to aunitary operator onL2(R){\displaystyle L^{2}(\mathbb {R} )}. This extension is important in part because, unlike the case ofL1{\displaystyle L^{1}}, the Fourier transform is anautomorphism of the spaceL2(R){\displaystyle L^{2}(\mathbb {R} )}.

In such cases, the Fourier transform can be obtained explicitly byregularizing the integral, and then passing to a limit. In practice, the integral is often regarded as animproper integral instead of a proper Lebesgue integral, but sometimes for convergence one needs to useweak limit orprincipal value instead of the (pointwise) limits implicit in an improper integral.Titchmarsh (1986) andDym & McKean (1985) each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure. A general principle in working with theL2{\displaystyle L^{2}} Fourier transform is that Gaussians are dense inL1∩L2{\displaystyle L^{1}\cap L^{2}}, and the various features of the Fourier transform, such as its unitarity, are easily inferred for Gaussians. Many of the properties of the Fourier transform can then be proven from two facts about Gaussians:[13]

A feature of theL1{\displaystyle L^{1}} Fourier transform is that it is a homomorphism of Banach algebras fromL1{\displaystyle L^{1}} equipped with the convolution operation to the Banach algebra of continuous functions under theL∞{\displaystyle L^{\infty }} (supremum) norm. The conventions chosen in this article are those ofharmonic analysis, and are characterized as the unique conventions such that the Fourier transform is bothunitary onL2 and an algebra homomorphism fromL1 toL∞, without renormalizing the Lebesgue measure.[14]

Angular frequency (ω)

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When the independent variable (x{\displaystyle x}) representstime (often denoted byt{\displaystyle t}), the transform variable (ξ{\displaystyle \xi }) representsfrequency (often denoted byf{\displaystyle f}). For example, if time is measured inseconds, then frequency is inhertz. The Fourier transform can also be written in terms ofangular frequency,ω=2πξ,{\displaystyle \omega =2\pi \xi ,} whose units areradians per second.

The substitutionξ=ω2π{\displaystyle \xi ={\tfrac {\omega }{2\pi }}} intoEq.1 produces this convention, where functionf^{\displaystyle {\widehat {f}}} is relabeledf1^:{\displaystyle {\widehat {f_{1}}}:}f3^(ω)≜∫−∞∞f(x)⋅e−iωxdx=f1^(ω2π),f(x)=12π∫−∞∞f3^(ω)⋅eiωxdω.{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )&\triangleq \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}Unlike theEq.1 definition, the Fourier transform is no longer aunitary transformation, and there is less symmetry between the formulas for the transform and its inverse. Those properties are restored by splitting the2π{\displaystyle 2\pi } factor evenly between the transform and its inverse, which leads to another convention:f2^(ω)≜12π∫−∞∞f(x)⋅e−iωxdx=12π  f1^(ω2π),f(x)=12π∫−∞∞f2^(ω)⋅eiωxdω.{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )&\triangleq {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}Variations of all three conventions can be created by conjugating the complex-exponentialkernel of both the forward and the reverse transform. The signs must be opposites.

Summary of popular forms of the Fourier transform, one-dimensional
ordinary frequencyξ (Hz)unitaryf1^(ξ) ≜ ∫−∞∞f(x)e−i2πξxdx=2π  f2^(2πξ)=f3^(2πξ)f(x)=∫−∞∞f1^(ξ)ei2πxξdξ{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i2\pi \xi x}\,dx={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{-\infty }^{\infty }{\widehat {f_{1}}}(\xi )\,e^{i2\pi x\xi }\,d\xi \end{aligned}}}
angular frequencyω (rad/s)unitaryf2^(ω) ≜ 12π ∫−∞∞f(x)e−iωxdx=12π  f1^(ω2π)=12π  f3^(ω)f(x)=12π ∫−∞∞f2^(ω)eiωxdω{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}
non-unitaryf3^(ω) ≜ ∫−∞∞f(x)e−iωxdx=f1^(ω2π)=2π  f2^(ω)f(x)=12π∫−∞∞f3^(ω)eiωxdω{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}
Generalization forn-dimensional functions
ordinary frequencyξ (Hz)unitaryf1^(ξ) ≜ ∫Rnf(x)e−i2πξ⋅xdx=(2π)n2f2^(2πξ)=f3^(2πξ)f(x)=∫Rnf1^(ξ)ei2πξ⋅xdξ{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{\mathbb {R} ^{n}}{\widehat {f_{1}}}(\xi )e^{i2\pi \xi \cdot x}\,d\xi \end{aligned}}}
angular frequencyω (rad/s)unitaryf2^(ω) ≜ 1(2π)n2∫Rnf(x)e−iω⋅xdx=1(2π)n2f1^(ω2π)=1(2π)n2f3^(ω)f(x)=1(2π)n2∫Rnf2^(ω)eiω⋅xdω{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{2}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}
non-unitaryf3^(ω) ≜ ∫Rnf(x)e−iω⋅xdx=f1^(ω2π)=(2π)n2f2^(ω)f(x)=1(2π)n∫Rnf3^(ω)eiω⋅xdω{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{3}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}

Background

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History

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Main articles:Fourier analysis § History, andFourier series § History

In 1822, Fourier claimed (seeJoseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be expanded into a series of sines.[15] That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

Complex sinusoids

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The redsinusoid can be described by peak amplitude (1), peak-to-peak (2),RMS (3), andwavelength (4). The red and blue sinusoids have a phase difference ofθ.

In general, the coefficientsf^(ξ){\displaystyle {\widehat {f}}(\xi )} are complex numbers, which have two equivalent forms (seeEuler's formula):f^(ξ)=Aeiθ⏟polar coordinate form=Acos⁡(θ)+iAsin⁡(θ)⏟rectangular coordinate form.{\displaystyle {\widehat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}

The product withei2πξx{\displaystyle e^{i2\pi \xi x}} (Eq.2) has these forms:f^(ξ)⋅ei2πξx=Aeiθ⋅ei2πξx=Aei(2πξx+θ)⏟polar coordinate form=Acos⁡(2πξx+θ)+iAsin⁡(2πξx+θ)⏟rectangular coordinate form.{\displaystyle {\begin{aligned}{\widehat {f}}(\xi )\cdot e^{i2\pi \xi x}&=Ae^{i\theta }\cdot e^{i2\pi \xi x}\\&=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}\\&=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.\end{aligned}}}which conveys bothamplitude andphase of frequencyξ.{\displaystyle \xi .} Likewise, the intuitive interpretation ofEq.1 is that multiplyingf(x){\displaystyle f(x)} bye−i2πξx{\displaystyle e^{-i2\pi \xi x}} has the effect of subtractingξ{\displaystyle \xi } from every frequency component of functionf(x).{\displaystyle f(x).}[note 4] Only the component that was at frequencyξ{\displaystyle \xi } can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see§ Example)

It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.

Negative frequency

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See also:Negative frequency § Simplifying the Fourier transform

Euler's formula introduces the possibility of negativeξ.{\displaystyle \xi .}  AndEq.1 is defined∀ξ∈R.{\displaystyle \forall \xi \in \mathbb {R} .} Only certain complex-valuedf(x){\displaystyle f(x)} have transformsf^=0, ∀ ξ<0{\displaystyle {\widehat {f}}=0,\ \forall \ \xi <0} (SeeAnalytic signal. A simple example isei2πξ0x (ξ0>0).{\displaystyle e^{i2\pi \xi _{0}x}\ (\xi _{0}>0).})  But negative frequency is necessary to characterize all other complex-valuedf(x),{\displaystyle f(x),} found insignal processing,partial differential equations,radar,nonlinear optics,quantum mechanics, and others.

For a real-valuedf(x),{\displaystyle f(x),}Eq.1 has the symmetry propertyf^(−ξ)=f^∗(ξ){\displaystyle {\widehat {f}}(-\xi )={\widehat {f}}^{*}(\xi )} (see§ Conjugation below). This redundancy enablesEq.2 to distinguishf(x)=cos⁡(2πξ0x){\displaystyle f(x)=\cos(2\pi \xi _{0}x)} fromei2πξ0x.{\displaystyle e^{i2\pi \xi _{0}x}.}  But of course it cannot tell us the actual sign ofξ0,{\displaystyle \xi _{0},} becausecos⁡(2πξ0x){\displaystyle \cos(2\pi \xi _{0}x)} andcos⁡(2π(−ξ0)x){\displaystyle \cos(2\pi (-\xi _{0})x)} are indistinguishable on just the real numbers line.

Fourier transform for periodic functions

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The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral inEq.1 to be defined the function must beabsolutely integrable. Instead it is common to useFourier series. It is possible to extend the definition to include periodic functions by viewing them astempered distributions.

This makes it possible to see a connection between theFourier series and the Fourier transform for periodic functions that have aconvergent Fourier series. Iff(x){\displaystyle f(x)} is aperiodic function, with periodP{\displaystyle P}, that has a convergent Fourier series, then:f^(ξ)=∑n=−∞∞cn⋅δ(ξ−nP),{\displaystyle {\widehat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}wherecn{\displaystyle c_{n}} are the Fourier series coefficients off{\displaystyle f}, andδ{\displaystyle \delta } is theDirac delta function. In other words, the Fourier transform is aDirac comb function whoseteeth are multiplied by the Fourier series coefficients.

Sampling the Fourier transform

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For broader coverage of this topic, seePoisson summation formula.

The Fourier transform of anintegrable functionf{\displaystyle f} can be sampled at regular intervals of arbitrary length1P.{\displaystyle {\tfrac {1}{P}}.} These samples can be deduced from one cycle of a periodic functionfP{\displaystyle f_{P}} which hasFourier series coefficients proportional to those samples by thePoisson summation formula:fP(x)≜∑n=−∞∞f(x+nP)=1P∑k=−∞∞f^(kP)ei2πkPx,∀k∈Z{\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\widehat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }

The integrability off{\displaystyle f} ensures the periodic summation converges. Therefore, the samplesf^(kP){\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)} can be determined by Fourier series analysis:f^(kP)=∫PfP(x)⋅e−i2πkPxdx.{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}

Whenf(x){\displaystyle f(x)} hascompact support,fP(x){\displaystyle f_{P}(x)} has a finite number of terms within the interval of integration. Whenf(x){\displaystyle f(x)} does not have compact support, numerical evaluation offP(x){\displaystyle f_{P}(x)} requires an approximation, such as taperingf(x){\displaystyle f(x)} or truncating the number of terms.

Units

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See also:Spectral density § Units

The frequency variable must have inverse units to the units of the original function's domain (typically namedt{\displaystyle t} orx{\displaystyle x}). For example, ift{\displaystyle t} is measured in seconds,ξ{\displaystyle \xi } should be in cycles per second orhertz. If the scale of time is in units of2π{\displaystyle 2\pi } seconds, then another Greek letterω{\displaystyle \omega } is typically used instead to representangular frequency (whereω=2πξ{\displaystyle \omega =2\pi \xi }) in units ofradians per second. If usingx{\displaystyle x} for units of length, thenξ{\displaystyle \xi } must be in inverse length, e.g.,wavenumbers. That is to say, there are two versions of the real line: one which is therange oft{\displaystyle t} and measured in units oft,{\displaystyle t,} and the other which is the range ofξ{\displaystyle \xi } and measured in inverse units to the units oft.{\displaystyle t.} These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In general,ξ{\displaystyle \xi } must always be taken to be alinear form on the space of its domain, which is to say that the second real line is thedual space of the first real line. See the article onlinear algebra for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to generalsymmetry groups, including the case of Fourier series.

That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants.

In other conventions, the Fourier transform hasi in the exponent instead of−i, and vice versa for the inversion formula. This convention is common in modern physics[16] and is the default forWolfram Alpha, and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means thatf^(ξ){\displaystyle {\hat {f}}(\xi )} is the amplitude of the wave e−i2πξx{\displaystyle e^{-i2\pi \xi x}} instead of the wave ei2πξx{\displaystyle e^{i2\pi \xi x}}(the former, with its minus sign, is often seen in the time dependence forsinusoidal plane-wave solutions of the electromagnetic wave equation, or in thetime dependence for quantum wave functions). Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involvei have it replaced by−i. Inelectrical engineering the letterj is typically used for theimaginary unit instead ofi becausei is used for current.

When usingdimensionless units, the constant factors might not be written in the transform definition. For instance, inprobability theory, the characteristic functionΦ of the probability density functionf of a random variableX of continuous type is defined without a negative sign in the exponential, and since the units ofx are ignored, there is no 2π either:ϕ(λ)=∫−∞∞f(x)eiλxdx.{\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}

In probability theory and mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because many random variables are not of continuous type, and do not possess a density function, and one must treat not functions butdistributions, i.e., measures which possess "atoms".

From the higher point of view ofgroup characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on alocally compact Abelian group.

Properties

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Letf(x){\displaystyle f(x)} andh(x){\displaystyle h(x)} representintegrable functionsLebesgue-measurable on the real line satisfying:∫−∞∞|f(x)|dx<∞.{\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}We denote the Fourier transforms of these functions asf^(ξ){\displaystyle {\hat {f}}(\xi )} andh^(ξ){\displaystyle {\hat {h}}(\xi )} respectively.

Basic properties

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The Fourier transform has the following basic properties:[17]

Linearity

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a f(x)+b h(x)  ⟺F  a f^(ξ)+b h^(ξ); a,b∈C{\displaystyle a\ f(x)+b\ h(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\ {\widehat {f}}(\xi )+b\ {\widehat {h}}(\xi );\quad \ a,b\in \mathbb {C} }

Time shifting

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f(x−x0)  ⟺F  e−i2πx0ξ f^(ξ); x0∈R{\displaystyle f(x-x_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi x_{0}\xi }\ {\widehat {f}}(\xi );\quad \ x_{0}\in \mathbb {R} }

Frequency shifting

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ei2πξ0xf(x)  ⟺F  f^(ξ−ξ0); ξ0∈R{\displaystyle e^{i2\pi \xi _{0}x}f(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(\xi -\xi _{0});\quad \ \xi _{0}\in \mathbb {R} }

Time scaling

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f(ax)  ⟺F  1|a|f^(ξa); a≠0{\displaystyle f(ax)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right);\quad \ a\neq 0}The casea=−1{\displaystyle a=-1} leads to thetime-reversal property:f(−x)  ⟺F  f^(−ξ){\displaystyle f(-x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(-\xi )}

Symmetry

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When the real and imaginary parts of a complex function are decomposed into theireven and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[18]

Time domainf=fRE+fRO+i fIE+i fIO⏟⇕F⇕F  ⇕F  ⇕F  ⇕FFrequency domainf^=f^RE+i f^IO⏞+i f^IE+f^RO{\displaystyle {\begin{array}{rlcccccccc}{\mathsf {Time\ domain}}&f&=&f_{_{\text{RE}}}&+&f_{_{\text{RO}}}&+&i\ f_{_{\text{IE}}}&+&\underbrace {i\ f_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}&{\widehat {f}}&=&{\widehat {f}}_{_{\text{RE}}}&+&\overbrace {i\ {\widehat {f}}_{_{\text{IO}}}\,} &+&i\ {\widehat {f}}_{_{\text{IE}}}&+&{\widehat {f}}_{_{\text{RO}}}\end{array}}}

From this, various relationships are apparent, for example:

Conjugation

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(f(x))∗  ⟺F  (f^(−ξ))∗{\displaystyle {\bigl (}f(x){\bigr )}^{*}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ \left({\widehat {f}}(-\xi )\right)^{*}}(Note: the ∗ denotescomplex conjugation.)

In particular, iff{\displaystyle f} isreal, thenf^{\displaystyle {\widehat {f}}} iseven symmetric (akaHermitian function):f^(−ξ)=(f^(ξ))∗.{\displaystyle {\widehat {f}}(-\xi )={\bigl (}{\widehat {f}}(\xi ){\bigr )}^{*}.}

And iff{\displaystyle f} is purely imaginary, thenf^{\displaystyle {\widehat {f}}} isodd symmetric:f^(−ξ)=−(f^(ξ))∗.{\displaystyle {\widehat {f}}(-\xi )=-({\widehat {f}}(\xi ))^{*}.}

Real and imaginary parts

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Re⁡{f(x)}  ⟺F  12(f^(ξ)+(f^(−ξ))∗){\displaystyle \operatorname {Re} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2}}\left({\widehat {f}}(\xi )+{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}Im⁡{f(x)}  ⟺F  12i(f^(ξ)−(f^(−ξ))∗){\displaystyle \operatorname {Im} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2i}}\left({\widehat {f}}(\xi )-{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}

Zero frequency component

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Substitutingξ=0{\displaystyle \xi =0} in the definition, we obtain:f^(0)=∫−∞∞f(x)dx.{\displaystyle {\widehat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}

The integral off{\displaystyle f} over its domain is known as the average value orDC bias of the function.

Uniform continuity and the Riemann–Lebesgue lemma

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Therectangular function isLebesgue integrable.
Thesinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.

The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties.

The Fourier transformf^{\displaystyle {\hat {f}}} of any integrable functionf{\displaystyle f} isuniformly continuous and[19][20]‖f^‖∞≤‖f‖1{\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}

By theRiemann–Lebesgue lemma,[21]f^(ξ)→0 as |ξ|→∞.{\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}

However,f^{\displaystyle {\hat {f}}} need not be integrable. For example, the Fourier transform of therectangular function, which is integrable, is thesinc function, which is notLebesgue integrable, because itsimproper integrals behave analogously to thealternating harmonic series, in converging to a sum without beingabsolutely convergent.

It is not generally possible to write theinverse transform as aLebesgue integral. However, when bothf{\displaystyle f} andf^{\displaystyle {\hat {f}}} are integrable, the inverse equalityf(x)=∫−∞∞f^(ξ)ei2πxξdξ{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi } holds for almost everyx. As a result, the Fourier transform isinjective onL1(R).

Plancherel theorem and Parseval's theorem

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Main articles:Plancherel theorem andParseval's theorem

Letf(x) andg(x) be integrable, and letf̂(ξ) andĝ(ξ) be their Fourier transforms. Iff(x) andg(x) are alsosquare-integrable, then the Parseval formula follows:[22]⟨f,g⟩L2=∫−∞∞f(x)g(x)¯dx=∫−∞∞f^(ξ)g^(ξ)¯dξ,{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}where the bar denotescomplex conjugation.

ThePlancherel theorem, which follows from the above, states that[23]‖f‖L22=∫−∞∞|f(x)|2dx=∫−∞∞|f^(ξ)|2dξ.{\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to aunitary operator onL2(R). OnL1(R) ∩L2(R), this extension agrees with original Fourier transform defined onL1(R), thus enlarging the domain of the Fourier transform toL1(R) +L2(R) (and consequently toLp(R) for1 ≤p ≤ 2). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves theenergy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.

SeePontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Convolution theorem

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Main article:Convolution theorem

The Fourier transform translates betweenconvolution and multiplication of functions. Iff(x) andg(x) are integrable functions with Fourier transformsf̂(ξ) andĝ(ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transformsf̂(ξ) andĝ(ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if:h(x)=(f∗g)(x)=∫−∞∞f(y)g(x−y)dy,{\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}where∗ denotes the convolution operation, then:h^(ξ)=f^(ξ)g^(ξ).{\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}

Inlinear time invariant (LTI) system theory, it is common to interpretg(x) as theimpulse response of an LTI system with inputf(x) and outputh(x), since substituting theunit impulse forf(x) yieldsh(x) =g(x). In this case,ĝ(ξ) represents thefrequency response of the system.

Conversely, iff(x) can be decomposed as the product of two square integrable functionsp(x) andq(x), then the Fourier transform off(x) is given by the convolution of the respective Fourier transformsp̂(ξ) andq̂(ξ).

Cross-correlation theorem

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Main articles:Cross-correlation andWiener–Khinchin_theorem

In an analogous manner, it can be shown that ifh(x) is thecross-correlation off(x) andg(x):h(x)=(f⋆g)(x)=∫−∞∞f(y)¯g(x+y)dy{\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}then the Fourier transform ofh(x) is:h^(ξ)=f^(ξ)¯g^(ξ).{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}

As a special case, theautocorrelation of functionf(x) is:h(x)=(f⋆f)(x)=∫−∞∞f(y)¯f(x+y)dy{\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}for whichh^(ξ)=f^(ξ)¯f^(ξ)=|f^(ξ)|2.{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}

Differentiation

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Supposef(x) is an absolutely continuous differentiable function, and bothf and its derivativef′ are integrable. Then the Fourier transform of the derivative is given byf′^(ξ)=F{ddxf(x)}=i2πξf^(ξ).{\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}More generally, the Fourier transformation of thenth derivativef(n) is given byf(n)^(ξ)=F{dndxnf(x)}=(i2πξ)nf^(ξ).{\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}

Analogously,F{dndξnf^(ξ)}=(i2πx)nf(x){\displaystyle {\mathcal {F}}\left\{{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi )\right\}=(i2\pi x)^{n}f(x)}, soF{xnf(x)}=(i2π)ndndξnf^(ξ).{\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}

By applying the Fourier transform and using these formulas, someordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "f(x) is smoothif and only iff̂(ξ) quickly falls to 0 for|ξ| → ∞." By using the analogous rules for the inverse Fourier transform, one can also say "f(x) quickly falls to 0 for|x| → ∞ if and only iff̂(ξ) is smooth."

Eigenfunctions

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See also:Mehler kernel andHermite polynomials § Hermite functions as eigenfunctions of the Fourier transform

The Fourier transform is a linear transform which has eigenfunctions obeyingF[ψ]=λψ,{\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,} withλ∈C.{\displaystyle \lambda \in \mathbb {C} .}

A set of eigenfunctions is found by noting that the homogeneous differential equation[U(12πddx)+U(x)]ψ(x)=0{\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0} leads to eigenfunctionsψ(x){\displaystyle \psi (x)} of the Fourier transformF{\displaystyle {\mathcal {F}}} as long as the form of the equation remains invariant under Fourier transform.[note 5] In other words, every solutionψ(x){\displaystyle \psi (x)} and its Fourier transformψ^(ξ){\displaystyle {\hat {\psi }}(\xi )} obey the same equation. Assuminguniqueness of the solutions, every solutionψ(x){\displaystyle \psi (x)} must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform ifU(x){\displaystyle U(x)} can be expanded in a power series in which for all terms the same factor of either one of±1,±i{\displaystyle \pm 1,\pm i} arises from the factorsin{\displaystyle i^{n}} introduced by thedifferentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowableU(x)=x{\displaystyle U(x)=x} leads to thestandard normal distribution.[24]

More generally, a set of eigenfunctions is also found by noting that thedifferentiation rules imply that theordinary differential equation[W(i2πddx)+W(x)]ψ(x)=Cψ(x){\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}withC{\displaystyle C} constant andW(x){\displaystyle W(x)} being a non-constant even function remains invariant in form when applying the Fourier transformF{\displaystyle {\mathcal {F}}} to both sides of the equation. The simplest example is provided byW(x)=x2{\displaystyle W(x)=x^{2}} which is equivalent to considering the Schrödinger equation for thequantum harmonic oscillator.[25] The corresponding solutions provide an important choice of an orthonormal basis forL2(R) and are given by the "physicist's"Hermite functions. Equivalently one may useψn(x)=24n!e−πx2Hen(2xπ),{\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}whereHen(x) are the "probabilist's"Hermite polynomials, defined asHen(x)=(−1)ne12x2(ddx)ne−12x2.{\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}

Under this convention for the Fourier transform, we have thatψ^n(ξ)=(−i)nψn(ξ).{\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}

In other words, the Hermite functions form a completeorthonormal system ofeigenfunctions for the Fourier transform onL2(R).[17][26] However, this choice of eigenfunctions is not unique. Because ofF4=id{\displaystyle {\mathcal {F}}^{4}=\mathrm {id} } there are only four differenteigenvalues of the Fourier transform (the fourth roots of unity ±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.[27] As a consequence of this, it is possible to decomposeL2(R) as a direct sum of four spacesH0,H1,H2, andH3 where the Fourier transform acts onHek simply by multiplication byik.

Since the complete set of Hermite functionsψn provides a resolution of the identity they diagonalize the Fourier operator, i.e. the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed:F[f](ξ)=∫dxf(x)∑n≥0(−i)nψn(x)ψn(ξ) .{\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}

This approach to define the Fourier transform was first proposed byNorbert Wiener.[28] Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely thefractional Fourier transform used in time–frequency analysis.[29] Inphysics, this transform was introduced byEdward Condon.[30] This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the rightconventions. Consequently, under the proper conditions it may be expected to result from a self-adjoint generatorN{\displaystyle N} via[31]F[ψ]=e−itNψ.{\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}

The operatorN{\displaystyle N} is thenumber operator of the quantum harmonic oscillator written as[32][33]N≡12(x−∂∂x)(x+∂∂x)=12(−∂2∂x2+x2−1).{\displaystyle N\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right).}

It can be interpreted as thegenerator offractional Fourier transforms for arbitrary values oft, and of the conventional continuous Fourier transformF{\displaystyle {\mathcal {F}}} for the particular valuet=π/2,{\displaystyle t=\pi /2,} with theMehler kernel implementing the correspondingactive transform. The eigenfunctions ofN{\displaystyle N} are theHermite functionsψn(x){\displaystyle \psi _{n}(x)} which are therefore also eigenfunctions ofF.{\displaystyle {\mathcal {F}}.}

Upon extending the Fourier transform todistributions theDirac comb is also an eigenfunction of the Fourier transform.

Inversion and periodicity

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Further information:Fourier inversion theorem andFractional Fourier transform

Under suitable conditions on the functionf{\displaystyle f}, it can be recovered from its Fourier transformf^{\displaystyle {\hat {f}}}. Indeed, denoting the Fourier transform operator byF{\displaystyle {\mathcal {F}}}, soFf:=f^{\displaystyle {\mathcal {F}}f:={\hat {f}}}, then for suitable functions, applying the Fourier transform twice simply flips the function:(F2f)(x)=f(−x){\displaystyle \left({\mathcal {F}}^{2}f\right)(x)=f(-x)}, which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yieldsF4(f)=f{\displaystyle {\mathcal {F}}^{4}(f)=f}, so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times:F3(f^)=f{\displaystyle {\mathcal {F}}^{3}\left({\hat {f}}\right)=f}. In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining theparity operatorP{\displaystyle {\mathcal {P}}} such that(Pf)(x)=f(−x){\displaystyle ({\mathcal {P}}f)(x)=f(-x)}, we have:F0=id,F1=F,F2=P,F3=F−1=P∘F=F∘P,F4=id{\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equalityalmost everywhere?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of theFourier inversion theorem.

This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in thetime–frequency domain (considering time as thex-axis and frequency as they-axis), and the Fourier transform can be generalized to thefractional Fourier transform, which involves rotations by other angles. This can be further generalized tolinear canonical transformations, which can be visualized as the action of thespecial linear groupSL2(R) on the time–frequency plane, with the preserved symplectic form corresponding to theuncertainty principle, below. This approach is particularly studied insignal processing, undertime–frequency analysis.

Connection with the Heisenberg group

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TheHeisenberg group is a certaingroup ofunitary operators on theHilbert spaceL2(R) of square integrable complex valued functionsf on the real line, generated by the translations(Ty f)(x) =f (x +y) and multiplication byei2πξx,(Mξ f)(x) =ei2πξxf (x). These operators do not commute, as their (group) commutator is(Mξ−1Ty−1MξTyf)(x)=ei2πξyf(x){\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{i2\pi \xi y}f(x)}which is multiplication by the constant (independent ofx)ei2πξy ∈U(1) (thecircle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensionalLie group of triples(x,ξ,z) ∈R2 ×U(1), with the group law(x1,ξ1,t1)⋅(x2,ξ2,t2)=(x1+x2,ξ1+ξ2,t1t2ei2π(x1ξ1+x2ξ2+x1ξ2)).{\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{i2\pi \left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}

Denote the Heisenberg group byH1. The above procedure describes not only the group structure, but also a standardunitary representation ofH1 on a Hilbert space, which we denote byρ :H1 →B(L2(R)). Define the linear automorphism ofR2 byJ(xξ)=(−ξx){\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}so thatJ2 = −I. ThisJ can be extended to a unique automorphism ofH1:j(x,ξ,t)=(−ξ,x,te−i2πξx).{\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-i2\pi \xi x}\right).}

According to theStone–von Neumann theorem, the unitary representationsρ andρ ∘j are unitarily equivalent, so there is a unique intertwinerW ∈U(L2(R)) such thatρ∘j=WρW∗.{\displaystyle \rho \circ j=W\rho W^{*}.}This operatorW is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.[34] For example, the square of the Fourier transform,W2, is an intertwiner associated withJ2 = −I, and so we have(W2f)(x) =f (−x) is the reflection of the original functionf.

Complex domain

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Theintegral for the Fourier transformf^(ξ)=∫−∞∞e−i2πξtf(t)dt{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}can be studied forcomplex values of its argumentξ. Depending on the properties off, this might not converge off the real axis at all, or it might converge to acomplexanalytic function for all values ofξ =σ +iτ, or something in between.[35]

ThePaley–Wiener theorem says thatf is smooth (i.e.,n-times differentiable for all positive integersn) and compactly supported if and only iff̂ (σ +iτ) is aholomorphic function for which there exists aconstanta > 0 such that for anyintegern ≥ 0,|ξnf^(ξ)|≤Cea|τ|{\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi )\right\vert \leq Ce^{a\vert \tau \vert }}for some constantC. (In this case,f is supported on[−a,a].) This can be expressed by saying thatf̂ is anentire function which israpidly decreasing inσ (for fixedτ) and of exponential growth inτ (uniformly inσ).[36]

(Iff is not smooth, but onlyL2, the statement still holds providedn = 0.[37]) The space of such functions of acomplex variable is called the Paley—Wiener space. This theorem has been generalised to semisimpleLie groups.[38]

Iff is supported on the half-linet ≥ 0, thenf is said to be "causal" because theimpulse response function of a physically realisablefilter must have this property, as no effect can precede its cause.Paley and Wiener showed that thenf̂ extends to aholomorphic function on the complex lower half-planeτ < 0 which tends to zero asτ goes to infinity.[39] The converse is false and it is not known how to characterise the Fourier transform of a causal function.[40]

Laplace transform

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See also:Laplace transform § Fourier transform

The Fourier transformf̂(ξ) is related to theLaplace transformF(s), which is also used for the solution ofdifferential equations and the analysis offilters.

It may happen that a functionf for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of thecomplex plane.

For example, iff(t) is of exponential growth, i.e.,|f(t)|<Cea|t|{\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}for some constantsC,a ≥ 0, then[41]f^(iτ)=∫−∞∞e2πτtf(t)dt,{\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}convergent for all2πτ < −a, is thetwo-sided Laplace transform off.

The more usual version ("one-sided") of the Laplace transform isF(s)=∫0∞f(t)e−stdt.{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

Iff is also causal, and analytical, then:f^(iτ)=F(−2πτ).{\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).} Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variables =i2πξ.

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements. Two particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle. Such designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.

Furthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response. There, only the delay circuit in-between admits a classical Fourier description, which is critical. Both the circuits to the side are unstable, and do not admit a convergent Fourier decomposition. However, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.

In modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea ofharmonic analysis.

Inversion

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Still withξ=σ+iτ{\displaystyle \xi =\sigma +i\tau }, iff^{\displaystyle {\widehat {f}}} is complex analytic fora ≤τ ≤b, then

∫−∞∞f^(σ+ia)ei2πξtdσ=∫−∞∞f^(σ+ib)ei2πξtdσ{\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{i2\pi \xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{i2\pi \xi t}\,d\sigma }byCauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.[42]

Theorem: Iff(t) = 0 fort < 0, and|f(t)| <Cea|t| for some constantsC,a > 0, thenf(t)=∫−∞∞f^(σ+iτ)ei2πξtdσ,{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{i2\pi \xi t}\,d\sigma ,}for anyτ < −⁠a/2π⁠.

This theorem implies theMellin inversion formula for the Laplace transformation,[41]f(t)=1i2π∫b−i∞b+i∞F(s)estds{\displaystyle f(t)={\frac {1}{i2\pi }}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}for anyb >a, whereF(s) is the Laplace transform off(t).

The hypotheses can be weakened, as in the results of Carleson and Hunt, tof(t)e−at beingL1, provided thatf be of bounded variation in a closed neighborhood oft (cf.Dini test), the value off att be taken to be thearithmetic mean of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.[43]

L2 versions of these inversion formulas are also available.[44]

Fourier transform on Euclidean space

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The Fourier transform can be defined in any arbitrary number of dimensionsn. As with the one-dimensional case, there are many conventions. For an integrable functionf(x), this article takes the definition:f^(ξ)=F(f)(ξ)=∫Rnf(x)e−i2πξ⋅xdx{\displaystyle {\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} }wherex andξ aren-dimensionalvectors, andx ·ξ is thedot product of the vectors. Alternatively,ξ can be viewed as belonging to thedual vector spaceRn⋆{\displaystyle \mathbb {R} ^{n\star }}, in which case the dot product becomes thecontraction ofx andξ, usually written as⟨x,ξ⟩.

All of the basic properties listed above hold for then-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and theRiemann–Lebesgue lemma holds.[21]

Uncertainty principle

[edit]
Further information:Uncertainty principle

Generally speaking, the more concentratedf(x) is, the more spread out its Fourier transformf̂(ξ) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function inx, its Fourier transform stretches out inξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of anuncertainty principle by viewing a function and its Fourier transform asconjugate variables with respect to thesymplectic form on thetime–frequency domain: from the point of view of thelinear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves thesymplectic form.

Supposef(x) is an integrable andsquare-integrable function. Without loss of generality, assume thatf(x) is normalized:∫−∞∞|f(x)|2dx=1.{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}

It follows from thePlancherel theorem thatf̂(ξ) is also normalized.

The spread aroundx = 0 may be measured by thedispersion about zero defined by[45]D0(f)=∫−∞∞x2|f(x)|2dx.{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}

In probability terms, this is thesecond moment of|f(x)|2 about zero.

The uncertainty principle states that, iff(x) is absolutely continuous and the functionsx·f(x) andf′(x) are square integrable, thenD0(f)D0(f^)≥116π2.{\displaystyle D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}.}

The equality is attained only in the casef(x)=C1e−πx2σ2∴f^(ξ)=σC1e−πσ2ξ2{\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}whereσ > 0 is arbitrary andC1 =⁠4√2/√σ⁠ so thatf isL2-normalized. In other words, wheref is a (normalized)Gaussian function with varianceσ2/2π, centered at zero, and its Fourier transform is a Gaussian function with varianceσ−2/2π. Gaussian functions are examples ofSchwartz functions (see the discussion on tempered distributions below).

In fact, this inequality implies that:(∫−∞∞(x−x0)2|f(x)|2dx)(∫−∞∞(ξ−ξ0)2|f^(ξ)|2dξ)≥116π2,∀x0,ξ0∈R.{\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}},\quad \forall x_{0},\xi _{0}\in \mathbb {R} .}Inquantum mechanics, themomentum and positionwave functions are Fourier transform pairs, up to a factor of thePlanck constant. With this constant properly taken into account, the inequality above becomes the statement of theHeisenberg uncertainty principle.[46]

A stronger uncertainty principle is theHirschman uncertainty principle, which is expressed as:H(|f|2)+H(|f^|2)≥log⁡(e2){\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}whereH(p) is thedifferential entropy of theprobability density functionp(x):H(p)=−∫−∞∞p(x)log⁡(p(x))dx{\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.

Sine and cosine transforms

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Main article:Sine and cosine transforms

Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable functionf for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically[47])λ byf(t)=∫0∞(a(λ)cos⁡(2πλt)+b(λ)sin⁡(2πλt))dλ.{\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functionsa andb can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):a(λ)=2∫−∞∞f(t)cos⁡(2πλt)dt{\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}andb(λ)=2∫−∞∞f(t)sin⁡(2πλt)dt.{\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}

Older literature refers to the two transform functions, the Fourier cosine transform,a, and the Fourier sine transform,b.

The functionf can be recovered from the sine and cosine transform usingf(t)=2∫0∞∫−∞∞f(τ)cos⁡(2πλ(τ−t))dτdλ.{\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}together with trigonometric identities. This is referred to as Fourier's integral formula.[41][48][49][50]

Spherical harmonics

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Let the set ofhomogeneousharmonicpolynomials of degreek onRn be denoted byAk. The setAk consists of thesolid spherical harmonics of degreek. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, iff(x) =e−π|x|2P(x) for someP(x) inAk, thenf̂(ξ) =i−kf(ξ). Let the setHk be the closure inL2(Rn) of linear combinations of functions of the formf(|x|)P(x) whereP(x) is inAk. The spaceL2(Rn) is then a direct sum of the spacesHk and the Fourier transform maps each spaceHk to itself and is possible to characterize the action of the Fourier transform on each spaceHk.[21]

Letf(x) =f0(|x|)P(x) (withP(x) inAk), thenf^(ξ)=F0(|ξ|)P(ξ){\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}whereF0(r)=2πi−kr−n+2k−22∫0∞f0(s)Jn+2k−22(2πrs)sn+2k2ds.{\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}

HereJ(n + 2k − 2)/2 denotes theBessel function of the first kind with order⁠n + 2k − 2/2⁠. Whenk = 0 this gives a useful formula for the Fourier transform of a radial function.[51] This is essentially theHankel transform. Moreover, there is a simple recursion relating the casesn + 2 andn[52] allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.

Restriction problems

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In higher dimensions it becomes interesting to studyrestriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a generalclass of square integrable functions. As such, the restriction of the Fourier transform of anL2(Rn) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems inLp for1 <p < 2. It is possible in some cases to define the restriction of a Fourier transform to a setS, providedS has non-zero curvature. The case whenS is the unit sphere inRn is of particular interest. In this case the Tomas–Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere inRn is a bounded operator onLp provided1 ≤p ≤⁠2n + 2/n + 3⁠.

One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable setsER indexed byR ∈ (0,∞): such as balls of radiusR centered at the origin, or cubes of side2R. For a given integrable functionf, consider the functionfR defined by:fR(x)=∫ERf^(ξ)ei2πx⋅ξdξ,x∈Rn.{\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}

Suppose in addition thatf ∈Lp(Rn). Forn = 1 and1 <p < ∞, if one takesER = (−R,R), thenfR converges tof inLp asR tends to infinity, by the boundedness of theHilbert transform. Naively one may hope the same holds true forn > 1. In the case thatER is taken to be a cube with side lengthR, then convergence still holds. Another natural candidate is the Euclidean ballER = {ξ : |ξ| <R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded inLp(Rn). Forn ≥ 2 it is a celebrated theorem ofCharles Fefferman that the multiplier for the unit ball is never bounded unlessp = 2.[28] In fact, whenp ≠ 2, this shows that not only mayfR fail to converge tof inLp, but for some functionsf ∈Lp(Rn),fR is not even an element ofLp.

Fourier transform on function spaces

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See also:Riesz–Thorin theorem

The definition of the Fourier transform naturally extends fromL1(R){\displaystyle L^{1}(\mathbb {R} )} toL1(Rn){\displaystyle L^{1}(\mathbb {R} ^{n})}. That is, iff∈L1(Rn){\displaystyle f\in L^{1}(\mathbb {R} ^{n})} then the Fourier transformF:L1(Rn)→L∞(Rn){\displaystyle {\mathcal {F}}:L^{1}(\mathbb {R} ^{n})\to L^{\infty }(\mathbb {R} ^{n})} is given byf(x)↦f^(ξ)=∫Rnf(x)e−i2πξ⋅xdx,∀ξ∈Rn.{\displaystyle f(x)\mapsto {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx,\quad \forall \xi \in \mathbb {R} ^{n}.}This operator isbounded assupξ∈Rn|f^(ξ)|≤∫Rn|f(x)|dx,{\displaystyle \sup _{\xi \in \mathbb {R} ^{n}}\left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}which shows that itsoperator norm is bounded by1. TheRiemann–Lebesgue lemma shows that iff∈L1(Rn){\displaystyle f\in L^{1}(\mathbb {R} ^{n})} then its Fourier transform actually belongs to thespace of continuous functions which vanish at infinity, i.e.,f^∈C0(Rn)⊂L∞(Rn){\displaystyle {\hat {f}}\in C_{0}(\mathbb {R} ^{n})\subset L^{\infty }(\mathbb {R} ^{n})}.[53] Furthermore, theimage ofL1{\displaystyle L^{1}} underF{\displaystyle {\mathcal {F}}} is a strict subset ofC0(Rn){\displaystyle C_{0}(\mathbb {R} ^{n})}.

Similarly to the case of one variable, the Fourier transform can be defined onL2(Rn){\displaystyle L^{2}(\mathbb {R} ^{n})}. The Fourier transform inL2(Rn){\displaystyle L^{2}(\mathbb {R} ^{n})} is no longer given by an ordinary Lebesgue integral, although it can be computed by animproper integral, i.e.,f^(ξ)=limR→∞∫|x|≤Rf(x)e−i2πξ⋅xdx{\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-i2\pi \xi \cdot x}\,dx}where the limit is taken in theL2 sense.[54][55]

Furthermore,F:L2(Rn)→L2(Rn){\displaystyle {\mathcal {F}}:L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})} is aunitary operator.[56] For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for anyf,g ∈L2(Rn) we have∫Rnf(x)Fg(x)dx=∫RnFf(x)g(x)dx.{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}

In particular, the image ofL2(Rn) is itself under the Fourier transform.

On otherLp

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For1<p<2{\displaystyle 1<p<2}, the Fourier transform can be defined onLp(R){\displaystyle L^{p}(\mathbb {R} )} byMarcinkiewicz interpolation, which amounts to decomposing such functions into a fat tail part inL2 plus a fat body part inL1. In each of these spaces, the Fourier transform of a function inLp(Rn) is inLq(Rn), whereq =⁠p/p − 1⁠ is theHölder conjugate ofp (by theHausdorff–Young inequality). However, except forp = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions inLp for the range2 <p < ∞ requires the study of distributions.[57] In fact, it can be shown that there are functions inLp withp > 2 so that the Fourier transform is not defined as a function.[21]

Tempered distributions

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Main article:Distribution (mathematics) § Tempered distributions and Fourier transform
See also:Rigged Hilbert space

One might consider enlarging the domain of the Fourier transform fromL1+L2{\displaystyle L^{1}+L^{2}} by consideringgeneralized functions, or distributions. A distribution onRn{\displaystyle \mathbb {R} ^{n}} is a continuous linear functional on the spaceCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})} of compactly supported smooth functions (i.e.bump functions), equipped with a suitable topology. SinceCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})} is dense inL2(Rn){\displaystyle L^{2}(\mathbb {R} ^{n})}, thePlancherel theorem allows one to extend the definition of the Fourier transform to general functions inL2(Rn){\displaystyle L^{2}(\mathbb {R} ^{n})} by continuity arguments. The strategy is then to consider the action of the Fourier transform onCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})} and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not mapCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})} toCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}. In fact the Fourier transform of an element inCc∞(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})} can not vanish on an open set; see the above discussion on the uncertainty principle.[58][59]

The Fourier transform can also be defined fortempered distributionsS′(Rn){\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}, dual to the space ofSchwartz functionsS(Rn){\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}. A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives, henceCc∞(Rn)⊂S(Rn){\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})\subset {\mathcal {S}}(\mathbb {R} ^{n})} and:F:Cc∞(Rn)→S(Rn)∖Cc∞(Rn).{\displaystyle {\mathcal {F}}:C_{c}^{\infty }(\mathbb {R} ^{n})\rightarrow S(\mathbb {R} ^{n})\setminus C_{c}^{\infty }(\mathbb {R} ^{n}).} The Fourier transform is anautomorphism of the Schwartz space and, by duality, also an automorphism of the space of tempered distributions.[21][60] The tempered distributions include well-behaved functions of polynomial growth, distributions of compact support as well as all the integrable functions mentioned above.

For the definition of the Fourier transform of a tempered distribution, letf{\displaystyle f} andg{\displaystyle g} be integrable functions, and letf^{\displaystyle {\hat {f}}} andg^{\displaystyle {\hat {g}}} be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,[21]∫Rnf^(x)g(x)dx=∫Rnf(x)g^(x)dx.{\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}

Every integrable functionf{\displaystyle f} defines (induces) a distributionTf{\displaystyle T_{f}} by the relationTf(ϕ)=∫Rnf(x)ϕ(x)dx,∀ϕ∈S(Rn).{\displaystyle T_{f}(\phi )=\int _{\mathbb {R} ^{n}}f(x)\phi (x)\,dx,\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ^{n}).}So it makes sense to define the Fourier transform of a tempered distributionTf∈S′(R){\displaystyle T_{f}\in {\mathcal {S}}'(\mathbb {R} )} by the duality:⟨T^f,ϕ⟩=⟨Tf,ϕ^⟩,∀ϕ∈S(Rn).{\displaystyle \langle {\widehat {T}}_{f},\phi \rangle =\langle T_{f},{\widehat {\phi }}\rangle ,\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ^{n}).}Extending this to all tempered distributionsT{\displaystyle T} gives the general definition of the Fourier transform.

Distributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

Generalizations

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Fourier–Stieltjes transform on measurable spaces

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See also:Bochner–Minlos theorem,Riesz–Markov–Kakutani representation theorem, andFourier series § Fourier-Stieltjes series

The Fourier transform of afiniteBorel measureμ onRn is given by the continuous function:[61]μ^(ξ)=∫Rne−i2πx⋅ξdμ,{\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu ,}and called theFourier-Stieltjes transform due to its connection with theRiemann-Stieltjes integral representation of(Radon) measures.[62] Ifμ{\displaystyle \mu } is theprobability distribution of arandom variableX{\displaystyle X} then its Fourier–Stieltjes transform is, by definition, acharacteristic function.[63] If, in addition, the probability distribution has aprobability density function, this definition is subject to the usual Fourier transform.[64] Stated more generally, whenμ{\displaystyle \mu } isabsolutely continuous with respect to the Lebesgue measure, i.e.,dμ=f(x)dx,{\displaystyle d\mu =f(x)dx,}thenμ^(ξ)=f^(ξ),{\displaystyle {\hat {\mu }}(\xi )={\hat {f}}(\xi ),}and the Fourier-Stieltjes transform reduces to the usual definition of the Fourier transform. That is, the notable difference with the Fourier transform of integrable functions is that the Fourier-Stieltjes transform need not vanish at infinity, i.e., theRiemann–Lebesgue lemma fails for measures.[65]

Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.

One example of a finite Borel measure that is not a function is theDirac measure.[66] Its Fourier transform is a constant function (whose value depends on the form of the Fourier transform used).

Locally compact abelian groups

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Main article:Pontryagin duality

The Fourier transform may be generalized to anylocally compact abelian group, i.e., anabelian group that is also alocally compact Hausdorff space such that the group operation is continuous. IfG is a locally compact abelian group, it has a translation invariant measureμ, calledHaar measure. For a locally compact abelian groupG, the set of irreducible, i.e. one-dimensional, unitary representations are called itscharacters. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by thecompact-open topology on the space of all continuous functions fromG{\displaystyle G} to thecircle group), the set of charactersĜ is itself a locally compact abelian group, called thePontryagin dual ofG. For a functionf inL1(G), its Fourier transform is defined by[57]f^(ξ)=∫Gξ(x)f(x)dμfor any ξ∈G^.{\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \quad {\text{for any }}\xi \in {\hat {G}}.}

The Riemann–Lebesgue lemma holds in this case;f̂(ξ) is a function vanishing at infinity onĜ.

The Fourier transform onT = R/Z is an example; hereT is a locally compact abelian group, and the Haar measureμ onT can be thought of as the Lebesgue measure on [0,1). Consider the representation ofT on the complex planeC that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible sinceC is 1-dim){ek:T→GL1(C)=C∗∣k∈Z}{\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}} whereek(x)=ei2πkx{\displaystyle e_{k}(x)=e^{i2\pi kx}} forx∈T{\displaystyle x\in T}.

The character of such representation, that is the trace ofek(x){\displaystyle e_{k}(x)} for eachx∈T{\displaystyle x\in T} andk∈Z{\displaystyle k\in Z}, isei2πkx{\displaystyle e^{i2\pi kx}} itself. In the case of representation of finite group, the character table of the groupG are rows of vectors such that each row is the character of one irreducible representation ofG, and these vectors form an orthonormal basis of the space of class functions that map fromG toC by Schur's lemma. Now the groupT is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the functionek(x){\displaystyle e_{k}(x)} ofx∈T,{\displaystyle x\in T,} and the inner product between two class functions (all functions being class functions sinceT is abelian)f,g∈L2(T,dμ){\displaystyle f,g\in L^{2}(T,d\mu )} is defined as⟨f,g⟩=1|T|∫[0,1)f(y)g¯(y)dμ(y){\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)} with the normalizing factor|T|=1{\displaystyle |T|=1}. The sequence{ek∣k∈Z}{\displaystyle \{e_{k}\mid k\in Z\}} is an orthonormal basis of the space of class functionsL2(T,dμ){\displaystyle L^{2}(T,d\mu )}.

For any representationV of a finite groupG,χv{\displaystyle \chi _{v}} can be expressed as the span∑i⟨χv,χvi⟩χvi{\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}} (Vi{\displaystyle V_{i}} are the irreps ofG), such that⟨χv,χvi⟩=1|G|∑g∈Gχv(g)χ¯vi(g){\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}. Similarly forG=T{\displaystyle G=T} andf∈L2(T,dμ){\displaystyle f\in L^{2}(T,d\mu )},f(x)=∑k∈Zf^(k)ek{\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}. The Pontriagin dualT^{\displaystyle {\hat {T}}} is{ek}(k∈Z){\displaystyle \{e_{k}\}(k\in Z)} and forf∈L2(T,dμ){\displaystyle f\in L^{2}(T,d\mu )},f^(k)=1|T|∫[0,1)f(y)e−i2πkydy{\textstyle {\hat {f}}(k)={\frac {1}{|T|}}\int _{[0,1)}f(y)e^{-i2\pi ky}dy} is its Fourier transform forek∈T^{\displaystyle e_{k}\in {\hat {T}}}.

Gelfand transform

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Main article:Gelfand representation

The Fourier transform is also a special case ofGelfand transform. In this particular context, it is closely related to the Pontryagin duality map defined above.

Given an abelianlocally compactHausdorfftopological groupG, as before we consider spaceL1(G), defined using a Haar measure. With convolution as multiplication,L1(G) is an abelianBanach algebra. It also has aninvolution * given byf∗(g)=f(g−1)¯.{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.}

Taking the completion with respect to the largest possiblyC*-norm gives its envelopingC*-algebra, called the groupC*-algebraC*(G) ofG. (AnyC*-norm onL1(G) is bounded by theL1 norm, therefore their supremum exists.)

Given any abelianC*-algebraA, the Gelfand transform gives an isomorphism betweenA andC0(A^), whereA^ is the multiplicative linear functionals, i.e. one-dimensional representations, onA with the weak-* topology. The map is simply given bya↦(φ↦φ(a)){\displaystyle a\mapsto {\bigl (}\varphi \mapsto \varphi (a){\bigr )}}It turns out that the multiplicative linear functionals ofC*(G), after suitable identification, are exactly the characters ofG, and the Gelfand transform, when restricted to the dense subsetL1(G) is the Fourier–Pontryagin transform.

Compact non-abelian groups

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The Fourier transform can also be defined for functions on a non-abelian group, provided that the group iscompact. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators.[67] The Fourier transform on compact groups is a major tool inrepresentation theory[68] andnon-commutative harmonic analysis.

LetG be a compactHausdorfftopological group. LetΣ denote the collection of all isomorphism classes of finite-dimensional irreducibleunitary representations, along with a definite choice of representationU(σ) on theHilbert spaceHσ of finite dimensiondσ for eachσ ∈ Σ. Ifμ is a finiteBorel measure onG, then the Fourier–Stieltjes transform ofμ is the operator onHσ defined by⟨μ^ξ,η⟩Hσ=∫G⟨U¯g(σ)ξ,η⟩dμ(g){\displaystyle \left\langle {\hat {\mu }}\xi ,\eta \right\rangle _{H_{\sigma }}=\int _{G}\left\langle {\overline {U}}_{g}^{(\sigma )}\xi ,\eta \right\rangle \,d\mu (g)}whereU(σ) is the complex-conjugate representation ofU(σ) acting onHσ. Ifμ isabsolutely continuous with respect to theleft-invariant probability measureλ onG,represented asdμ=fdλ{\displaystyle d\mu =f\,d\lambda }for somef ∈L1(λ), one identifies the Fourier transform off with the Fourier–Stieltjes transform ofμ.

The mappingμ↦μ^{\displaystyle \mu \mapsto {\hat {\mu }}}defines an isomorphism between theBanach spaceM(G) of finite Borel measures (seerca space) and a closed subspace of the Banach spaceC∞(Σ) consisting of all sequencesE = (Eσ) indexed byΣ of (bounded) linear operatorsEσ :Hσ →Hσ for which the norm‖E‖=supσ∈Σ‖Eσ‖{\displaystyle \|E\|=\sup _{\sigma \in \Sigma }\left\|E_{\sigma }\right\|}is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism ofC*-algebras into a subspace ofC∞(Σ). Multiplication onM(G) is given byconvolution of measures and the involution * defined byf∗(g)=f(g−1)¯,{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}},}andC∞(Σ) has a naturalC*-algebra structure as Hilbert space operators.

ThePeter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: iff ∈L2(G), thenf(g)=∑σ∈Σdσtr⁡(f^(σ)Ug(σ)){\displaystyle f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} \left({\hat {f}}(\sigma )U_{g}^{(\sigma )}\right)}where the summation is understood as convergent in theL2 sense.

The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development ofnoncommutative geometry.[citation needed] In this context, a categorical generalization of the Fourier transform to noncommutative groups isTannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.

Alternatives

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Insignal processing terms, a function (of time) is a representation of a signal with perfecttime resolution, but no frequency information, while the Fourier transform has perfectfrequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), andstanding waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notablytransients, or any signal of finite extent.

As alternatives to the Fourier transform, intime–frequency analysis, one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as theshort-time Fourier transform,fractional Fourier transform, Synchrosqueezing Fourier transform,[69] or other functions to represent signals, as inwavelet transforms andchirplet transforms, with the wavelet analog of the (continuous) Fourier transform being thecontinuous wavelet transform.[29]

Example

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The following figures provide a visual illustration of how the Fourier transform's integral measures whether a frequency is present in a particular function. The first image depicts the functionf(t)=cos⁡(2π 3t) e−πt2,{\displaystyle f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}},} which is a 3 Hz cosine wave (the first term) shaped by aGaussianenvelope function (the second term) that smoothly turns the wave on and off. The next 2 images show the productf(t)e−i2π3t,{\displaystyle f(t)e^{-i2\pi 3t},} which must be integrated to calculate the Fourier transform at +3 Hz. The real part of the integrand has a non-negative average value, because the alternating signs off(t){\displaystyle f(t)} andRe⁡(e−i2π3t){\displaystyle \operatorname {Re} (e^{-i2\pi 3t})} oscillate at the same rate and in phase, whereasf(t){\displaystyle f(t)} andIm⁡(e−i2π3t){\displaystyle \operatorname {Im} (e^{-i2\pi 3t})} oscillate at the same rate but with orthogonal phase. The absolute value of the Fourier transform at +3 Hz is 0.5, which is relatively large. When added to the Fourier transform at -3 Hz (which is identical because we started with a real signal), we find that the amplitude of the 3 Hz frequency component is 1.

Original function, which has a strong 3 Hz component. Real and imaginary parts of the integrand of its Fourier transform at +3 Hz.

However, when you try to measure a frequency that is not present, both the real and imaginary component of the integral vary rapidly between positive and negative values. For instance, the red curve is looking for 5 Hz. The absolute value of its integral is nearly zero, indicating that almost no 5 Hz component was in the signal. The general situation is usually more complicated than this, but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a functionf(t).{\displaystyle f(t).}

  • Real and imaginary parts of the integrand for its Fourier transform at +5 Hz.
    Real and imaginary parts of the integrand for its Fourier transform at +5 Hz.
  • Magnitude of its Fourier transform, with +3 and +5 Hz labeled.
    Magnitude of its Fourier transform, with +3 and +5 Hz labeled.

To re-enforce an earlier point, the reason for the response at  ξ=−3{\displaystyle \xi =-3} Hz  is because  cos⁡(2π3t){\displaystyle \cos(2\pi 3t)}  and  cos⁡(2π(−3)t){\displaystyle \cos(2\pi (-3)t)}  are indistinguishable. The transform of  ei2π3t⋅e−πt2{\displaystyle e^{i2\pi 3t}\cdot e^{-\pi t^{2}}}  would have just one response, whose amplitude is the integral of the smooth envelope:e−πt2,{\displaystyle e^{-\pi t^{2}},}  whereas  Re⁡(f(t)⋅e−i2π3t){\displaystyle \operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})} is  e−πt2(1+cos⁡(2π6t))/2.{\displaystyle e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.}

Applications

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See also:Spectral density § Applications
Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation ofdifferentiation in the time domain corresponds to multiplication by the frequency,[note 6] so somedifferential equations are easier to analyze in the frequency domain. Also,convolution in the time domain corresponds to ordinary multiplication in the frequency domain (seeConvolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain.Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.

Analysis of differential equations

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Perhaps the most important use of the Fourier transformation is to solvepartial differential equations.Many of the equations of the mathematical physics of the nineteenth century can be treated this way. Fourier studied the heat equation, which in one dimension and in dimensionless units is∂2y(x,t)∂2x=∂y(x,t)∂t.{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial y(x,t)}{\partial t}}.}The example we will give, a slightly more difficult one, is the wave equation in one dimension,∂2y(x,t)∂2x=∂2y(x,t)∂2t.{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial ^{2}y(x,t)}{\partial ^{2}t}}.}

As usual, the problem is not to find a solution: there are infinitely many. The problem is that of the so-called "boundary problem": find a solution which satisfies the "boundary conditions"y(x,0)=f(x),∂y(x,0)∂t=g(x).{\displaystyle y(x,0)=f(x),\qquad {\frac {\partial y(x,0)}{\partial t}}=g(x).}

Here,f andg are given functions. For the heat equation, only one boundary condition can be required (usually the first one). But for the wave equation, there are still infinitely many solutionsy which satisfy the first boundary condition. But when one imposes both conditions, there is only one possible solution.

It is easier to find the Fourier transformŷ of the solution than to find the solution directly. This is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function. Afterŷ is determined, we can apply the inverse Fourier transformation to findy.

Fourier's method is as follows. First, note that any function of the formscos⁡(2πξ(x±t)) or sin⁡(2πξ(x±t)){\displaystyle \cos {\bigl (}2\pi \xi (x\pm t){\bigr )}{\text{ or }}\sin {\bigl (}2\pi \xi (x\pm t){\bigr )}}satisfies the wave equation. These are called the elementary solutions.

Second, note that therefore any integraly(x,t)=∫0∞dξ[a+(ξ)cos⁡(2πξ(x+t))+a−(ξ)cos⁡(2πξ(x−t))+b+(ξ)sin⁡(2πξ(x+t))+b−(ξ)sin⁡(2πξ(x−t))]{\displaystyle {\begin{aligned}y(x,t)=\int _{0}^{\infty }d\xi {\Bigl [}&a_{+}(\xi )\cos {\bigl (}2\pi \xi (x+t){\bigr )}+a_{-}(\xi )\cos {\bigl (}2\pi \xi (x-t){\bigr )}+{}\\&b_{+}(\xi )\sin {\bigl (}2\pi \xi (x+t){\bigr )}+b_{-}(\xi )\sin \left(2\pi \xi (x-t)\right){\Bigr ]}\end{aligned}}}satisfies the wave equation for arbitrarya+,a−,b+,b−. This integral may be interpreted as a continuous linear combination of solutions for the linear equation.

Now this resembles the formula for the Fourier synthesis of a function. In fact, this is the real inverse Fourier transform ofa± andb± in the variablex.

The third step is to examine how to find the specific unknown coefficient functionsa± andb± that will lead toy satisfying the boundary conditions. We are interested in the values of these solutions att = 0. So we will sett = 0. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variablex) of both sides and obtain2∫−∞∞y(x,0)cos⁡(2πξx)dx=a++a−{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\cos(2\pi \xi x)\,dx=a_{+}+a_{-}}and2∫−∞∞y(x,0)sin⁡(2πξx)dx=b++b−.{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\sin(2\pi \xi x)\,dx=b_{+}+b_{-}.}

Similarly, taking the derivative ofy with respect tot and then applying the Fourier sine and cosine transformations yields2∫−∞∞∂y(u,0)∂tsin⁡(2πξx)dx=(2πξ)(−a++a−){\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\sin(2\pi \xi x)\,dx=(2\pi \xi )\left(-a_{+}+a_{-}\right)}and2∫−∞∞∂y(u,0)∂tcos⁡(2πξx)dx=(2πξ)(b+−b−).{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\cos(2\pi \xi x)\,dx=(2\pi \xi )\left(b_{+}-b_{-}\right).}

These are four linear equations for the four unknownsa± andb±, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.

In summary, we chose a set of elementary solutions, parametrized byξ, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameterξ. But this integral was in the form of a Fourier integral. The next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functionsf andg. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative. The last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functionsa± andb± in terms of the given boundary conditionsf andg.

From a higher point of view, Fourier's procedure can be reformulated more conceptually. Since there are two variables, we will use the Fourier transformation in bothx andt rather than operate as Fourier did, who only transformed in the spatial variables. Note thatŷ must be considered in the sense of a distribution sincey(x,t) is not going to beL1: as a wave, it will persist through time and thus is not a transient phenomenon. But it will be bounded and so its Fourier transform can be defined as a distribution. The operational properties of the Fourier transformation that are relevant to this equation are that it takes differentiation inx to multiplication byi2πξ and differentiation with respect tot to multiplication byi2πf wheref is the frequency. Then the wave equation becomes an algebraic equation inŷ:ξ2y^(ξ,f)=f2y^(ξ,f).{\displaystyle \xi ^{2}{\hat {y}}(\xi ,f)=f^{2}{\hat {y}}(\xi ,f).}This is equivalent to requiringŷ(ξ,f) = 0 unlessξ = ±f. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviouslyf̂ =δ(ξ ±f) will be solutions. Applying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier. But from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conicξ2 −f2 = 0.

We may as well consider the distributions supported on the conic that are given by distributions of one variable on the lineξ =f plus distributions on the lineξ = −f as follows: ifΦ is any test function,∬y^ϕ(ξ,f)dξdf=∫s+ϕ(ξ,ξ)dξ+∫s−ϕ(ξ,−ξ)dξ,{\displaystyle \iint {\hat {y}}\phi (\xi ,f)\,d\xi \,df=\int s_{+}\phi (\xi ,\xi )\,d\xi +\int s_{-}\phi (\xi ,-\xi )\,d\xi ,}wheres+, ands−, are distributions of one variable.

Then Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (putΦ(ξ,f) =ei2π(xξ+tf), which is clearly of polynomial growth):y(x,0)=∫{s+(ξ)+s−(ξ)}ei2πξx+0dξ{\displaystyle y(x,0)=\int {\bigl \{}s_{+}(\xi )+s_{-}(\xi ){\bigr \}}e^{i2\pi \xi x+0}\,d\xi }and∂y(x,0)∂t=∫{s+(ξ)−s−(ξ)}i2πξei2πξx+0dξ.{\displaystyle {\frac {\partial y(x,0)}{\partial t}}=\int {\bigl \{}s_{+}(\xi )-s_{-}(\xi ){\bigr \}}i2\pi \xi e^{i2\pi \xi x+0}\,d\xi .}

Now, as before, applying the one-variable Fourier transformation in the variablex to these functions ofx yields two equations in the two unknown distributionss± (which can be taken to be ordinary functions if the boundary conditions areL1 orL2).

From a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform. Closed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate. For practical calculations, other methods are often used.

The twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well.

Fourier-transform spectroscopy

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Main article:Fourier-transform spectroscopy

The Fourier transform is also used innuclear magnetic resonance (NMR) and in other kinds ofspectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used inmagnetic resonance imaging (MRI) andmass spectrometry.

Quantum mechanics

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The Fourier transform is useful inquantum mechanics in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs ofcomplementary variables, connected by theHeisenberg uncertainty principle. For example, in one dimension, the spatial variableq of, say, a particle, can only be measured by the quantum mechanical "position operator" at the cost of losing information about the momentump of the particle. Therefore, the physical state of the particle can either be described by a function, called "the wave function", ofq or by a function ofp but not by a function of both variables. The variablep is called the conjugate variable toq. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to bothp andq simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with ap-axis and aq-axis called thephase space.

In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, theq-axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis. Nevertheless, choosing thep-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle. Both representations of the wavefunction are related by a Fourier transform, such thatϕ(p)=∫dqψ(q)e−ipq/h,{\displaystyle \phi (p)=\int dq\,\psi (q)e^{-ipq/h},}or, equivalently,ψ(q)=∫dpϕ(p)eipq/h.{\displaystyle \psi (q)=\int dp\,\phi (p)e^{ipq/h}.}

Physically realisable states areL2, and so by thePlancherel theorem, their Fourier transforms are alsoL2. (Note that sinceq is in units of distance andp is in units of momentum, the presence of the Planck constant in the exponent makes the exponentdimensionless, as it should be.)

Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum. Infinitely many different polarisations are possible, and all are equally valid. Being able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberguncertainty principle.

The other use of the Fourier transform in both quantum mechanics andquantum field theory is to solve the applicable wave equation. In non-relativistic quantum mechanics, theSchrödinger equation for a time-varying wave function in one-dimension, not subject to external forces, is−∂2∂x2ψ(x,t)=ih2π∂∂tψ(x,t).{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}

This is the same as the heat equation except for the presence of the imaginary uniti. Fourier methods can be used to solve this equation.

In the presence of a potential, given by the potential energy functionV(x), the equation becomes−∂2∂x2ψ(x,t)+V(x)ψ(x,t)=ih2π∂∂tψ(x,t).{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}

The "elementary solutions", as we referred to them above, are the so-called "stationary states" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution ofψ given its values fort = 0. Neither of these approaches is of much practical use in quantum mechanics. Boundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.

In relativistic quantum mechanics, the Schrödinger equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered. A simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,(∂2∂x2+1)ψ(x,t)=∂2∂t2ψ(x,t).{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+1\right)\psi (x,t)={\frac {\partial ^{2}}{\partial t^{2}}}\psi (x,t).}

This is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods). This is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as "second quantization". Fourier methods have been adapted to also deal with non-trivial interactions.

Finally, thenumber operator of thequantum harmonic oscillator can be interpreted, for example via theMehler kernel, as thegenerator of theFourier transformF{\displaystyle {\mathcal {F}}}.[32]

Signal processing

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The Fourier transform is used for the spectral analysis of time-series. The subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself. Even if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time. The Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.

The autocorrelation functionR of a functionf is defined byRf(τ)=limT→∞12T∫−TTf(t)f(t+τ)dt.{\displaystyle R_{f}(\tau )=\lim _{T\rightarrow \infty }{\frac {1}{2T}}\int _{-T}^{T}f(t)f(t+\tau )\,dt.}

This function is a function of the time-lagτ elapsing between the values off to be correlated.

For most functionsf that occur in practice,R is a bounded even function of the time-lagτ and for typical noisy signals it turns out to be uniformly continuous with a maximum atτ = 0.

The autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values off separated by a time lag. This is a way of searching for the correlation off with its own past. It is useful even for other statistical tasks besides the analysis of signals. For example, iff(t) represents the temperature at timet, one expects a strong correlation with the temperature at a time lag of 24 hours.

It possesses a Fourier transform,Pf(ξ)=∫−∞∞Rf(τ)e−i2πξτdτ.{\displaystyle P_{f}(\xi )=\int _{-\infty }^{\infty }R_{f}(\tau )e^{-i2\pi \xi \tau }\,d\tau .}

This Fourier transform is called thepower spectral density function off. (Unless all periodic components are first filtered out fromf, this integral will diverge, but it is easy to filter out such periodicities.)

The power spectrum, as indicated by this density functionP, measures the amount of variance contributed to the data by the frequencyξ. In electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal. This process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series (ANOVA).

Knowledge of which frequencies are "important" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses. It can also be useful for the scientific analysis of the phenomena responsible for producing the data.

The power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.

Spectral analysis is carried out for visual signals as well. The power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool.

Other notations

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Other common notations forf^(ξ){\displaystyle {\hat {f}}(\xi )} include:f~(ξ), F(ξ), F(f)(ξ), (Ff)(ξ), F(f), F{f}, F(f(t)), F{f(t)}.{\displaystyle {\tilde {f}}(\xi ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}{\bigl (}f(t){\bigr )},\ {\mathcal {F}}{\bigl \{}f(t){\bigr \}}.}

In the sciences and engineering it is also common to make substitutions like these:ξ→f,x→t,f→x,f^→X.{\displaystyle \xi \rightarrow f,\quad x\rightarrow t,\quad f\rightarrow x,\quad {\hat {f}}\rightarrow X.}

So the transform pairf(x) ⟺F f^(ξ){\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )} can becomex(t) ⟺F X(f){\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}

A disadvantage of the capital letter notation is when expressing a transform such asf⋅g^{\displaystyle {\widehat {f\cdot g}}} orf′^,{\displaystyle {\widehat {f'}},} which become the more awkwardF{f⋅g}{\displaystyle {\mathcal {F}}\{f\cdot g\}} andF{f′}.{\displaystyle {\mathcal {F}}\{f'\}.}

In some contexts such as particle physics, the same symbolf{\displaystyle f} may be used for both for a function as well as it Fourier transform, with the two only distinguished by theirargument I.e.f(k1+k2){\displaystyle f(k_{1}+k_{2})} would refer to the Fourier transform because of the momentum argument, whilef(x0+πr→){\displaystyle f(x_{0}+\pi {\vec {r}})} would refer to the original function because of the positional argument. Although tildes may be used as inf~{\displaystyle {\tilde {f}}} to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a moreLorentz invariant form, such asdk~=dk(2π)32ω{\displaystyle {\tilde {dk}}={\frac {dk}{(2\pi )^{3}2\omega }}}, so care must be taken. Similarly,f^{\displaystyle {\hat {f}}} often denotes theHilbert transform off{\displaystyle f}.

The interpretation of the complex functionf̂(ξ) may be aided by expressing it inpolar coordinate formf^(ξ)=A(ξ)eiφ(ξ){\displaystyle {\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}}in terms of the two real functionsA(ξ) andφ(ξ) where:A(ξ)=|f^(ξ)|,{\displaystyle A(\xi )=\left|{\hat {f}}(\xi )\right|,}is theamplitude andφ(ξ)=arg⁡(f^(ξ)),{\displaystyle \varphi (\xi )=\arg \left({\hat {f}}(\xi )\right),}is thephase (seearg function).

Then the inverse transform can be written:f(x)=∫−∞∞A(ξ) ei(2πξx+φ(ξ))dξ,{\displaystyle f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i{\bigl (}2\pi \xi x+\varphi (\xi ){\bigr )}}\,d\xi ,}which is a recombination of all the frequency components off(x). Each component is a complexsinusoid of the forme2πixξ whose amplitude isA(ξ) and whose initialphase angle (atx = 0) isφ(ξ).

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denotedF andF(f) is used to denote the Fourier transform of the functionf. This mapping is linear, which means thatF can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the functionf) can be used to writeFf instead ofF(f). Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the valueξ for its variable, and this is denoted either asFf(ξ) or as(Ff)(ξ). Notice that in the former case, it is implicitly understood thatF is applied first tof and then the resulting function is evaluated atξ, not the other way around.

In mathematics and various applied sciences, it is often necessary to distinguish between a functionf and the value off when its variable equalsx, denotedf(x). This means that a notation likeF(f(x)) formally can be interpreted as the Fourier transform of the values off atx. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example,F(rect⁡(x))=sinc⁡(ξ){\displaystyle {\mathcal {F}}{\bigl (}\operatorname {rect} (x){\bigr )}=\operatorname {sinc} (\xi )}is sometimes used to express that the Fourier transform of arectangular function is asinc function, orF(f(x+x0))=F(f(x))ei2πx0ξ{\displaystyle {\mathcal {F}}{\bigl (}f(x+x_{0}){\bigr )}={\mathcal {F}}{\bigl (}f(x){\bigr )}\,e^{i2\pi x_{0}\xi }}is used to express the shift property of the Fourier transform.

Notice, that the last example is only correct under the assumption that the transformed function is a function ofx, not ofx0.

As discussed above, thecharacteristic function of a random variable is the same as theFourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is definedE(eit⋅X)=∫eit⋅xdμX(x).{\displaystyle E\left(e^{it\cdot X}\right)=\int e^{it\cdot x}\,d\mu _{X}(x).}

As in the case of the "non-unitary angular frequency" convention above, the factor of 2π appears in neither the normalizing constant nor the exponent. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.

Computation methods

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The appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function. In this section we consider both functions of a continuous variable,f(x),{\displaystyle f(x),} and functions of a discrete variable (i.e. ordered pairs ofx{\displaystyle x} andf{\displaystyle f} values). For discrete-valuedx,{\displaystyle x,} the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (ξ{\displaystyle \xi } orω{\displaystyle \omega }). When the sinusoids are harmonically related (i.e. when thex{\displaystyle x}-values are spaced at integer multiples of an interval), the transform is calleddiscrete-time Fourier transform (DTFT).

Discrete Fourier transforms and fast Fourier transforms

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Sampling the DTFT at equally-spaced values of frequency is the most common modern method of computation. Efficient procedures, depending on the frequency resolution needed, are described atDiscrete-time Fourier transform § Sampling the DTFT. Thediscrete Fourier transform (DFT), used there, is usually computed by afast Fourier transform (FFT) algorithm.

Analytic integration of closed-form functions

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Tables ofclosed-form Fourier transforms, such as§ Square-integrable functions, one-dimensional and§ Table of discrete-time Fourier transforms, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (ξ{\displaystyle \xi } orω{\displaystyle \omega }).[70] When mathematically possible, this provides a transform for a continuum of frequency values.

Many computer algebra systems such asMatlab andMathematica that are capable ofsymbolic integration are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform ofcos(6πt)e−πt2 one might enter the commandintegrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf intoWolfram Alpha.[note 7]

Numerical integration of closed-form continuous functions

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Discrete sampling of the Fourier transform can also be done bynumerical integration of the definition at each value of frequency for which transform is desired.[71][72][73] The numerical integration approach works on a much broader class of functions than the analytic approach.

Numerical integration of a series of ordered pairs

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If the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.[74] The DTFT is a common subcase of this more general situation.

Tables of important Fourier transforms

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The following tables record some closed-form Fourier transforms. For functionsf(x) andg(x) denote their Fourier transforms byf̂ andĝ. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.

Functional relationships, one-dimensional

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The Fourier transforms in this table may be found inErdélyi (1954) orKammler (2000, appendix).

FunctionFourier transform
unitary, ordinary frequency		Fourier transform
unitary, angular frequency		Fourier transform
non-unitary, angular frequency		Remarks
f(x){\displaystyle f(x)\,}f^(ξ)≜f1^(ξ)=∫−∞∞f(x)e−i2πξxdx{\displaystyle {\begin{aligned}&{\widehat {f}}(\xi )\triangleq {\widehat {f_{1}}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}f^(ω)≜f2^(ω)=12π∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{2}}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}f^(ω)≜f3^(ω)=∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{3}}}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}Definitions
101af(x)+bg(x){\displaystyle a\,f(x)+b\,g(x)\,}af^(ξ)+bg^(ξ){\displaystyle a\,{\widehat {f}}(\xi )+b\,{\widehat {g}}(\xi )\,}af^(ω)+bg^(ω){\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}af^(ω)+bg^(ω){\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}Linearity
102f(x−a){\displaystyle f(x-a)\,}e−i2πξaf^(ξ){\displaystyle e^{-i2\pi \xi a}{\widehat {f}}(\xi )\,}e−iaωf^(ω){\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}e−iaωf^(ω){\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}Shift in time domain
103f(x)eiax{\displaystyle f(x)e^{iax}\,}f^(ξ−a2π){\displaystyle {\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}f^(ω−a){\displaystyle {\widehat {f}}(\omega -a)\,}f^(ω−a){\displaystyle {\widehat {f}}(\omega -a)\,}Shift in frequency domain, dual of 102
104f(ax){\displaystyle f(ax)\,}1|a|f^(ξa){\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right)\,}1|a|f^(ωa){\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}1|a|f^(ωa){\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}Scaling in the time domain. If|a| is large, thenf(ax) is concentrated around 0 and
1|a|f^(ωa){\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}
spreads out and flattens.
105fn^(x){\displaystyle {\widehat {f_{n}}}(x)\,}f1^(x) ⟷F1 f(−ξ){\displaystyle {\widehat {f_{1}}}(x)\ {\stackrel {{\mathcal {F}}_{1}}{\longleftrightarrow }}\ f(-\xi )\,}f2^(x) ⟷F2 f(−ω){\displaystyle {\widehat {f_{2}}}(x)\ {\stackrel {{\mathcal {F}}_{2}}{\longleftrightarrow }}\ f(-\omega )\,}f3^(x) ⟷F3 2πf(−ω){\displaystyle {\widehat {f_{3}}}(x)\ {\stackrel {{\mathcal {F}}_{3}}{\longleftrightarrow }}\ 2\pi f(-\omega )\,}The same transform is applied twice, butx replaces the frequency variable (ξ orω) after the first transform.
106dnf(x)dxn{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}(i2πξ)nf^(ξ){\displaystyle (i2\pi \xi )^{n}{\widehat {f}}(\xi )\,}(iω)nf^(ω){\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}(iω)nf^(ω){\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}nth-order derivative.

Asf is aSchwartz function

106.5∫−∞xf(τ)dτ{\displaystyle \int _{-\infty }^{x}f(\tau )d\tau }f^(ξ)i2πξ+Cδ(ξ){\displaystyle {\frac {{\widehat {f}}(\xi )}{i2\pi \xi }}+C\,\delta (\xi )}f^(ω)iω+2πCδ(ω){\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+{\sqrt {2\pi }}C\delta (\omega )}f^(ω)iω+2πCδ(ω){\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+2\pi C\delta (\omega )}Integration.[75] Note:δ{\displaystyle \delta } is theDirac delta function andC{\displaystyle C} is the average (DC) value off(x){\displaystyle f(x)} such that∫−∞∞(f(x)−C)dx=0{\displaystyle \int _{-\infty }^{\infty }(f(x)-C)\,dx=0}
107xnf(x){\displaystyle x^{n}f(x)\,}(i2π)ndnf^(ξ)dξn{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\widehat {f}}(\xi )}{d\xi ^{n}}}\,}indnf^(ω)dωn{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}indnf^(ω)dωn{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}This is the dual of 106
108(f∗g)(x){\displaystyle (f*g)(x)\,}f^(ξ)g^(ξ){\displaystyle {\widehat {f}}(\xi ){\widehat {g}}(\xi )\,}2π f^(ω)g^(ω){\displaystyle {\sqrt {2\pi }}\ {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}f^(ω)g^(ω){\displaystyle {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}The notationf ∗g denotes theconvolution off andg — this rule is theconvolution theorem
109f(x)g(x){\displaystyle f(x)g(x)\,}(f^∗g^)(ξ){\displaystyle \left({\widehat {f}}*{\widehat {g}}\right)(\xi )\,}12π(f^∗g^)(ω){\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}12π(f^∗g^)(ω){\displaystyle {\frac {1}{2\pi }}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}This is the dual of 108
110Forf(x) purely realf^(−ξ)=f^(ξ)¯{\displaystyle {\widehat {f}}(-\xi )={\overline {{\widehat {f}}(\xi )}}\,}f^(−ω)=f^(ω)¯{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}f^(−ω)=f^(ω)¯{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}Hermitian symmetry.z indicates thecomplex conjugate.
113Forf(x) purely imaginaryf^(−ξ)=−f^(ξ)¯{\displaystyle {\widehat {f}}(-\xi )=-{\overline {{\widehat {f}}(\xi )}}\,}f^(−ω)=−f^(ω)¯{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}f^(−ω)=−f^(ω)¯{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}z indicates thecomplex conjugate.
114f(x)¯{\displaystyle {\overline {f(x)}}}f^(−ξ)¯{\displaystyle {\overline {{\widehat {f}}(-\xi )}}}f^(−ω)¯{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}f^(−ω)¯{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}Complex conjugation, generalization of 110 and 113
115f(x)cos⁡(ax){\displaystyle f(x)\cos(ax)}f^(ξ−a2π)+f^(ξ+a2π)2{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}f^(ω−a)+f^(ω+a)2{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}\,}f^(ω−a)+f^(ω+a)2{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}}This follows from rules 101 and 103 usingEuler's formula:
cos⁡(ax)=eiax+e−iax2.{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}
116f(x)sin⁡(ax){\displaystyle f(x)\sin(ax)}f^(ξ−a2π)−f^(ξ+a2π)2i{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}f^(ω−a)−f^(ω+a)2i{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}f^(ω−a)−f^(ω+a)2i{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}This follows from 101 and 103 usingEuler's formula:
sin⁡(ax)=eiax−e−iax2i.{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}

Square-integrable functions, one-dimensional

[edit]

The Fourier transforms in this table may be found inCampbell & Foster (1948),Erdélyi (1954), orKammler (2000, appendix).

FunctionFourier transform
unitary, ordinary frequency		Fourier transform
unitary, angular frequency		Fourier transform
non-unitary, angular frequency		Remarks
f(x){\displaystyle f(x)\,}f^(ξ)≜f^1(ξ)=∫−∞∞f(x)e−i2πξxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}f^(ω)≜f^2(ω)=12π∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}f^(ω)≜f^3(ω)=∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}Definitions
201rect⁡(ax){\displaystyle \operatorname {rect} (ax)\,}1|a|sinc⁡(ξa){\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\xi }{a}}\right)}12πa2sinc⁡(ω2πa){\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}1|a|sinc⁡(ω2πa){\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}Therectangular pulse and thenormalizedsinc function, here defined assinc(x) =⁠sin(πx)/πx⁠
202sinc⁡(ax){\displaystyle \operatorname {sinc} (ax)\,}1|a|rect⁡(ξa){\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\xi }{a}}\right)\,}12πa2rect⁡(ω2πa){\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}1|a|rect⁡(ω2πa){\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}Dual of rule 201. Therectangular function is an ideallow-pass filter, and thesinc function is thenon-causal impulse response of such a filter. Thesinc function is defined here assinc(x) =⁠sin(πx)/πx⁠
203sinc2⁡(ax){\displaystyle \operatorname {sinc} ^{2}(ax)}1|a|tri⁡(ξa){\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\xi }{a}}\right)}12πa2tri⁡(ω2πa){\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}1|a|tri⁡(ω2πa){\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}The functiontri(x) is thetriangular function
204tri⁡(ax){\displaystyle \operatorname {tri} (ax)}1|a|sinc2⁡(ξa){\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}12πa2sinc2⁡(ω2πa){\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}1|a|sinc2⁡(ω2πa){\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}Dual of rule 203.
205e−axu(x){\displaystyle e^{-ax}u(x)\,}1a+i2πξ{\displaystyle {\frac {1}{a+i2\pi \xi }}}12π(a+iω){\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}1a+iω{\displaystyle {\frac {1}{a+i\omega }}}The functionu(x) is theHeaviside unit step function anda > 0.
206e−αx2{\displaystyle e^{-\alpha x^{2}}\,}παe−(πξ)2α{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}12αe−ω24α{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}παe−ω24α{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}This shows that, for the unitary Fourier transforms, theGaussian functione−αx2 is its own Fourier transform for some choice ofα. For this to be integrable we must haveRe(α) > 0.
208e−a|x|{\displaystyle e^{-a|x|}\,}2aa2+4π2ξ2{\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}2πaa2+ω2{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {a}{a^{2}+\omega ^{2}}}}2aa2+ω2{\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}ForRe(a) > 0. That is, the Fourier transform of atwo-sided decaying exponential function is aLorentzian function.
209sech⁡(ax){\displaystyle \operatorname {sech} (ax)\,}πasech⁡(π2aξ){\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}1aπ2sech⁡(π2aω){\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}πasech⁡(π2aω){\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}Hyperbolic secant is its own Fourier transform
210e−a2x22Hn(ax){\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}2π(−i)nae−2π2ξ2a2Hn(2πξa){\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)}(−i)nae−ω22a2Hn(ωa){\displaystyle {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}(−i)n2πae−ω22a2Hn(ωa){\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}Hn is thenth-orderHermite polynomial. Ifa = 1 then the Gauss–Hermite functions areeigenfunctions of the Fourier transform operator. For a derivation, seeHermite polynomial. The formula reduces to 206 forn = 0.

Distributions, one-dimensional

[edit]

The Fourier transforms in this table may be found inErdélyi (1954) orKammler (2000, appendix).

FunctionFourier transform
unitary, ordinary frequency		Fourier transform
unitary, angular frequency		Fourier transform
non-unitary, angular frequency		Remarks
f(x){\displaystyle f(x)\,}f^(ξ)≜f^1(ξ)=∫−∞∞f(x)e−i2πξxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}f^(ω)≜f^2(ω)=12π∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}f^(ω)≜f^3(ω)=∫−∞∞f(x)e−iωxdx{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}Definitions
3011{\displaystyle 1}δ(ξ){\displaystyle \delta (\xi )}2πδ(ω){\displaystyle {\sqrt {2\pi }}\,\delta (\omega )}2πδ(ω){\displaystyle 2\pi \delta (\omega )}The distributionδ(ξ) denotes theDirac delta function.
302δ(x){\displaystyle \delta (x)\,}1{\displaystyle 1}12π{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}1{\displaystyle 1}Dual of rule 301.
303eiax{\displaystyle e^{iax}}δ(ξ−a2π){\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)}2πδ(ω−a){\displaystyle {\sqrt {2\pi }}\,\delta (\omega -a)}2πδ(ω−a){\displaystyle 2\pi \delta (\omega -a)}This follows from 103 and 301.
304cos⁡(ax){\displaystyle \cos(ax)}δ(ξ−a2π)+δ(ξ+a2π)2{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}}2πδ(ω−a)+δ(ω+a)2{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}π(δ(ω−a)+δ(ω+a)){\displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)}This follows from rules 101 and 303 usingEuler's formula:
cos⁡(ax)=eiax+e−iax2.{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}
305sin⁡(ax){\displaystyle \sin(ax)}δ(ξ−a2π)−δ(ξ+a2π)2i{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}2πδ(ω−a)−δ(ω+a)2i{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}−iπ(δ(ω−a)−δ(ω+a)){\displaystyle -i\pi {\bigl (}\delta (\omega -a)-\delta (\omega +a){\bigr )}}This follows from 101 and 303 using
sin⁡(ax)=eiax−e−iax2i.{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}
306cos⁡(ax2){\displaystyle \cos \left(ax^{2}\right)}πacos⁡(π2ξ2a−π4){\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}12acos⁡(ω24a−π4){\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}πacos⁡(ω24a−π4){\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}This follows from 101 and 207 using
cos⁡(ax2)=eiax2+e−iax22.{\displaystyle \cos(ax^{2})={\frac {e^{iax^{2}}+e^{-iax^{2}}}{2}}.}
307sin⁡(ax2){\displaystyle \sin \left(ax^{2}\right)}−πasin⁡(π2ξ2a−π4){\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}−12asin⁡(ω24a−π4){\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}−πasin⁡(ω24a−π4){\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}This follows from 101 and 207 using
sin⁡(ax2)=eiax2−e−iax22i.{\displaystyle \sin(ax^{2})={\frac {e^{iax^{2}}-e^{-iax^{2}}}{2i}}.}
308e−πiαx2{\displaystyle e^{-\pi i\alpha x^{2}}\,}1αe−iπ4eiπξ2α{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\pi \xi ^{2}}{\alpha }}}}12παe−iπ4eiω24πα{\displaystyle {\frac {1}{\sqrt {2\pi \alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}1αe−iπ4eiω24πα{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}Here it is assumedα{\displaystyle \alpha } is real. For the case that alpha is complex see table entry 206 above.
309xn{\displaystyle x^{n}\,}(i2π)nδ(n)(ξ){\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )}in2πδ(n)(ω){\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}2πinδ(n)(ω){\displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )}Here,n is anatural number andδ(n)(ξ) is thenth distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform allpolynomials.
310δ(n)(x){\displaystyle \delta ^{(n)}(x)}(i2πξ)n{\displaystyle (i2\pi \xi )^{n}}(iω)n2π{\displaystyle {\frac {(i\omega )^{n}}{\sqrt {2\pi }}}}(iω)n{\displaystyle (i\omega )^{n}}Dual of rule 309.δ(n)(ξ) is thenth distribution derivative of the Dirac delta function. This rule follows from 106 and 302.
3111x{\displaystyle {\frac {1}{x}}}−iπsgn⁡(ξ){\displaystyle -i\pi \operatorname {sgn}(\xi )}−iπ2sgn⁡(ω){\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}−iπsgn⁡(ω){\displaystyle -i\pi \operatorname {sgn}(\omega )}Heresgn(ξ) is thesign function. Note that⁠1/x⁠ is not a distribution. It is necessary to use theCauchy principal value when testing againstSchwartz functions. This rule is useful in studying theHilbert transform.
3121xn:=(−1)n−1(n−1)!dndxnlog⁡|x|{\displaystyle {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}}}−iπ(−i2πξ)n−1(n−1)!sgn⁡(ξ){\displaystyle -i\pi {\frac {(-i2\pi \xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}−iπ2(−iω)n−1(n−1)!sgn⁡(ω){\displaystyle -i{\sqrt {\frac {\pi }{2}}}\,{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}−iπ(−iω)n−1(n−1)!sgn⁡(ω){\displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}⁠1/xn⁠ is thehomogeneous distribution defined by the distributional derivative
(−1)n−1(n−1)!dndxnlog⁡|x|{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|}
313|x|α{\displaystyle |x|^{\alpha }}−2sin⁡(πα2)Γ(α+1)|2πξ|α+1{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}}−22πsin⁡(πα2)Γ(α+1)|ω|α+1{\displaystyle {\frac {-2}{\sqrt {2\pi }}}\,{\frac {\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}−2sin⁡(πα2)Γ(α+1)|ω|α+1{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}This formula is valid for0 >α > −1. Forα > 0 some singular terms arise at the origin that can be found by differentiating 320. IfReα > −1, then|x|α is a locally integrable function, and so a tempered distribution. The functionα ↦ |x|α is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted|x|α forα ≠ −1, −3, ... (Seehomogeneous distribution.)
1|x|{\displaystyle {\frac {1}{\sqrt {|x|}}}}1|ξ|{\displaystyle {\frac {1}{\sqrt {|\xi |}}}}1|ω|{\displaystyle {\frac {1}{\sqrt {|\omega |}}}}2π|ω|{\displaystyle {\frac {\sqrt {2\pi }}{\sqrt {|\omega |}}}}Special case of 313.
314sgn⁡(x){\displaystyle \operatorname {sgn}(x)}1iπξ{\displaystyle {\frac {1}{i\pi \xi }}}2π1iω{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }}}2iω{\displaystyle {\frac {2}{i\omega }}}The dual of rule 311. This time the Fourier transforms need to be considered as aCauchy principal value.
315u(x){\displaystyle u(x)}12(1iπξ+δ(ξ)){\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)}π2(1iπω+δ(ω)){\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}π(1iπω+δ(ω)){\displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}The functionu(x) is the Heavisideunit step function; this follows from rules 101, 301, and 314.
316∑n=−∞∞δ(x−nT){\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)}1T∑k=−∞∞δ(ξ−kT){\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)}2πT∑k=−∞∞δ(ω−2πkT){\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}2πT∑k=−∞∞δ(ω−2πkT){\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}This function is known as theDirac comb function. This result can be derived from 302 and 102, together with the fact that
∑n=−∞∞einx=2π∑k=−∞∞δ(x+2πk){\displaystyle {\begin{aligned}&\sum _{n=-\infty }^{\infty }e^{inx}\\={}&2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)\end{aligned}}}
as distributions.
317J0(x){\displaystyle J_{0}(x)}2rect⁡(πξ)1−4π2ξ2{\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}2πrect⁡(ω2)1−ω2{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}2rect⁡(ω2)1−ω2{\displaystyle {\frac {2\,\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}The functionJ0(x) is the zeroth orderBessel function of first kind.
318Jn(x){\displaystyle J_{n}(x)}2(−i)nTn(2πξ)rect⁡(πξ)1−4π2ξ2{\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}2π(−i)nTn(ω)rect⁡(ω2)1−ω2{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}2(−i)nTn(ω)rect⁡(ω2)1−ω2{\displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}This is a generalization of 317. The functionJn(x) is thenth orderBessel function of first kind. The functionTn(x) is theChebyshev polynomial of the first kind.
319log⁡|x|{\displaystyle \log \left|x\right|}−121|ξ|−γδ(ξ){\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)}−π2|ω|−2πγδ(ω){\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right)}−π|ω|−2πγδ(ω){\displaystyle -{\frac {\pi }{\left|\omega \right|}}-2\pi \gamma \delta \left(\omega \right)}γ is theEuler–Mascheroni constant. It is necessary to use a finite part integral when testing⁠1/|ξ|⁠ or⁠1/|ω|⁠againstSchwartz functions. The details of this might change the coefficient of the delta function.
320(∓ix)−α{\displaystyle \left(\mp ix\right)^{-\alpha }}(2π)αΓ(α)u(±ξ)(±ξ)α−1{\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}}2πΓ(α)u(±ω)(±ω)α−1{\displaystyle {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}2πΓ(α)u(±ω)(±ω)α−1{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}This formula is valid for1 >α > 0. Use differentiation to derive formula for higher exponents.u is the Heaviside function.

Two-dimensional functions

[edit]
FunctionFourier transform
unitary, ordinary frequency		Fourier transform
unitary, angular frequency		Fourier transform
non-unitary, angular frequency		Remarks
400f(x,y){\displaystyle f(x,y)}f^(ξx,ξy)≜∬f(x,y)e−i2π(ξxx+ξyy)dxdy{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\triangleq \\&\iint f(x,y)e^{-i2\pi (\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}}f^(ωx,ωy)≜12π∬f(x,y)e−i(ωxx+ωyy)dxdy{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&{\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}f^(ωx,ωy)≜∬f(x,y)e−i(ωxx+ωyy)dxdy{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}The variablesξx,ξy,ωx,ωy are real numbers. The integrals are taken over the entire plane.
401e−π(a2x2+b2y2){\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}1|ab|e−π(ξx2a2+ξy2b2){\displaystyle {\frac {1}{|ab|}}e^{-\pi \left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}}12π|ab|e−14π(ωx2a2+ωy2b2){\displaystyle {\frac {1}{2\pi \,|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}1|ab|e−14π(ωx2a2+ωy2b2){\displaystyle {\frac {1}{|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}Both functions are Gaussians, which may not have unit volume.
402circ⁡(x2+y2){\displaystyle \operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)}J1(2πξx2+ξy2)ξx2+ξy2{\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}J1(ωx2+ωy2)ωx2+ωy2{\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}2πJ1(ωx2+ωy2)ωx2+ωy2{\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}The function is defined bycirc(r) = 1 for0 ≤r ≤ 1, and is 0 otherwise. The result is the amplitude distribution of theAiry disk, and is expressed usingJ1 (the order-1Bessel function of the first kind).[76]
4031x2+y2{\displaystyle {\frac {1}{\sqrt {x^{2}+y^{2}}}}}1ξx2+ξy2{\displaystyle {\frac {1}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}1ωx2+ωy2{\displaystyle {\frac {1}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}2πωx2+ωy2{\displaystyle {\frac {2\pi }{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}This is theHankel transform ofr−1, a 2-D Fourier "self-transform".[77]
404ix+iy{\displaystyle {\frac {i}{x+iy}}}1ξx+iξy{\displaystyle {\frac {1}{\xi _{x}+i\xi _{y}}}}1ωx+iωy{\displaystyle {\frac {1}{\omega _{x}+i\omega _{y}}}}2πωx+iωy{\displaystyle {\frac {2\pi }{\omega _{x}+i\omega _{y}}}}

Formulas for generaln-dimensional functions

[edit]
FunctionFourier transform
unitary, ordinary frequency		Fourier transform
unitary, angular frequency		Fourier transform
non-unitary, angular frequency		Remarks
500f(x){\displaystyle f(\mathbf {x} )\,}f1^(ξ)≜∫Rnf(x)e−i2πξ⋅xdx{\displaystyle {\begin{aligned}&{\hat {f_{1}}}({\boldsymbol {\xi }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}f2^(ω)≜1(2π)n2∫Rnf(x)e−iω⋅xdx{\displaystyle {\begin{aligned}&{\hat {f_{2}}}({\boldsymbol {\omega }})\triangleq \\&{\frac {1}{{(2\pi )}^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}f3^(ω)≜∫Rnf(x)e−iω⋅xdx{\displaystyle {\begin{aligned}&{\hat {f_{3}}}({\boldsymbol {\omega }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}
501χ[0,1](|x|)(1−|x|2)δ{\displaystyle \chi _{[0,1]}(|\mathbf {x} |)\left(1-|\mathbf {x} |^{2}\right)^{\delta }}Γ(δ+1)πδ|ξ|n2+δJn2+δ(2π|ξ|){\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }\,|{\boldsymbol {\xi }}|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(2\pi |{\boldsymbol {\xi }}|)}2δΓ(δ+1)|ω|n2+δJn2+δ(|ω|){\displaystyle 2^{\delta }\,{\frac {\Gamma (\delta +1)}{\left|{\boldsymbol {\omega }}\right|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(|{\boldsymbol {\omega }}|)}Γ(δ+1)πδ|ω2π|−n2−δJn2+δ(|ω|){\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }}}\left|{\frac {\boldsymbol {\omega }}{2\pi }}\right|^{-{\frac {n}{2}}-\delta }J_{{\frac {n}{2}}+\delta }(\!|{\boldsymbol {\omega }}|\!)}The functionχ[0, 1] is theindicator function of the interval[0, 1]. The functionΓ(x) is the gamma function. The functionJ⁠n/2⁠ +δ is a Bessel function of the first kind, with order⁠n/2⁠ +δ. Takingn = 2 andδ = 0 produces 402.[78]
502|x|−α,0<Re⁡α<n.{\displaystyle |\mathbf {x} |^{-\alpha },\quad 0<\operatorname {Re} \alpha <n.}(2π)αcn,α|ξ|−(n−α){\displaystyle {\frac {(2\pi )^{\alpha }}{c_{n,\alpha }}}|{\boldsymbol {\xi }}|^{-(n-\alpha )}}(2π)n2cn,α|ω|−(n−α){\displaystyle {\frac {(2\pi )^{\frac {n}{2}}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}(2π)ncn,α|ω|−(n−α){\displaystyle {\frac {(2\pi )^{n}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}SeeRiesz potential where the constant is given by
cn,α=πn22αΓ(α2)Γ(n−α2).{\displaystyle c_{n,\alpha }=\pi ^{\frac {n}{2}}2^{\alpha }{\frac {\Gamma \left({\frac {\alpha }{2}}\right)}{\Gamma \left({\frac {n-\alpha }{2}}\right)}}.}
The formula also holds for allα ≠n,n + 2, ... by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. Seehomogeneous distribution.[note 8]
5031|σ|(2π)n2e−12xTσ−Tσ−1x{\displaystyle {\frac {1}{\left|{\boldsymbol {\sigma }}\right|\left(2\pi \right)^{\frac {n}{2}}}}e^{-{\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }{\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}\mathbf {x} }}e−2π2ξTσσTξ{\displaystyle e^{-2\pi ^{2}{\boldsymbol {\xi }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\xi }}}}(2π)−n2e−12ωTσσTω{\displaystyle (2\pi )^{-{\frac {n}{2}}}e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}e−12ωTσσTω{\displaystyle e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}This is the formula for amultivariate normal distribution normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page,Σ =σσT andΣ−1 =σ−Tσ−1
504e−2πα|x|{\displaystyle e^{-2\pi \alpha |\mathbf {x} |}}cnα(α2+|ξ|2)n+12{\displaystyle {\frac {c_{n}\alpha }{\left(\alpha ^{2}+|{\boldsymbol {\xi }}|^{2}\right)^{\frac {n+1}{2}}}}}cn(2π)n+22α(4π2α2+|ω|2)n+12{\displaystyle {\frac {c_{n}(2\pi )^{\frac {n+2}{2}}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}cn(2π)n+1α(4π2α2+|ω|2)n+12{\displaystyle {\frac {c_{n}(2\pi )^{n+1}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}Here[79]
cn=Γ(n+12)πn+12,{\displaystyle c_{n}={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\pi ^{\frac {n+1}{2}}}},}Re(α) > 0

See also

[edit]

Notes

[edit]
  1. ^Depending on the application aLebesgue integral,distributional, or other approach may be most appropriate.
  2. ^Vretblad (2000) provides solid justification for these formal procedures without going too deeply intofunctional analysis or thetheory of distributions.
  3. ^Inrelativistic quantum mechanics one encounters vector-valued Fourier transforms of multi-component wave functions. Inquantum field theory, operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for exampleGreiner & Reinhardt (1996).
  4. ^A possible source of confusion is thefrequency-shifting property; i.e. the transform of functionf(x)e−i2πξ0x{\displaystyle f(x)e^{-i2\pi \xi _{0}x}} isf^(ξ+ξ0).{\displaystyle {\widehat {f}}(\xi +\xi _{0}).}  The value of this function at  ξ=0{\displaystyle \xi =0}  isf^(ξ0),{\displaystyle {\widehat {f}}(\xi _{0}),} meaning that a frequencyξ0{\displaystyle \xi _{0}} has been shifted to zero (also seeNegative frequency).
  5. ^The operatorU(12πddx){\displaystyle U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)} is defined by replacingx{\displaystyle x} by12πddx{\displaystyle {\frac {1}{2\pi }}{\frac {d}{dx}}} in theTaylor expansion ofU(x).{\displaystyle U(x).}
  6. ^Up to an imaginary constant factor whose magnitude depends on what Fourier transform convention is used.
  7. ^The direct commandfourier transform of cos(6*pi*t) exp(−pi*t^2) would also work for Wolfram Alpha, although the options for the convention (seeFourier transform § Other conventions) must be changed away from the default option, which is actually equivalent tointegrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf.
  8. ^InGelfand & Shilov 1964, p. 363, with the non-unitary conventions of this table, the transform of|x|λ{\displaystyle |\mathbf {x} |^{\lambda }} is given to be
    2λ+nπ12nΓ(λ+n2)Γ(−λ2)|ω|−λ−n{\displaystyle 2^{\lambda +n}\pi ^{{\tfrac {1}{2}}n}{\frac {\Gamma \left({\frac {\lambda +n}{2}}\right)}{\Gamma \left(-{\frac {\lambda }{2}}\right)}}|{\boldsymbol {\omega }}|^{-\lambda -n}}
    from which this follows, withλ=−α{\displaystyle \lambda =-\alpha }.

Citations

[edit]
  1. ^Pinsky 2002, p. 91.
  2. ^Fourier 1822, p. 525
  3. ^Fourier 1878, p. 408
  4. ^Jordan (1883) proves on pp. 216–226 theFourier integral theorem before studying Fourier series.
  5. ^Titchmarsh 1986, p. 1
  6. ^Rahman 2011, p. 10.
  7. ^Oppenheim, Schafer & Buck 1999, p. 58
  8. ^Khare, Butola & Rajora 2023, pp. 13–14
  9. ^Kaiser 1994, p. 29
  10. ^Rahman 2011, p. 11
  11. ^Dym & McKean 1985
  12. ^Stade 2005, pp. 298–299.
  13. ^Howe 1980.
  14. ^Folland 1989
  15. ^Fourier 1822
  16. ^Arfken 1985
  17. ^abPinsky 2002
  18. ^Proakis, John G.;Manolakis, Dimitris G. (1996).Digital Signal Processing: Principles, Algorithms, and Applications (3rd ed.). Prentice Hall. p. 291.ISBN 978-0-13-373762-2.
  19. ^Katznelson 2004, p. 134.
  20. ^Stein & Weiss 1971, p. 2.
  21. ^abcdefStein & Weiss 1971
  22. ^Rudin 1987, p. 187
  23. ^Rudin 1987, p. 186
  24. ^Folland 1992, p. 216
  25. ^Wolf 1979, p. 307ff
  26. ^Folland 1989, p. 53
  27. ^Celeghini, Gadella & del Olmo 2021
  28. ^abDuoandikoetxea 2001
  29. ^abBoashash 2003
  30. ^Condon 1937
  31. ^Wolf 1979, p. 320
  32. ^abWolf 1979, p. 312
  33. ^Folland 1989, p. 52
  34. ^Howe 1980
  35. ^Paley & Wiener 1934
  36. ^Gelfand & Vilenkin 1964
  37. ^Kirillov & Gvishiani 1982
  38. ^Clozel & Delorme 1985, pp. 331–333
  39. ^de Groot & Mazur 1984, p. 146
  40. ^Champeney 1987, p. 80
  41. ^abcKolmogorov & Fomin 1999
  42. ^Wiener 1949
  43. ^Champeney 1987, p. 63
  44. ^Widder & Wiener 1938, p. 537
  45. ^Pinsky 2002, chpt. 2.4.3 The Uncertainty Principle
  46. ^Stein & Shakarchi 2003, chpt. 5.4 The Heisenberg uncertainty principle
  47. ^Chatfield 2004, p. 113
  48. ^Fourier 1822, p. 441
  49. ^Poincaré 1895, p. 102
  50. ^Whittaker & Watson 1927, p. 188
  51. ^Grafakos 2004
  52. ^Grafakos & Teschl 2013
  53. ^Stein & Weiss 1971, pp. 1–2.
  54. ^More generally, one can take a sequence of functions that are in the intersection ofL1 andL2 and that converges tof in theL2-norm, and define the Fourier transform off as theL2 -limit of the Fourier transforms of these functions.
  55. ^"Applied Fourier Analysis and Elements of Modern Signal Processing Lecture 3"(PDF). January 12, 2016. Retrieved2019-10-11.
  56. ^Stein & Weiss 1971, Thm. 2.3.
  57. ^abKatznelson 2004.
  58. ^Mallat 2009, p. 45.
  59. ^Strichartz 1994, p. 150.
  60. ^Hunter 2014.
  61. ^Pinsky 2002, p. 256.
  62. ^Edwards 1982, pp. 53, 67, 72–73.
  63. ^Katznelson 2004, p. 173
    The typical conventions in probability theory takeeiξx instead ofe−i2πξx.
  64. ^Billingsley 1995, p. 345.
  65. ^Katznelson 2004, pp. 155, 164.
  66. ^Edwards 1982, p. 53.
  67. ^Hewitt & Ross 1970, Chapter 8
  68. ^Knapp 2001
  69. ^Correia, L. B.; Justo, J. F.; Angélico, B. A. (2024). "Polynomial Adaptive Synchrosqueezing Fourier Transform: A method to optimize multiresolution".Digital Signal Processing.150: 104526.Bibcode:2024DSPRJ.15004526C.doi:10.1016/j.dsp.2024.104526.
  70. ^Gradshteyn et al. 2015
  71. ^Press et al. 1992
  72. ^Bailey & Swarztrauber 1994
  73. ^Lado 1971
  74. ^Simonen & Olkkonen 1985
  75. ^"The Integration Property of the Fourier Transform".The Fourier Transform .com. 2015 [2010].Archived from the original on 2022-01-26. Retrieved2023-08-20.
  76. ^Stein & Weiss 1971, Thm. IV.3.3
  77. ^Easton 2010
  78. ^Stein & Weiss 1971, Thm. 4.15
  79. ^Stein & Weiss 1971, p. 6

References

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External links

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