Fourier transform

File:CQT-piano-chord.png

An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.

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

File:Fourier transform time and frequency domains (small).gif

The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.

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

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

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

Definition

The Fourier transform of a complex-valued (Lebesgue) integrable function <math>f(x)</math> on the real line, is the complex valued function <math>\hat{f}(\xi)</math>, defined by the integralTemplate:Sfn Template:Equation box 1 Evaluating the Fourier transform for all values of <math>\xi</math> produces the frequency-domain function, and it converges at all frequencies to a continuous function tending to zero at infinity. If <math>f(x)</math> decays with all derivatives, i.e., <math display="block">\lim_{|x|\to\infty} f^{(n)}(x) = 0, \quad \forall n\in \mathbb{N},</math> then <math>\widehat f</math> converges for all frequencies and, by the Riemann–Lebesgue lemma, <math>\widehat f</math> also decays with all derivatives.

First introduced in Fourier's Analytical Theory of Heat.,<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref><ref>Template:Harvtxt proves on pp. 216–226 the Fourier integral theorem before studying Fourier series.</ref><ref>Template:Harvnb</ref> the corresponding inversion formula for "sufficiently nice" functions is given by the Fourier inversion theorem, i.e., Template:Equation box 1 The functions <math>f</math> and <math>\widehat{f}</math> are referred to as a Fourier transform pair.<ref>Template:Harvnb.</ref>  A common notation for designating transform pairs is:<ref>Template:Harvnb</ref> <math display="block">f(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \widehat f(\xi),</math>   for example   <math>\operatorname{rect}(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \operatorname{sinc}(\xi).</math>

By analogy, the Fourier series can be regarded as an abstract Fourier transform on the group <math>\mathbb{Z}</math> of integers. That is, the synthesis of a sequence of complex numbers <math>c_n</math> is defined by the Fourier transform <math display="block">f(x) = \sum_{n=-\infty}^\infty c_n\, e^{i 2\pi \tfrac{n}{P}x},</math> such that <math>c_n</math> are given by the inversion formula, i.e., the analysis <math display="block">c_n = \frac{1}{P} \int_{-P/2}^{P/2} f(x) \, e^{-i 2\pi \frac{n}{P}x} \, dx,</math> for some complex-valued, <math>P</math>-periodic function <math>f(x)</math> defined on a bounded interval <math>[-P/2, P/2] \in \mathbb{R}</math>. When <math>P\to\infty,</math> the constituent frequencies are a continuum: <math>\tfrac{n}{P} \to \xi \in \mathbb R,</math><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> and <math>c_n \to \hat{f}(\xi)\in\mathbb{C}</math>.<ref>Template:Harvnb</ref>

In other words, on the finite interval <math>[-P/2, P/2]</math> the function <math>f(x)</math> has a discrete decomposition in the periodic functions <math>e^{i2\pi x n/P}</math>. On the infinite interval <math>(-\infty,\infty)</math> the function <math>f(x)</math> has a continuous decomposition in periodic functions <math>e^{i2\pi x \xi}</math>.

Lebesgue integrable functions

A measurable function <math>f:\mathbb R\to\mathbb C</math> is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite: <math display="block">\|f\|_1 = \int_{\mathbb R}|f(x)|\,dx < \infty.</math> If <math>f</math> is Lebesgue integrable then the Fourier transform, given by Template:EquationNote, is well-defined for all <math>\xi\in\mathbb R</math>.Template:Sfn Furthermore, <math>\widehat f\in L^\infty\cap C(\mathbb R)</math> is bounded, uniformly continuous and (by the Riemann–Lebesgue lemma) zero at infinity.

The space <math>L^1(\mathbb R)</math> is the space of measurable functions for which the norm <math>\|f\|_1</math> is finite, modulo the equivalence relation of equality almost everywhere. The Fourier transform on <math>L^1(\mathbb R)</math> is one-to-one. However, there is no easy characterization of the image, and thus no easy characterization of the inverse transform. In particular, Template:EquationNote is no longer valid, as it was stated only under the hypothesis that <math>f(x)</math> decayed with all derivatives.

While Template:EquationNote defines the Fourier transform for (complex-valued) functions in <math>L^1(\mathbb R)</math>, it is not well-defined for other integrability classes, most importantly the space of square-integrable functions <math>L^2(\mathbb R)</math>. For example, the function <math>f(x)=(1+x^2)^{-1/2}</math> is in <math>L^2</math> but not <math>L^1</math> and therefore the Lebesgue integral Template:EquationNote does not exist. However, the Fourier transform on the dense subspace <math>L^1\cap L^2(\mathbb R) \subset L^2(\mathbb R)</math> admits a unique continuous extension to a unitary operator on <math>L^2(\mathbb R)</math>. This extension is important in part because, unlike the case of <math>L^1</math>, the Fourier transform is an automorphism of the space <math>L^2(\mathbb R)</math>.

In such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit. In practice, the integral is often regarded as an improper integral instead of a proper Lebesgue integral, but sometimes for convergence one needs to use weak limit or principal value instead of the (pointwise) limits implicit in an improper integral. Template:Harvtxt and Template:Harvtxt each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure. A general principle in working with the <math>L^2</math> Fourier transform is that Gaussians are dense in <math>L^1\cap L^2</math>, 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:Template:Sfn

• that <math>e^{-\pi x^2}</math> is its own Fourier transform; and
• that the Gaussian integral <math>\int_{-\infty}^\infty e^{-\pi x^2}\,dx = 1.</math>

A feature of the <math>L^1</math> Fourier transform is that it is a homomorphism of Banach algebras from <math>L^1</math> equipped with the convolution operation to the Banach algebra of continuous functions under the <math>L^\infty</math> (supremum) norm. The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on Template:Math and an algebra homomorphism from Template:Math to Template:Math, without renormalizing the Lebesgue measure.<ref>Template:Harvnb</ref>

Angular frequency (ω)

When the independent variable (<math>x</math>) represents time (often denoted by <math>t</math>), the transform variable (<math>\xi</math>) represents frequency (often denoted by <math>f</math>). For example, if time is measured in seconds, then frequency is in hertz. The Fourier transform can also be written in terms of angular frequency, <math>\omega = 2\pi \xi,</math> whose units are radians per second.

The substitution <math>\xi = \tfrac{\omega}{2 \pi}</math> into Template:EquationNote produces this convention, where function <math>\widehat f</math> is relabeled <math>\widehat {f_1}:</math> <math display="block">\begin{align} \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{align} </math> Unlike the Template:EquationNote definition, the Fourier transform is no longer a unitary transformation, and there is less symmetry between the formulas for the transform and its inverse. Those properties are restored by splitting the <math>2 \pi</math> factor evenly between the transform and its inverse, which leads to another convention: <math display="block">\begin{align} \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{align}</math> Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites.

Summary of popular forms of the Fourier transform, one-dimensional

ordinary frequency Template:Mvar (Hz)	unitary	<math>\begin{align} \widehat{f_1}(\xi)\ &\triangleq\ \int_{-\infty}^{\infty} f(x)\, e^{-i 2\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^{i 2\pi x \xi}\, d\xi \end{align}</math>
angular frequency Template:Mvar (rad/s)	unitary	<math>\begin{align} \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{align}</math>
	non-unitary	<math>\begin{align} \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{align}</math>

Generalization for Template:Math-dimensional functions

ordinary frequency Template:Mvar (Hz)	unitary	<math>\begin{align} \widehat{f_1}(\xi)\ &\triangleq\ \int_{\mathbb{R}^n} f(x) e^{-i 2\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^{i 2\pi \xi\cdot x}\, d\xi \end{align}</math>
angular frequency Template:Mvar (rad/s)	unitary	<math>\begin{align} \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{align}</math>
	non-unitary	<math>\begin{align} \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{align}</math>

Background

History

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In 1822, Fourier claimed (see Template:Slink) that any function, whether continuous or discontinuous, can be expanded into a series of sines.<ref>Template:Harvnb</ref> 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

Template:Multiple image

In general, the coefficients <math>\widehat f(\xi)</math> are complex numbers, which have two equivalent forms (see Euler's formula): <math display="block"> \widehat f(\xi) = \underbrace{A e^{i \theta}}_{\text{polar coordinate form}} = \underbrace{A \cos(\theta) + i A \sin(\theta)}_{\text{rectangular coordinate form}}.</math>

The product with <math>e^{i 2 \pi \xi x}</math> (Template:EquationNote) has these forms: <math display="block">\begin{aligned}\widehat f(\xi)\cdot e^{i 2 \pi \xi x} &= A e^{i \theta} \cdot e^{i 2 \pi \xi x}\\ &= \underbrace{A e^{i (2 \pi \xi x+\theta)}}_{\text{polar coordinate form}}\\ &= \underbrace{A\cos(2\pi \xi x +\theta) + i A\sin(2\pi \xi x +\theta)}_{\text{rectangular coordinate form}}.\end{aligned}</math> which conveys both amplitude and phase of frequency <math>\xi.</math> Likewise, the intuitive interpretation of Template:EquationNote is that multiplying <math>f(x)</math> by <math>e^{-i 2\pi \xi x}</math> has the effect of subtracting <math>\xi</math> from every frequency component of function <math>f(x).</math><ref group="note">A possible source of confusion is the frequency-shifting property; i.e. the transform of function <math>f(x)e^{-i 2\pi \xi_0 x}</math> is <math>\widehat{f}(\xi+\xi_0).</math>  The value of this function at  <math>\xi=0</math>  is <math>\widehat{f}(\xi_0),</math> meaning that a frequency <math>\xi_0</math> has been shifted to zero (also see Negative frequency).</ref> Only the component that was at frequency <math>\xi</math> 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 Template:Slink)

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

Euler's formula introduces the possibility of negative <math>\xi.</math>  And Template:EquationNote is defined <math>\forall \xi \in \mathbb{R}.</math> Only certain complex-valued <math> f(x)</math> have transforms <math> \widehat f =0, \ \forall \ \xi < 0</math> (See Analytic signal. A simple example is <math> e^{i 2 \pi \xi_0 x}\ (\xi_0 > 0).</math>)  But negative frequency is necessary to characterize all other complex-valued <math> f(x),</math> found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.

For a real-valued <math> f(x),</math> Template:EquationNote has the symmetry property <math>\widehat f(-\xi) = \widehat {f}^* (\xi)</math> (see Template:Slink below). This redundancy enables Template:EquationNote to distinguish <math>f(x) = \cos(2 \pi \xi_0 x)</math> from <math>e^{i2 \pi \xi_0 x}.</math>  But of course it cannot tell us the actual sign of <math>\xi_0,</math> because <math>\cos(2 \pi \xi_0 x)</math> and <math>\cos(2 \pi (-\xi_0) x)</math> are indistinguishable on just the real numbers line.

Fourier transform for periodic functions

The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Template:EquationNote to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.

This makes it possible to see a connection between the Fourier series and the Fourier transform for periodic functions that have a convergent Fourier series. If <math>f(x)</math> is a periodic function, with period <math>P</math>, that has a convergent Fourier series, then: <math display="block"> \widehat{f}(\xi) = \sum_{n=-\infty}^\infty c_n \cdot \delta \left(\xi - \tfrac{n}{P}\right), </math> where <math>c_n</math> are the Fourier series coefficients of <math>f</math>, and <math>\delta</math> is the Dirac delta function. In other words, the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.

Sampling the Fourier transform

Template:Broader The Fourier transform of an integrable function <math>f</math> can be sampled at regular intervals of arbitrary length <math>\tfrac{1}{P}.</math> These samples can be deduced from one cycle of a periodic function <math>f_P</math> which has Fourier series coefficients proportional to those samples by the Poisson summation formula: <math display="block">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}</math>

The integrability of <math>f</math> ensures the periodic summation converges. Therefore, the samples <math>\widehat f\left(\tfrac{k}{P}\right)</math> can be determined by Fourier series analysis: <math display="block">\widehat f\left(\tfrac{k}{P}\right) = \int_{P} f_P(x) \cdot e^{-i2\pi \frac{k}{P} x} \,dx.</math>

When <math>f(x)</math> has compact support, <math>f_P(x)</math> has a finite number of terms within the interval of integration. When <math>f(x)</math> does not have compact support, numerical evaluation of <math>f_P(x)</math> requires an approximation, such as tapering <math>f(x)</math> or truncating the number of terms.

Units

The frequency variable must have inverse units to the units of the original function's domain (typically named <math>t</math> or <math>x</math>). For example, if <math>t</math> is measured in seconds, <math>\xi</math> should be in cycles per second or hertz. If the scale of time is in units of <math>2\pi</math> seconds, then another Greek letter <math>\omega</math> is typically used instead to represent angular frequency (where <math>\omega=2\pi \xi</math>) in units of radians per second. If using <math>x</math> for units of length, then <math>\xi</math> must be in inverse length, e.g., wavenumbers. That is to say, there are two versions of the real line: one which is the range of <math>t</math> and measured in units of <math>t,</math> and the other which is the range of <math>\xi</math> and measured in inverse units to the units of <math>t.</math> 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, <math>\xi</math> must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to general symmetry 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 has Template:Mvar in the exponent instead of Template:Math, and vice versa for the inversion formula. This convention is common in modern physics<ref>Template:Harvnb</ref> and is the default for Wolfram 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 that <math>\hat f(\xi)</math> is the amplitude of the wave <math>e^{-i 2\pi \xi x}</math> instead of the wave <math>e^{i 2\pi \xi x}</math>(the former, with its minus sign, is often seen in the time dependence for sinusoidal plane-wave solutions of the electromagnetic wave equation, or in the time dependence for quantum wave functions). Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve Template:Math have it replaced by Template:Math. In electrical engineering the letter Template:Math is typically used for the imaginary unit instead of Template:Math because Template:Math is used for current.

When using dimensionless units, the constant factors might not be written in the transform definition. For instance, in probability theory, the characteristic function Template:Mvar of the probability density function Template:Mvar of a random variable Template:Mvar of continuous type is defined without a negative sign in the exponential, and since the units of Template:Mvar are ignored, there is no 2Template:Pi either: <math display="block">\phi (\lambda) = \int_{-\infty}^\infty f(x) e^{i\lambda x} \,dx.</math>

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 but distributions, i.e., measures which possess "atoms".

From the higher point of view of group 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 a locally compact Abelian group.

Properties

Let <math>f(x)</math> and <math>h(x)</math> represent integrable functions Lebesgue-measurable on the real line satisfying: <math display="block">\int_{-\infty}^\infty |f(x)| \, dx < \infty.</math> We denote the Fourier transforms of these functions as <math>\hat f(\xi)</math> and <math>\hat h(\xi)</math> respectively.

Basic properties

The Fourier transform has the following basic properties:<ref name="Pinsky-2002">Template:Harvnb</ref>

Linearity

<math display="block">a\ f(x) + b\ h(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ a\ \widehat f(\xi) + b\ \widehat h(\xi);\quad \ a,b \in \mathbb C</math>

Time shifting

<math display="block">f(x-x_0)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ e^{-i 2\pi x_0 \xi}\ \widehat f(\xi);\quad \ x_0 \in \mathbb R</math>

Frequency shifting

<math display="block">e^{i 2\pi \xi_0 x} f(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f(\xi - \xi_0);\quad \ \xi_0 \in \mathbb R</math>

Time scaling

<math display="block">f(ax)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \frac{1}{|a|}\widehat{f}\left(\frac{\xi}{a}\right);\quad \ a \ne 0 </math> The case <math>a=-1</math> leads to the time-reversal property: <math display="block">f(-x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f (-\xi)</math>

Template:Annotated image

Symmetry

When the real and imaginary parts of a complex function are decomposed into their even 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:<ref name="ProakisManolakis1996">Template:Cite book</ref>

<math> \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} </math>

From this, various relationships are apparent, for example:

• The transform of a real-valued function <math>(f_{_{RE}}+f_{_{RO}})</math> is the conjugate symmetric function <math>\hat f_{RE}+i\ \hat f_{IO}.</math> Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function <math>(i\ f_{_{IE}}+i\ f_{_{IO}})</math> is the conjugate antisymmetric function <math>\hat f_{RO}+i\ \hat f_{IE},</math> and the converse is true.
• The transform of a conjugate symmetric function <math>(f_{_{RE}}+i\ f_{_{IO}})</math> is the real-valued function <math>\hat f_{RE}+\hat f_{RO},</math> and the converse is true.
• The transform of a conjugate antisymmetric function <math>(f_{_{RO}}+i\ f_{_{IE}})</math> is the imaginary-valued function <math>i\ \hat f_{IE}+i\hat f_{IO},</math> and the converse is true.

Conjugation

<math display="block">\bigl(f(x)\bigr)^*\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \left(\widehat{f}(-\xi)\right)^*</math> (Note: the ∗ denotes complex conjugation.)

In particular, if <math>f</math> is real, then <math>\widehat f</math> is even symmetric (aka Hermitian function): <math display="block">\widehat{f}(-\xi)=\bigl(\widehat f(\xi)\bigr)^*.</math>

And if <math>f</math> is purely imaginary, then <math>\widehat f</math> is odd symmetric: <math display="block">\widehat f(-\xi) = -(\widehat f(\xi))^*.</math>

Real and imaginary parts

<math display="block">\operatorname{Re}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \tfrac{1}{2} \left( \widehat f(\xi) + \bigl(\widehat f (-\xi) \bigr)^* \right)</math> <math display="block">\operatorname{Im}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \tfrac{1}{2i} \left( \widehat f(\xi) - \bigl(\widehat f (-\xi) \bigr)^* \right)</math>

Zero frequency component

Substituting <math>\xi = 0</math> in the definition, we obtain: <math display="block">\widehat{f}(0) = \int_{-\infty}^{\infty} f(x)\,dx.</math>

The integral of <math>f</math> over its domain is known as the average value or DC bias of the function.

Uniform continuity and the Riemann–Lebesgue lemma

File:Rectangular function.svg

The rectangular function is Lebesgue integrable.

File:Sinc function (normalized).svg

The sinc 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 transform <math>\hat{f}</math> of any integrable function <math>f</math> is uniformly continuous andTemplate:SfnTemplate:Sfn <math display="block">\left\|\hat{f}\right\|_\infty \leq \left\|f\right\|_1</math>

By the Riemann–Lebesgue lemma,<ref name="Stein-Weiss-1971">Template:Harvnb</ref> <math display="block">\hat{f}(\xi) \to 0\text{ as }|\xi| \to \infty.</math>

However, <math>\hat{f}</math> need not be integrable. For example, the Fourier transform of the rectangular function, which is integrable, is the sinc function, which is not Lebesgue integrable, because its improper integrals behave analogously to the alternating harmonic series, in converging to a sum without being absolutely convergent.

It is not generally possible to write the inverse transform as a Lebesgue integral. However, when both <math>f</math> and <math>\hat{f}</math> are integrable, the inverse equality <math display="block">f(x) = \int_{-\infty}^\infty \hat f(\xi) e^{i 2\pi x \xi} \, d\xi</math> holds for almost every Template:Mvar. As a result, the Fourier transform is injective on Template:Math.

Plancherel theorem and Parseval's theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let Template:Math and Template:Math be integrable, and let Template:Math and Template:Math be their Fourier transforms. If Template:Math and Template:Math are also square-integrable, then the Parseval formula follows:<ref>Template:Harvnb</ref> <math display="block">\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,</math> where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that<ref>Template:Harvnb</ref> <math display="block">\|f\|^2_{L^{2}} = \int_{-\infty}^\infty \left| f(x) \right|^2\,dx = \int_{-\infty}^\infty \left| \hat{f}(\xi) \right|^2\,d\xi. </math>

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on Template:Math. On Template:Math, this extension agrees with original Fourier transform defined on Template:Math, thus enlarging the domain of the Fourier transform to Template:Math (and consequently to Template:Math for Template:Math). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy 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.

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

Convolution theorem

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The Fourier transform translates between convolution and multiplication of functions. If Template:Math and Template:Math are integrable functions with Fourier transforms Template:Math and Template:Math respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms Template:Math and Template:Math (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if: <math display="block">h(x) = (f*g)(x) = \int_{-\infty}^\infty f(y)g(x - y)\,dy,</math> where Template:Math denotes the convolution operation, then: <math display="block">\hat{h}(\xi) = \hat{f}(\xi)\, \hat{g}(\xi).</math>

In linear time invariant (LTI) system theory, it is common to interpret Template:Math as the impulse response of an LTI system with input Template:Math and output Template:Math, since substituting the unit impulse for Template:Math yields Template:Math. In this case, Template:Math represents the frequency response of the system.

Conversely, if Template:Math can be decomposed as the product of two square integrable functions Template:Math and Template:Math, then the Fourier transform of Template:Math is given by the convolution of the respective Fourier transforms Template:Math and Template:Math.

Cross-correlation theorem

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In an analogous manner, it can be shown that if Template:Math is the cross-correlation of Template:Math and Template:Math: <math display="block">h(x) = (f \star g)(x) = \int_{-\infty}^\infty \overline{f(y)}g(x + y)\,dy</math> then the Fourier transform of Template:Math is: <math display="block">\hat{h}(\xi) = \overline{\hat{f}(\xi)} \, \hat{g}(\xi).</math>

As a special case, the autocorrelation of function Template:Math is: <math display="block">h(x) = (f \star f)(x) = \int_{-\infty}^\infty \overline{f(y)}f(x + y)\,dy</math> for which <math display="block">\hat{h}(\xi) = \overline{\hat{f}(\xi)}\hat{f}(\xi) = \left|\hat{f}(\xi)\right|^2.</math>

Differentiation

Suppose Template:Math is an absolutely continuous differentiable function, and both Template:Math and its derivative Template:Math are integrable. Then the Fourier transform of the derivative is given by <math display="block">\widehat{f'\,}(\xi) = \mathcal{F}\left\{ \frac{d}{dx} f(x)\right\} = i 2\pi \xi\hat{f}(\xi).</math> More generally, the Fourier transformation of the Template:Mvarth derivative Template:Math is given by <math display="block">\widehat{f^{(n)}}(\xi) = \mathcal{F}\left\{ \frac{d^n}{dx^n} f(x) \right\} = (i 2\pi \xi)^n\hat{f}(\xi).</math>

Analogously, <math>\mathcal{F}\left\{ \frac{d^n}{d\xi^n} \hat{f}(\xi)\right\} = (i 2\pi x)^n f(x)</math>, so <math>\mathcal{F}\left\{ x^n f(x)\right\} = \left(\frac{i}{2\pi}\right)^n \frac{d^n}{d\xi^n} \hat{f}(\xi).</math>

By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "Template:Math is smooth if and only if Template:Math quickly falls to 0 for Template:Math." By using the analogous rules for the inverse Fourier transform, one can also say "Template:Math quickly falls to 0 for Template:Math if and only if Template:Math is smooth."

Eigenfunctions

Template:See also The Fourier transform is a linear transform which has eigenfunctions obeying <math>\mathcal{F}[\psi] = \lambda \psi,</math> with <math> \lambda \in \mathbb{C}.</math>

A set of eigenfunctions is found by noting that the homogeneous differential equation <math display="block">\left[ U\left( \frac{1}{2\pi}\frac{d}{dx} \right) + U( x ) \right] \psi(x) = 0</math> leads to eigenfunctions <math>\psi(x)</math> of the Fourier transform <math>\mathcal{F}</math> as long as the form of the equation remains invariant under Fourier transform.<ref group=note>The operator <math>U\left( \frac{1}{2\pi}\frac{d}{dx} \right)</math> is defined by replacing <math>x</math> by <math>\frac{1}{2\pi}\frac{d}{dx}</math> in the Taylor expansion of <math>U(x).</math></ref> In other words, every solution <math>\psi(x)</math> and its Fourier transform <math>\hat\psi(\xi)</math> obey the same equation. Assuming uniqueness of the solutions, every solution <math>\psi(x)</math> must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if <math>U(x)</math> can be expanded in a power series in which for all terms the same factor of either one of <math>\pm 1, \pm i</math> arises from the factors <math>i^n</math> introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable <math>U(x)=x</math> leads to the standard normal distribution.<ref>Template:Harvnb</ref>

More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation <math display="block">\left[ W\left( \frac{i}{2\pi}\frac{d}{dx} \right) + W(x) \right] \psi(x) = C \psi(x)</math> with <math>C</math> constant and <math>W(x)</math> being a non-constant even function remains invariant in form when applying the Fourier transform <math>\mathcal{F}</math> to both sides of the equation. The simplest example is provided by <math>W(x) = x^2</math> which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator.<ref>Template:Harvnb</ref> The corresponding solutions provide an important choice of an orthonormal basis for Template:Math and are given by the "physicist's" Hermite functions. Equivalently one may use <math display="block">\psi_n(x) = \frac{\sqrt[4]{2}}{\sqrt{n!}} e^{-\pi x^2}\mathrm{He}_n\left(2x\sqrt{\pi}\right),</math> where Template:Math are the "probabilist's" Hermite polynomials, defined as <math display="block">\mathrm{He}_n(x) = (-1)^n e^{\frac{1}{2}x^2}\left(\frac{d}{dx}\right)^n e^{-\frac{1}{2}x^2}.</math>

Under this convention for the Fourier transform, we have that <math display="block">\hat\psi_n(\xi) = (-i)^n \psi_n(\xi).</math>

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on Template:Math.<ref name="Pinsky-2002" /><ref>Template:Harvnb</ref> However, this choice of eigenfunctions is not unique. Because of <math>\mathcal{F}^4 = \mathrm{id}</math> there are only four different eigenvalues of the Fourier transform (the fourth roots of unity ±1 and ±Template:Mvar) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.<ref>Template:Harvnb</ref> As a consequence of this, it is possible to decompose Template:Math as a direct sum of four spaces Template:Math, Template:Math, Template:Math, and Template:Math where the Fourier transform acts on Template:Math simply by multiplication by Template:Math.

Since the complete set of Hermite functions Template:Math 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: <math display="block">\mathcal{F}[f](\xi) = \int dx f(x) \sum_{n \geq 0} (-i)^n \psi_n(x) \psi_n(\xi) ~.</math>

This approach to define the Fourier transform was first proposed by Norbert Wiener.<ref name="Duoandikoetxea-2001">Template:Harvnb</ref> 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 the fractional Fourier transform used in time–frequency analysis.<ref name="Boashash-2003">Template:Harvnb</ref> In physics, this transform was introduced by Edward Condon.<ref>Template:Harvnb</ref> This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right conventions. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator <math>N</math> via<ref>Template:Harvnb</ref> <math display="block">\mathcal{F}[\psi] = e^{-i t N} \psi.</math>

The operator <math>N</math> is the number operator of the quantum harmonic oscillator written as<ref name="auto">Template:Harvnb</ref><ref>Template:Harvnb</ref> <math display="block">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).</math>

It can be interpreted as the generator of fractional Fourier transforms for arbitrary values of Template:Mvar, and of the conventional continuous Fourier transform <math>\mathcal{F}</math> for the particular value <math>t = \pi/2,</math> with the Mehler kernel implementing the corresponding active transform. The eigenfunctions of <math> N</math> are the Hermite functions <math>\psi_n(x)</math> which are therefore also eigenfunctions of <math>\mathcal{F}.</math>

Upon extending the Fourier transform to distributions the Dirac comb is also an eigenfunction of the Fourier transform.

Inversion and periodicity

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

More precisely, defining the parity operator <math>\mathcal{P}</math> such that <math>(\mathcal{P} f)(x) = f(-x)</math>, we have: <math display="block">\begin{align}

 \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{align}</math> These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality almost 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 the Fourier 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 the time–frequency domain (considering time as the Template:Mvar-axis and frequency as the Template:Mvar-axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves rotations by other angles. This can be further generalized to linear canonical transformations, which can be visualized as the action of the special linear group Template:Math on the time–frequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. This approach is particularly studied in signal processing, under time–frequency analysis.

Connection with the Heisenberg group

The Heisenberg group is a certain group of unitary operators on the Hilbert space Template:Math of square integrable complex valued functions Template:Mvar on the real line, generated by the translations Template:Math and multiplication by Template:Math, Template:Math. These operators do not commute, as their (group) commutator is <math display="block">\left(M^{-1}_\xi T^{-1}_y M_\xi T_yf\right)(x) = e^{i 2\pi\xi y}f(x)</math> which is multiplication by the constant (independent of Template:Mvar) Template:Math (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional Lie group of triples Template:Math, with the group law <math display="block">\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^{i 2\pi \left(x_1 \xi_1 + x_2 \xi_2 + x_1 \xi_2\right)}\right).</math>

Denote the Heisenberg group by Template:Math. The above procedure describes not only the group structure, but also a standard unitary representation of Template:Math on a Hilbert space, which we denote by Template:Math. Define the linear automorphism of Template:Math by <math display="block">J \begin{pmatrix}

  x \\
 \xi

\end{pmatrix} = \begin{pmatrix}

 -\xi \\
   x

\end{pmatrix}</math> so that Template:Math. This Template:Mvar can be extended to a unique automorphism of Template:Math: <math display="block">j\left(x, \xi, t\right) = \left(-\xi, x, te^{-i 2\pi\xi x}\right).</math>

According to the Stone–von Neumann theorem, the unitary representations Template:Mvar and Template:Math are unitarily equivalent, so there is a unique intertwiner Template:Math such that <math display="block">\rho \circ j = W \rho W^*.</math> This operator Template:Mvar is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.<ref>Template:Harvnb</ref> For example, the square of the Fourier transform, Template:Math, is an intertwiner associated with Template:Math, and so we have Template:Math is the reflection of the original function Template:Mvar.

Complex domain

The integral for the Fourier transform <math display="block"> \hat f (\xi) = \int _{-\infty}^\infty e^{-i 2\pi \xi t} f(t) \, dt </math> can be studied for complex values of its argument Template:Mvar. Depending on the properties of Template:Mvar, this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of Template:Math, or something in between.<ref>Template:Harvnb</ref>

The Paley–Wiener theorem says that Template:Mvar is smooth (i.e., Template:Mvar-times differentiable for all positive integers Template:Mvar) and compactly supported if and only if Template:Math is a holomorphic function for which there exists a constant Template:Math such that for any integer Template:Math, <math display="block"> \left\vert \xi ^n \hat f(\xi) \right\vert \leq C e^{a\vert\tau\vert} </math> for some constant Template:Mvar. (In this case, Template:Mvar is supported on Template:Math.) This can be expressed by saying that Template:Math is an entire function which is rapidly decreasing in Template:Mvar (for fixed Template:Mvar) and of exponential growth in Template:Mvar (uniformly in Template:Mvar).<ref>Template:Harvnb</ref>

(If Template:Mvar is not smooth, but only Template:Math, the statement still holds provided Template:Math.<ref>Template:Harvnb</ref>) The space of such functions of a complex variable is called the Paley—Wiener space. This theorem has been generalised to semisimple Lie groups.<ref>Template:Harvnb</ref>

If Template:Mvar is supported on the half-line Template:Math, then Template:Mvar is said to be "causal" because the impulse response function of a physically realisable filter must have this property, as no effect can precede its cause. Paley and Wiener showed that then Template:Math extends to a holomorphic function on the complex lower half-plane Template:Math which tends to zero as Template:Mvar goes to infinity.<ref>Template:Harvnb</ref> The converse is false and it is not known how to characterise the Fourier transform of a causal function.<ref>Template:Harvnb</ref>

Laplace transform

Template:See also The Fourier transform Template:Math is related to the Laplace transform Template:Math, which is also used for the solution of differential equations and the analysis of filters.

It may happen that a function Template:Mvar 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 the complex plane.

For example, if Template:Math is of exponential growth, i.e., <math display="block"> \vert f(t) \vert < C e^{a\vert t\vert} </math> for some constants Template:Math, then<ref name="Kolmogorov-Fomin-1999">Template:Harvnb</ref> <math display="block"> \hat f (i\tau) = \int _{-\infty}^\infty e^{ 2\pi \tau t} f(t) \, dt, </math> convergent for all Template:Math, is the two-sided Laplace transform of Template:Mvar.

The more usual version ("one-sided") of the Laplace transform is <math display="block"> F(s) = \int_0^\infty f(t) e^{-st} \, dt.</math>

If Template:Mvar is also causal, and analytical, then: <math> \hat f(i\tau) = F(-2\pi\tau).</math> 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 variable Template:Math.

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 of harmonic analysis.

Inversion

Still with <math>\xi = \sigma+ i\tau</math>, if <math>\widehat f</math> is complex analytic for Template:Math, then

<math display="block"> \int _{-\infty}^\infty \hat f (\sigma + ia) e^{ i 2\pi \xi t} \, d\sigma = \int _{-\infty}^\infty \hat f (\sigma + ib) e^{ i 2\pi \xi t} \, d\sigma </math> by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.<ref>Template:Harvnb</ref>

Theorem: If Template:Math for Template:Math, and Template:Math for some constants Template:Math, then <math display="block"> f(t) = \int_{-\infty}^\infty \hat f(\sigma + i\tau) e^{i 2 \pi \xi t} \, d\sigma,</math> for any Template:Math.

This theorem implies the Mellin inversion formula for the Laplace transformation,<ref name="Kolmogorov-Fomin-1999" /> <math display="block"> f(t) = \frac 1 {i 2\pi} \int_{b-i\infty}^{b+i\infty} F(s) e^{st}\, ds</math> for any Template:Math, where Template:Math is the Laplace transform of Template:Math.

The hypotheses can be weakened, as in the results of Carleson and Hunt, to Template:Math being Template:Math, provided that Template:Mvar be of bounded variation in a closed neighborhood of Template:Mvar (cf. Dini test), the value of Template:Mvar at Template:Mvar be taken to be the arithmetic mean of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.<ref>Template:Harvnb</ref>

Template:Math versions of these inversion formulas are also available.<ref>Template:Harvnb</ref>

Fourier transform on Euclidean space

The Fourier transform can be defined in any arbitrary number of dimensions Template:Mvar. As with the one-dimensional case, there are many conventions. For an integrable function Template:Math, this article takes the definition: <math display="block">\hat{f}(\boldsymbol{\xi}) = \mathcal{F}(f)(\boldsymbol{\xi}) = \int_{\R^n} f(\mathbf{x}) e^{-i 2\pi \boldsymbol{\xi}\cdot\mathbf{x}} \, d\mathbf{x}</math> where Template:Math and Template:Math are Template:Mvar-dimensional vectors, and Template:Math is the dot product of the vectors. Alternatively, Template:Math can be viewed as belonging to the dual vector space <math>\R^{n\star}</math>, in which case the dot product becomes the contraction of Template:Math and Template:Math, usually written as Template:Math.

All of the basic properties listed above hold for the Template:Mvar-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.<ref name="Stein-Weiss-1971" />

Uncertainty principle

Generally speaking, the more concentrated Template:Math is, the more spread out its Fourier transform Template:Math must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in Template:Mvar, its Fourier transform stretches out in Template:Mvar. 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 an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.

Suppose Template:Math is an integrable and square-integrable function. Without loss of generality, assume that Template:Math is normalized: <math display="block">\int_{-\infty}^\infty |f(x)|^2 \,dx=1.</math>

It follows from the Plancherel theorem that Template:Math is also normalized.

The spread around Template:Math may be measured by the dispersion about zero defined by<ref>Template:Harvnb</ref> <math display="block">D_0(f)=\int_{-\infty}^\infty x^2|f(x)|^2\,dx.</math>

In probability terms, this is the second moment of Template:Math about zero.

The uncertainty principle states that, if Template:Math is absolutely continuous and the functions Template:Math and Template:Math are square integrable, then <math display="block">D_0(f)D_0(\hat{f}) \geq \frac{1}{16\pi^2}.</math>

The equality is attained only in the case <math display="block">\begin{align} 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{align} </math> where Template:Math is arbitrary and Template:Math so that Template:Mvar is Template:Math-normalized. In other words, where Template:Mvar is a (normalized) Gaussian function with variance Template:Math, centered at zero, and its Fourier transform is a Gaussian function with variance Template:Math. Gaussian functions are examples of Schwartz functions (see the discussion on tempered distributions below).

In fact, this inequality implies that: <math display="block">\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}.</math> In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, up to a factor of the Planck constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle.<ref>Template:Harvnb</ref>

A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as: <math display="block">H\left(\left|f\right|^2\right)+H\left(\left|\hat{f}\right|^2\right)\ge \log\left(\frac{e}{2}\right)</math> where Template:Math is the differential entropy of the probability density function Template:Math: <math display="block">H(p) = -\int_{-\infty}^\infty p(x)\log\bigl(p(x)\bigr) \, dx</math> 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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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 function Template:Mvar for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically<ref>Template:Harvnb</ref>) Template:Mvar by <math display="block">f(t) = \int_0^\infty \bigl( a(\lambda ) \cos( 2\pi \lambda t) + b(\lambda ) \sin( 2\pi \lambda t)\bigr) \, d\lambda.</math>

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions Template:Mvar and Template:Mvar can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised): <math display="block"> a (\lambda) = 2\int_{-\infty}^\infty f(t) \cos(2\pi\lambda t) \, dt</math> and <math display="block"> b (\lambda) = 2\int_{-\infty}^\infty f(t) \sin(2\pi\lambda t) \, dt. </math>

Older literature refers to the two transform functions, the Fourier cosine transform, Template:Mvar, and the Fourier sine transform, Template:Mvar.

The function Template:Mvar can be recovered from the sine and cosine transform using <math display="block"> f(t) = 2\int_0 ^{\infty} \int_{-\infty}^{\infty} f(\tau) \cos\bigl( 2\pi \lambda(\tau-t)\bigr) \, d\tau \, d\lambda.</math> together with trigonometric identities. This is referred to as Fourier's integral formula.<ref name="Kolmogorov-Fomin-1999" /><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree Template:Mvar on Template:Math be denoted by Template:Math. The set Template:Math consists of the solid spherical harmonics of degree Template:Mvar. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if Template:Math for some Template:Math in Template:Math, then Template:Math. Let the set Template:Math be the closure in Template:Math of linear combinations of functions of the form Template:Math where Template:Math is in Template:Math. The space Template:Math is then a direct sum of the spaces Template:Math and the Fourier transform maps each space Template:Math to itself and is possible to characterize the action of the Fourier transform on each space Template:Math.<ref name="Stein-Weiss-1971" />

Let Template:Math (with Template:Math in Template:Math), then <math display="block">\hat{f}(\xi)=F_0(|\xi|)P(\xi)</math> where <math display="block">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.</math>

Here Template:Math denotes the Bessel function of the first kind with order Template:Math. When Template:Math this gives a useful formula for the Fourier transform of a radial function.<ref>Template:Harvnb</ref> This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases Template:Math and Template:Mvar<ref>Template:Harvnb</ref> allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.

Restriction problems

In higher dimensions it becomes interesting to study restriction 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 general class of square integrable functions. As such, the restriction of the Fourier transform of an Template:Math function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in Template:Math for Template:Math. It is possible in some cases to define the restriction of a Fourier transform to a set Template:Mvar, provided Template:Mvar has non-zero curvature. The case when Template:Mvar is the unit sphere in Template:Math is of particular interest. In this case the Tomas–Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in Template:Math is a bounded operator on Template:Math provided Template:Math.

One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets Template:Math indexed by Template:Math: such as balls of radius Template:Mvar centered at the origin, or cubes of side Template:Math. For a given integrable function Template:Mvar, consider the function Template:Mvar defined by: <math display="block">f_R(x) = \int_{E_R}\hat{f}(\xi) e^{i 2\pi x\cdot\xi}\, d\xi, \quad x \in \mathbb{R}^n.</math>

Suppose in addition that Template:Math. For Template:Math and Template:Math, if one takes Template:Math, then Template:Mvar converges to Template:Mvar in Template:Math as Template:Mvar tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for Template:Math. In the case that Template:Mvar is taken to be a cube with side length Template:Mvar, then convergence still holds. Another natural candidate is the Euclidean ball Template:Math. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in Template:Math. For Template:Math it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless Template:Math.<ref name="Duoandikoetxea-2001" /> In fact, when Template:Math, this shows that not only may Template:Mvar fail to converge to Template:Mvar in Template:Math, but for some functions Template:Math, Template:Mvar is not even an element of Template:Math.

Fourier transform on function spaces

Template:See also The definition of the Fourier transform naturally extends from <math>L^1(\mathbb R)</math> to <math>L^1(\mathbb R^n)</math>. That is, if <math>f \in L^1(\mathbb{R}^n)</math> then the Fourier transform <math>\mathcal{F}:L^1(\mathbb{R}^n) \to L^\infty(\mathbb{R}^n)</math> is given by <math display="block">f(x)\mapsto \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x)e^{-i 2\pi \xi\cdot x}\,dx, \quad \forall \xi \in \mathbb{R}^n.</math> This operator is bounded as <math display="block">\sup_{\xi \in \mathbb{R}^n}\left\vert\hat{f}(\xi)\right\vert \leq \int_{\mathbb{R}^n} \vert f(x)\vert \,dx,</math> which shows that its operator norm is bounded by Template:Math. The Riemann–Lebesgue lemma shows that if <math>f\in L^1(\mathbb{R}^n)</math> then its Fourier transform actually belongs to the space of continuous functions which vanish at infinity, i.e., <math>\hat{f} \in C_{0}(\mathbb{R}^n)\subset L^{\infty}(\mathbb{R}^n)</math>.Template:Sfn Furthermore, the image of <math>L^1</math> under <math>\mathcal{F}</math> is a strict subset of <math>C_{0}(\mathbb{R}^n)</math>.

Similarly to the case of one variable, the Fourier transform can be defined on <math>L^2(\mathbb R^n)</math>. The Fourier transform in <math>L^2(\mathbb R^n)</math> is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, i.e., <math display="block">\hat{f}(\xi) = \lim_{R\to\infty}\int_{|x|\le R} f(x) e^{-i 2\pi\xi\cdot x}\,dx</math> where the limit is taken in the Template:Math sense.<ref>More generally, one can take a sequence of functions that are in the intersection of Template:Math and Template:Math and that converges to Template:Mvar in the Template:Math-norm, and define the Fourier transform of Template:Mvar as the Template:Math -limit of the Fourier transforms of these functions.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Furthermore, <math>\mathcal{F}:L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)</math> is a unitary operator.Template:Sfn 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 any Template:Math we have <math display="block">\int_{\mathbb{R}^n} f(x)\mathcal{F}g(x)\,dx = \int_{\mathbb{R}^n} \mathcal{F}f(x)g(x)\,dx. </math>

In particular, the image of Template:Math is itself under the Fourier transform.

On other L^p

For <math>1<p<2</math>, the Fourier transform can be defined on <math>L^p(\mathbb R)</math> by Marcinkiewicz interpolation, which amounts to decomposing such functions into a fat tail part in Template:Math plus a fat body part in Template:Math. In each of these spaces, the Fourier transform of a function in Template:Math is in Template:Math, where Template:Math is the Hölder conjugate of Template:Mvar (by the Hausdorff–Young inequality). However, except for Template:Math, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in Template:Math for the range Template:Math requires the study of distributions.Template:Sfn In fact, it can be shown that there are functions in Template:Math with Template:Math so that the Fourier transform is not defined as a function.<ref name="Stein-Weiss-1971" />

Tempered distributions

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also One might consider enlarging the domain of the Fourier transform from <math>L^1 + L^2</math> by considering generalized functions, or distributions. A distribution on <math>\mathbb{R}^n</math> is a continuous linear functional on the space <math>C_{c}^{\infty}(\mathbb{R}^n)</math> of compactly supported smooth functions (i.e. bump functions), equipped with a suitable topology. Since <math>C_{c}^{\infty}(\mathbb{R}^n)</math> is dense in <math>L^{2}(\mathbb{R}^n)</math>, the Plancherel theorem allows one to extend the definition of the Fourier transform to general functions in <math>L^{2}(\mathbb{R}^n)</math> by continuity arguments. The strategy is then to consider the action of the Fourier transform on <math>C_{c}^{\infty}(\mathbb{R}^n)</math> and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map <math>C_{c}^{\infty}(\mathbb{R}^n)</math> to <math>C_{c}^{\infty}(\mathbb{R}^n)</math>. In fact the Fourier transform of an element in <math>C_{c}^{\infty}(\mathbb{R}^n)</math> can not vanish on an open set; see the above discussion on the uncertainty principle.Template:SfnTemplate:Sfn

The Fourier transform can also be defined for tempered distributions <math>\mathcal S'(\mathbb R^n)</math>, dual to the space of Schwartz functions <math>\mathcal S(\mathbb R^n)</math>. A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives, hence <math>C_{c}^{\infty}(\mathbb{R}^n)\subset \mathcal S(\mathbb R^n)</math> and: <math display="block">\mathcal{F}: C_{c}^{\infty}(\mathbb{R}^n) \rightarrow S(\mathbb R^n) \setminus C_{c}^{\infty}(\mathbb{R}^n).</math> The Fourier transform is an automorphism of the Schwartz space and, by duality, also an automorphism of the space of tempered distributions.<ref name="Stein-Weiss-1971" />Template:Sfn 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, let <math>f</math> and <math>g</math> be integrable functions, and let <math>\hat{f}</math> and <math>\hat{g}</math> be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,<ref name="Stein-Weiss-1971" /> <math display="block">\int_{\mathbb{R}^n}\hat{f}(x)g(x)\,dx=\int_{\mathbb{R}^n}f(x)\hat{g}(x)\,dx.</math>

Every integrable function <math>f</math> defines (induces) a distribution <math>T_f</math> by the relation <math display="block">T_f(\phi)=\int_{\mathbb{R}^n}f(x)\phi(x)\,dx,\quad \forall \phi\in\mathcal S(\mathbb R^n).</math> So it makes sense to define the Fourier transform of a tempered distribution <math>T_{f}\in\mathcal S'(\mathbb R)</math> by the duality: <math display="block">\langle \widehat T_{f}, \phi\rangle = \langle T_{f},\widehat \phi\rangle,\quad \forall \phi\in\mathcal S(\mathbb R^n).</math> Extending this to all tempered distributions <math>T</math> 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

Fourier–Stieltjes transform on measurable spaces

Template:See also The Fourier transform of a finite Borel measure Template:Mvar on Template:Math is given by the continuous function:Template:Sfn <math display="block">\hat\mu(\xi)=\int_{\mathbb{R}^n} e^{-i 2\pi x \cdot \xi}\,d\mu,</math> and called the Fourier-Stieltjes transform due to its connection with the Riemann-Stieltjes integral representation of (Radon) measures.Template:Sfn If <math>\mu</math> is the probability distribution of a random variable <math>X</math> then its Fourier–Stieltjes transform is, by definition, a characteristic function.<ref>Template:Harvnb Template:Pb The typical conventions in probability theory take Template:Math instead of Template:Math.</ref> If, in addition, the probability distribution has a probability density function, this definition is subject to the usual Fourier transform.Template:Sfn Stated more generally, when <math>\mu</math> is absolutely continuous with respect to the Lebesgue measure, i.e., <math display="block"> d\mu = f(x)dx,</math> then <math display="block">\hat{\mu}(\xi)=\hat{f}(\xi),</math> 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., the Riemann–Lebesgue lemma fails for measures.Template:Sfn

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 the Dirac measure.Template:Sfn Its Fourier transform is a constant function (whose value depends on the form of the Fourier transform used).

Locally compact abelian groups

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Fourier transform may be generalized to any locally compact abelian group, i.e., an abelian group that is also a locally compact Hausdorff space such that the group operation is continuous. If Template:Mvar is a locally compact abelian group, it has a translation invariant measure Template:Mvar, called Haar measure. For a locally compact abelian group Template:Mvar, the set of irreducible, i.e. one-dimensional, unitary representations are called its characters. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the compact-open topology on the space of all continuous functions from <math>G</math> to the circle group), the set of characters Template:Mvar is itself a locally compact abelian group, called the Pontryagin dual of Template:Mvar. For a function Template:Mvar in Template:Math, its Fourier transform is defined byTemplate:Sfn <math display="block">\hat{f}(\xi) = \int_G \xi(x)f(x)\,d\mu\quad \text{for any }\xi \in \hat{G}.</math>

The Riemann–Lebesgue lemma holds in this case; Template:Math is a function vanishing at infinity on Template:Mvar.

The Fourier transform on Template:Nobr is an example; here Template:Mvar is a locally compact abelian group, and the Haar measure Template:Mvar on Template:Mvar can be thought of as the Lebesgue measure on [0,1). Consider the representation of Template:Mvar on the complex plane Template:Mvar that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since Template:Mvar is 1-dim) <math>\{e_{k}: T \rightarrow GL_{1}(C) = C^{*} \mid k \in Z\}</math> where <math>e_{k}(x) = e^{i 2\pi kx}</math> for <math>x\in T</math>.

The character of such representation, that is the trace of <math>e_{k}(x)</math> for each <math>x\in T</math> and <math>k\in Z</math>, is <math>e^{i 2\pi kx}</math> itself. In the case of representation of finite group, the character table of the group Template:Mvar are rows of vectors such that each row is the character of one irreducible representation of Template:Mvar, and these vectors form an orthonormal basis of the space of class functions that map from Template:Mvar to Template:Mvar by Schur's lemma. Now the group Template:Mvar is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function <math>e_{k}(x)</math> of <math>x\in T,</math> and the inner product between two class functions (all functions being class functions since Template:Mvar is abelian) <math>f,g \in L^{2}(T, d\mu)</math> is defined as <math display="inline">\langle f, g \rangle = \frac{1}{|T|}\int_{[0,1)}f(y)\overline{g}(y)d\mu(y)</math> with the normalizing factor <math>|T|=1</math>. The sequence <math>\{e_{k}\mid k\in Z\}</math> is an orthonormal basis of the space of class functions <math>L^{2}(T,d\mu)</math>.

For any representation Template:Mvar of a finite group Template:Mvar, <math>\chi_{v}</math> can be expressed as the span <math display="inline">\sum_{i} \left\langle \chi_{v},\chi_{v_{i}} \right\rangle \chi_{v_{i}}</math> (<math>V_{i}</math> are the irreps of Template:Mvar), such that <math display="inline">\left\langle \chi_{v}, \chi_{v_{i}} \right\rangle = \frac{1}{|G|}\sum_{g\in G}\chi_{v}(g)\overline{\chi}_{v_{i}}(g)</math>. Similarly for <math>G = T</math> and <math>f\in L^{2}(T, d\mu)</math>, <math display="inline">f(x) = \sum_{k\in Z}\hat{f}(k)e_{k}</math>. The Pontriagin dual <math>\hat{T}</math> is <math>\{e_{k}\}(k\in Z)</math> and for <math>f \in L^{2}(T, d\mu)</math>, <math display="inline">\hat{f}(k) = \frac{1}{|T|}\int_{[0,1)}f(y)e^{-i 2\pi ky}dy</math> is its Fourier transform for <math>e_{k} \in \hat{T}</math>.

Gelfand transform

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to the Pontryagin duality map defined above.

Given an abelian locally compact Hausdorff topological group Template:Mvar, as before we consider space Template:Math, defined using a Haar measure. With convolution as multiplication, Template:Math is an abelian Banach algebra. It also has an involution * given by <math display="block">f^*(g) = \overline{f\left(g^{-1}\right)}.</math>

Taking the completion with respect to the largest possibly Template:Math-norm gives its enveloping Template:Math-algebra, called the group Template:Math-algebra Template:Math of Template:Mvar. (Any Template:Math-norm on Template:Math is bounded by the Template:Math norm, therefore their supremum exists.)

Given any abelian Template:Math-algebra Template:Mvar, the Gelfand transform gives an isomorphism between Template:Mvar and Template:Math, where Template:Math is the multiplicative linear functionals, i.e. one-dimensional representations, on Template:Mvar with the weak-* topology. The map is simply given by <math display="block">a \mapsto \bigl( \varphi \mapsto \varphi(a) \bigr)</math> It turns out that the multiplicative linear functionals of Template:Math, after suitable identification, are exactly the characters of Template:Mvar, and the Gelfand transform, when restricted to the dense subset Template:Math is the Fourier–Pontryagin transform.

Compact non-abelian groups

The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. 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.<ref>Template:Harvnb</ref> The Fourier transform on compact groups is a major tool in representation theory<ref>Template:Harvnb</ref> and non-commutative harmonic analysis.

Let Template:Mvar be a compact Hausdorff topological group. Let Template:Math denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation Template:Math on the Hilbert space Template:Math of finite dimension Template:Math for each Template:Math. If Template:Mvar is a finite Borel measure on Template:Mvar, then the Fourier–Stieltjes transform of Template:Mvar is the operator on Template:Math defined by <math display="block">\left\langle \hat{\mu}\xi,\eta\right\rangle_{H_\sigma} = \int_G \left\langle \overline{U}^{(\sigma)}_g\xi,\eta\right\rangle\,d\mu(g)</math> where Template:Math is the complex-conjugate representation of Template:Math acting on Template:Math. If Template:Mvar is absolutely continuous with respect to the left-invariant probability measure Template:Mvar on Template:Mvar, represented as <math display="block">d\mu = f \, d\lambda</math> for some Template:Math, one identifies the Fourier transform of Template:Mvar with the Fourier–Stieltjes transform of Template:Mvar.

The mapping <math display="block">\mu\mapsto\hat{\mu}</math> defines an isomorphism between the Banach space Template:Math of finite Borel measures (see rca space) and a closed subspace of the Banach space Template:Math consisting of all sequences Template:Math indexed by Template:Math of (bounded) linear operators Template:Math for which the norm <math display="block">\|E\| = \sup_{\sigma\in\Sigma}\left\|E_\sigma\right\|</math> is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of C*-algebras into a subspace of Template:Math. Multiplication on Template:Math is given by convolution of measures and the involution * defined by <math display="block">f^*(g) = \overline{f\left(g^{-1}\right)},</math> and Template:Math has a natural Template:Math-algebra structure as Hilbert space operators.

The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if Template:Math, then <math display="block">f(g) = \sum_{\sigma\in\Sigma} d_\sigma \operatorname{tr}\left(\hat{f}(\sigma)U^{(\sigma)}_g\right)</math> where the summation is understood as convergent in the Template:Math sense.

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

Alternatives

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency 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), and standing 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, notably transients, or any signal of finite extent.

As alternatives to the Fourier transform, in time–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 the short-time Fourier transform, fractional Fourier transform, Synchrosqueezing Fourier transform,<ref>Template:Cite journal</ref> or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.<ref name="Boashash-2003" />

Example

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 function <math>f(t) = \cos(2\pi\ 3 t) \ e^{-\pi t^2},</math> which is a 3 Hz cosine wave (the first term) shaped by a Gaussian envelope function (the second term) that smoothly turns the wave on and off. The next 2 images show the product <math>f(t) e^{-i 2\pi 3 t},</math> 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 of <math>f(t)</math> and <math>\operatorname{Re}(e^{-i 2\pi 3 t})</math> oscillate at the same rate and in phase, whereas <math>f(t)</math> and <math>\operatorname{Im} (e^{-i 2\pi 3 t})</math> 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.

File:Onfreq.png

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 function <math> f(t).</math>

• Offfreq i2p.svg

  Real and imaginary parts of the integrand for its Fourier transform at +5 Hz.

• Fourier transform of oscillating function.svg

  Magnitude of its Fourier transform, with +3 and +5 Hz labeled.

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

Applications

File:Commutative diagram illustrating problem solving via the Fourier transform.svg

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 of differentiation in the time domain corresponds to multiplication by the frequency,<ref group="note">Up to an imaginary constant factor whose magnitude depends on what Fourier transform convention is used.</ref> so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution 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

Perhaps the most important use of the Fourier transformation is to solve partial 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 <math display="block">\frac{\partial^2 y(x, t)}{\partial^2 x} = \frac{\partial y(x, t)}{\partial t}.</math> The example we will give, a slightly more difficult one, is the wave equation in one dimension, <math display="block">\frac{\partial^2y(x, t)}{\partial^2 x} = \frac{\partial^2y(x, t)}{\partial^2t}.</math>

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" <math display="block">y(x, 0) = f(x),\qquad \frac{\partial y(x, 0)}{\partial t} = g(x).</math>

Here, Template:Mvar and Template:Mvar 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 solutions Template:Mvar 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 transform Template:Mvar 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. After Template:Mvar is determined, we can apply the inverse Fourier transformation to find Template:Mvar.

Fourier's method is as follows. First, note that any function of the forms <math display="block"> \cos\bigl(2\pi\xi(x\pm t)\bigr) \text{ or } \sin\bigl(2\pi\xi(x \pm t)\bigr)</math> satisfies the wave equation. These are called the elementary solutions.

Second, note that therefore any integral <math display="block">\begin{align}

 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{align}</math> satisfies the wave equation for arbitrary Template:Math. 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 of Template:Math and Template:Math in the variable Template:Mvar.

The third step is to examine how to find the specific unknown coefficient functions Template:Math and Template:Math that will lead to Template:Mvar satisfying the boundary conditions. We are interested in the values of these solutions at Template:Math. So we will set Template:Math. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable Template:Mvar) of both sides and obtain <math display="block"> 2\int_{-\infty}^\infty y(x,0) \cos(2\pi\xi x) \, dx = a_+ + a_-</math> and <math display="block">2\int_{-\infty}^\infty y(x,0) \sin(2\pi\xi x) \, dx = b_+ + b_-.</math>

Similarly, taking the derivative of Template:Mvar with respect to Template:Mvar and then applying the Fourier sine and cosine transformations yields <math display="block">2\int_{-\infty}^\infty \frac{\partial y(u,0)}{\partial t} \sin (2\pi\xi x) \, dx = (2\pi\xi)\left(-a_+ + a_-\right)</math> and <math display="block">2\int_{-\infty}^\infty \frac{\partial y(u,0)}{\partial t} \cos (2\pi\xi x) \, dx = (2\pi\xi)\left(b_+ - b_-\right).</math>

These are four linear equations for the four unknowns Template:Math and Template:Math, 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 by Template:Mvar, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter Template:Mvar. 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 functions Template:Mvar and Template:Mvar. 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 functions Template:Math and Template:Math in terms of the given boundary conditions Template:Mvar and Template:Mvar.

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 both Template:Mvar and Template:Mvar rather than operate as Fourier did, who only transformed in the spatial variables. Note that Template:Mvar must be considered in the sense of a distribution since Template:Math is not going to be Template:Math: 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 in Template:Mvar to multiplication by Template:Math and differentiation with respect to Template:Mvar to multiplication by Template:Math where Template:Mvar is the frequency. Then the wave equation becomes an algebraic equation in Template:Mvar: <math display="block">\xi^2 \hat y (\xi, f) = f^2 \hat y (\xi, f).</math> This is equivalent to requiring Template:Math unless Template:Math. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously Template:Math 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) conic Template:Math.

We may as well consider the distributions supported on the conic that are given by distributions of one variable on the line Template:Math plus distributions on the line Template:Math as follows: if Template:Mvar is any test function, <math display="block">\iint \hat y \phi(\xi,f) \, d\xi \, df = \int s_+ \phi(\xi,\xi) \, d\xi + \int s_- \phi(\xi,-\xi) \, d\xi,</math> where Template:Math, and Template:Math, are distributions of one variable.

Then Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put Template:Math, which is clearly of polynomial growth): <math display="block"> y(x,0) = \int\bigl\{s_+(\xi) + s_-(\xi)\bigr\} e^{i 2\pi \xi x+0} \, d\xi </math> and <math display="block"> \frac{\partial y(x,0)}{\partial t} = \int\bigl\{s_+(\xi) - s_-(\xi)\bigr\} i 2\pi \xi e^{i 2\pi\xi x+0} \, d\xi.</math>

Now, as before, applying the one-variable Fourier transformation in the variable Template:Mvar to these functions of Template:Mvar yields two equations in the two unknown distributions Template:Math (which can be taken to be ordinary functions if the boundary conditions are Template:Math or Template:Math).

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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, 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 in magnetic resonance imaging (MRI) and mass spectrometry.

Quantum mechanics

The Fourier transform is useful in quantum mechanics in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of complementary variables, connected by the Heisenberg uncertainty principle. For example, in one dimension, the spatial variable Template:Mvar of, say, a particle, can only be measured by the quantum mechanical "position operator" at the cost of losing information about the momentum Template:Mvar of the particle. Therefore, the physical state of the particle can either be described by a function, called "the wave function", of Template:Mvar or by a function of Template:Mvar but not by a function of both variables. The variable Template:Mvar is called the conjugate variable to Template:Mvar. 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 both Template:Mvar and Template:Mvar simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a Template:Mvar-axis and a Template:Mvar-axis called the phase 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, the Template:Mvar-axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis. Nevertheless, choosing the Template:Mvar-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 that <math display="block">\phi(p) = \int dq\, \psi (q) e^{-i pq/h} ,</math> or, equivalently, <math display="block">\psi(q) = \int dp \, \phi (p) e^{i pq/h}.</math>

Physically realisable states are Template:Math, and so by the Plancherel theorem, their Fourier transforms are also Template:Math. (Note that since Template:Mvar is in units of distance and Template:Mvar is in units of momentum, the presence of the Planck constant in the exponent makes the exponent dimensionless, 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 Heisenberg uncertainty principle.

The other use of the Fourier transform in both quantum mechanics and quantum field theory is to solve the applicable wave equation. In non-relativistic quantum mechanics, the Schrödinger equation for a time-varying wave function in one-dimension, not subject to external forces, is <math display="block">-\frac{\partial^2}{\partial x^2} \psi(x,t) = i \frac h{2\pi} \frac{\partial}{\partial t} \psi(x,t).</math>

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

In the presence of a potential, given by the potential energy function Template:Math, the equation becomes <math display="block">-\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).</math>

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 of Template:Mvar given its values for Template:Math. 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, <math display="block">\left (\frac{\partial^2}{\partial x^2} +1 \right) \psi(x,t) = \frac{\partial^2}{\partial t^2} \psi(x,t).</math>

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, the number operator of the quantum harmonic oscillator can be interpreted, for example via the Mehler kernel, as the generator of the Fourier transform <math>\mathcal{F}</math>.<ref name="auto"/>

Signal processing

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 function Template:Mvar of a function Template:Mvar is defined by <math display="block">R_f (\tau) = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T f(t) f(t+\tau) \, dt. </math>

This function is a function of the time-lag Template:Mvar elapsing between the values of Template:Mvar to be correlated.

For most functions Template:Mvar that occur in practice, Template:Mvar is a bounded even function of the time-lag Template:Mvar and for typical noisy signals it turns out to be uniformly continuous with a maximum at Template:Math.

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 of Template:Mvar separated by a time lag. This is a way of searching for the correlation of Template:Mvar with its own past. It is useful even for other statistical tasks besides the analysis of signals. For example, if Template:Math represents the temperature at time Template:Mvar, one expects a strong correlation with the temperature at a time lag of 24 hours.

It possesses a Fourier transform, <math display="block"> P_f(\xi) = \int_{-\infty}^\infty R_f (\tau) e^{-i 2\pi \xi\tau} \, d\tau. </math>

This Fourier transform is called the power spectral density function of Template:Mvar. (Unless all periodic components are first filtered out from Template:Mvar, this integral will diverge, but it is easy to filter out such periodicities.)

The power spectrum, as indicated by this density function Template:Mvar, measures the amount of variance contributed to the data by the frequency Template:Mvar. 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

Other common notations for <math>\hat f(\xi)</math> include: <math display="block">\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\}.</math>

In the sciences and engineering it is also common to make substitutions like these: <math display="block">\xi \rightarrow f, \quad x \rightarrow t, \quad f \rightarrow x,\quad \hat f \rightarrow X. </math>

So the transform pair <math>f(x)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \hat{f}(\xi)</math> can become <math>x(t)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ X(f)</math>

A disadvantage of the capital letter notation is when expressing a transform such as <math>\widehat{f\cdot g}</math> or <math>\widehat{f'},</math> which become the more awkward <math>\mathcal{F}\{f\cdot g\}</math> and <math>\mathcal{F} \{ f' \} . </math>

In some contexts such as particle physics, the same symbol <math>f</math> may be used for both for a function as well as it Fourier transform, with the two only distinguished by their argument I.e. <math>f(k_1 + k_2)</math> would refer to the Fourier transform because of the momentum argument, while <math>f(x_0 + \pi \vec r)</math> would refer to the original function because of the positional argument. Although tildes may be used as in <math>\tilde{f}</math> to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more Lorentz invariant form, such as <math>\tilde{dk} = \frac{dk}{(2\pi)^32\omega}</math>, so care must be taken. Similarly, <math>\hat f</math> often denotes the Hilbert transform of <math>f</math>.

The interpretation of the complex function Template:Math may be aided by expressing it in polar coordinate form <math display="block">\hat f(\xi) = A(\xi) e^{i\varphi(\xi)}</math> in terms of the two real functions Template:Math and Template:Math where: <math display="block">A(\xi) = \left|\hat f(\xi)\right|,</math> is the amplitude and <math display="block">\varphi (\xi) = \arg \left( \hat f(\xi) \right), </math> is the phase (see arg function).

Then the inverse transform can be written: <math display="block">f(x) = \int _{-\infty}^\infty A(\xi)\ e^{ i\bigl(2\pi \xi x +\varphi (\xi)\bigr)}\,d\xi,</math> which is a recombination of all the frequency components of Template:Math. Each component is a complex sinusoid of the form Template:Math whose amplitude is Template:Math and whose initial phase angle (at Template:Math) is Template:Math.

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted Template:Mathcal and Template:Math is used to denote the Fourier transform of the function Template:Mvar. This mapping is linear, which means that Template:Mathcal 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 function Template:Math) can be used to write Template:Math instead of Template:Math. 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 value Template:Mvar for its variable, and this is denoted either as Template:Math or as Template:Math. Notice that in the former case, it is implicitly understood that Template:Mathcal is applied first to Template:Mvar and then the resulting function is evaluated at Template:Mvar, not the other way around.

In mathematics and various applied sciences, it is often necessary to distinguish between a function Template:Mvar and the value of Template:Mvar when its variable equals Template:Mvar, denoted Template:Math. This means that a notation like Template:Math formally can be interpreted as the Fourier transform of the values of Template:Mvar at Template:Mvar. 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, <math display="block">\mathcal F\bigl( \operatorname{rect}(x) \bigr) = \operatorname{sinc}(\xi)</math> is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or <math display="block">\mathcal F\bigl(f(x + x_0)\bigr) = \mathcal F\bigl(f(x)\bigr)\, e^{i 2\pi x_0 \xi}</math> 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 of Template:Mvar, not of Template:Math.

As discussed above, the characteristic function of a random variable is the same as the Fourier–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 defined <math display="block">E\left(e^{it\cdot X}\right)=\int e^{it\cdot x} \, d\mu_X(x).</math>

As in the case of the "non-unitary angular frequency" convention above, the factor of 2Template:Pi 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

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, <math>f(x),</math> and functions of a discrete variable (i.e. ordered pairs of <math>x</math> and <math>f</math> values). For discrete-valued <math>x,</math> the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (<math>\xi</math> or <math>\omega</math>). When the sinusoids are harmonically related (i.e. when the <math>x</math>-values are spaced at integer multiples of an interval), the transform is called discrete-time Fourier transform (DTFT).

Discrete Fourier transforms and fast Fourier transforms

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 at Template:Slink. The discrete Fourier transform (DFT), used there, is usually computed by a fast Fourier transform (FFT) algorithm.

Analytic integration of closed-form functions

Tables of closed-form Fourier transforms, such as Template:Slink and Template:Slink, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (<math>\xi</math> or <math>\omega</math>).<ref name="Zwillinger-2014">Template:Harvnb</ref> When mathematically possible, this provides a transform for a continuum of frequency values.

Many computer algebra systems such as Matlab and Mathematica that are capable of symbolic integration are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform of Template:Math one might enter the command <syntaxhighlight lang="text" class="" style="" inline="1">integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf</syntaxhighlight> into Wolfram Alpha.<ref group=note>The direct command <syntaxhighlight lang="text" class="" style="" inline="1">fourier transform of cos(6*pi*t) exp(−pi*t^2)</syntaxhighlight> would also work for Wolfram Alpha, although the options for the convention (see Template:Section link) must be changed away from the default option, which is actually equivalent to <syntaxhighlight lang="text" class="" style="" inline="1">integrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf</syntaxhighlight>.</ref>

Numerical integration of closed-form continuous functions

Discrete sampling of the Fourier transform can also be done by numerical integration of the definition at each value of frequency for which transform is desired.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> The numerical integration approach works on a much broader class of functions than the analytic approach.

Numerical integration of a series of ordered pairs

If the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.<ref>Template:Harvnb</ref> The DTFT is a common subcase of this more general situation.

Tables of important Fourier transforms

The following tables record some closed-form Fourier transforms. For functions Template:Math and Template:Math denote their Fourier transforms by Template:Math and Template:Math. 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

The Fourier transforms in this table may be found in Template:Harvtxt or Template:Harvtxt.

	Function	Fourier transform Template:Br unitary, ordinary frequency	Fourier transform Template:Br unitary, angular frequency	Fourier transform Template:Br non-unitary, angular frequency	Remarks
	<math> f(x)\,</math>	<math>\begin{align} &\widehat{f}(\xi) \triangleq \widehat {f_1}(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>	<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_2}(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_3}(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	Definitions
101	<math> a\, f(x) + b\, g(x)\,</math>	<math> a\, \widehat{f}(\xi) + b\, \widehat{g}(\xi)\,</math>	<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>	<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>	Linearity
102	<math> f(x - a)\,</math>	<math> e^{-i 2\pi \xi a} \widehat{f}(\xi)\,</math>	<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>	<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>	Shift in time domain
103	<math> f(x)e^{iax}\,</math>	<math> \widehat{f} \left(\xi - \frac{a}{2\pi}\right)\,</math>	<math> \widehat{f}(\omega - a)\,</math>	<math> \widehat{f}(\omega - a)\,</math>	Shift in frequency domain, dual of 102
104	<math> f(a x)\,</math>	a|} \widehat{f}\left( \frac{\xi}{a} \right)\,</math>	a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>	a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>	Scaling in the time domain. If Template:Math is large, then Template:Math is concentrated around 0 andTemplate:Br<math> \frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\,</math>Template:Brspreads out and flattens.
105	<math> \widehat {f_n}(x)\,</math>	<math> \widehat {f_1}(x) \ \stackrel{\mathcal{F}_1}{\longleftrightarrow}\ f(-\xi)\,</math>	<math> \widehat {f_2}(x) \ \stackrel{\mathcal{F}_2}{\longleftrightarrow}\ f(-\omega)\,</math>	<math> \widehat {f_3}(x) \ \stackrel{\mathcal{F}_3}{\longleftrightarrow}\ 2\pi f(-\omega)\,</math>	The same transform is applied twice, but x replaces the frequency variable (ξ or ω) after the first transform.
106	<math> \frac{d^n f(x)}{dx^n}\,</math>	<math> (i 2\pi \xi)^n \widehat{f}(\xi)\,</math>	<math> (i\omega)^n \widehat{f}(\omega)\,</math>	<math> (i\omega)^n \widehat{f}(\omega)\,</math>	nTemplate:Superscript-order derivative. As Template:Math is a Schwartz function
106.5	<math>\int_{-\infty}^{x} f(\tau) d \tau</math>	<math>\frac{\widehat{f}(\xi)}{i 2 \pi \xi} + C \, \delta(\xi)</math>	<math>\frac{\widehat{f} (\omega)}{i\omega} + \sqrt{2 \pi} C \delta(\omega)</math>	<math>\frac{\widehat{f} (\omega)}{i\omega} + 2 \pi C \delta(\omega)</math>	citation	CitationClass=web }}</ref> Note: <math>\delta</math> is the Dirac delta function and <math>C</math> is the average (DC) value of <math>f(x)</math> such that <math>\int_{-\infty}^\infty (f(x) - C) \, dx = 0</math>
107	<math> x^n f(x)\,</math>	<math> \left (\frac{i}{2\pi}\right)^n \frac{d^n \widehat{f}(\xi)}{d\xi^n}\,</math>	<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>	<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>	This is the dual of 106
108	<math> (f * g)(x)\,</math>	<math> \widehat{f}(\xi) \widehat{g}(\xi)\,</math>	<math> \sqrt{2\pi}\ \widehat{f}(\omega) \widehat{g}(\omega)\,</math>	<math> \widehat{f}(\omega) \widehat{g}(\omega)\,</math>	The notation Template:Math denotes the convolution of Template:Mvar and Template:Mvar — this rule is the convolution theorem
109	<math> f(x) g(x)\,</math>	<math> \left(\widehat{f} * \widehat{g}\right)(\xi)\,</math>	<math> \frac{1}\sqrt{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>	<math> \frac{1}{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>	This is the dual of 108
110	For Template:Math purely real	<math> \widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}\,</math>	<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>	<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>	Hermitian symmetry. Template:Math indicates the complex conjugate.
113	For Template:Math purely imaginary	<math> \widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}\,</math>	<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>	<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>	Template:Math indicates the complex conjugate.
114	<math> \overline{f(x)}</math>	<math> \overline{\widehat{f}(-\xi)}</math>	<math> \overline{\widehat{f}(-\omega)}</math>	<math> \overline{\widehat{f}(-\omega)}</math>	Complex conjugation, generalization of 110 and 113
115	<math> f(x) \cos (a x)</math>	<math> \frac{ \widehat{f}\left(\xi - \frac{a}{2\pi}\right)+\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2}</math>	<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\,</math>	<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}</math>	This follows from rules 101 and 103 using Euler's formula:Template:Br<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
116	<math> f(x)\sin( ax)</math>	<math> \frac{\widehat{f}\left(\xi-\frac{a}{2\pi}\right)-\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>	<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>	<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>	This follows from 101 and 103 using Euler's formula:Template:Br<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>

Square-integrable functions, one-dimensional

The Fourier transforms in this table may be found in Template:Harvtxt, Template:Harvtxt, or Template:Harvtxt.

	Function	Fourier transform Template:Br unitary, ordinary frequency	Fourier transform Template:Br unitary, angular frequency	Fourier transform Template:Br non-unitary, angular frequency	Remarks
	<math> f(x)\,</math>	<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>	<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	Definitions
Template:Anchor 201	<math> \operatorname{rect}(a x) \,</math>	a|}\, \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>	<math> \frac{1}{\sqrt{2 \pi a^2}}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>	a|}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>	The rectangular pulse and the normalized sinc function, here defined as Template:Math
202	<math> \operatorname{sinc}(a x)\,</math>	a|}\, \operatorname{rect}\left(\frac{\xi}{a} \right)\,</math>	<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>	a|}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>	Dual of rule 201. The rectangular function is an ideal low-pass filter, and the sinc function is the non-causal impulse response of such a filter. The sinc function is defined here as Template:Math
203	<math> \operatorname{sinc}^2 (a x)</math>	a|}\, \operatorname{tri} \left( \frac{\xi}{a} \right) </math>	<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>	a|}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>	The function Template:Math is the triangular function
204	<math> \operatorname{tri} (a x)</math>	a|}\, \operatorname{sinc}^2 \left( \frac{\xi}{a} \right) \,</math>	<math> \frac{1}{\sqrt{2\pi a^2}} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>	a|} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>	Dual of rule 203.
205	<math> e^{- a x} u(x) \,</math>	<math> \frac{1}{a + i 2\pi \xi}</math>	<math> \frac{1}{\sqrt{2 \pi} (a + i \omega)}</math>	<math> \frac{1}{a + i \omega}</math>	The function Template:Math is the Heaviside unit step function and Template:Math.
206	<math> e^{-\alpha x^2}\,</math>	<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{(\pi \xi)^2}{\alpha}}</math>	<math> \frac{1}{\sqrt{2 \alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>	<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>	This shows that, for the unitary Fourier transforms, the Gaussian function Template:Math is its own Fourier transform for some choice of Template:Mvar. For this to be integrable we must have Template:Math.
208	x|} \,</math>	<math> \frac{2 a}{a^2 + 4 \pi^2 \xi^2} </math>	<math> \sqrt{\frac{2}{\pi}} \, \frac{a}{a^2 + \omega^2} </math>	<math> \frac{2a}{a^2 + \omega^{2}} </math>	For Template:Math. That is, the Fourier transform of a two-sided decaying exponential function is a Lorentzian function.
209	<math> \operatorname{sech}(a x) \,</math>	<math> \frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} \xi \right)</math>	<math> \frac{1}{a}\sqrt{\frac{\pi}{2}} \operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>	<math> \frac{\pi}{a}\operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>	Hyperbolic secant is its own Fourier transform
210	<math> e^{-\frac{a^2 x^2}2} H_n(a x)\,</math>	<math> \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)</math>	<math> \frac{(-i)^n}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a\right)</math>	<math> \frac{(-i)^n \sqrt{2\pi}}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a \right)</math>	Template:Math is the Template:Mvarth-order Hermite polynomial. If Template:Math then the Gauss–Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for Template:Math.

Distributions, one-dimensional

The Fourier transforms in this table may be found in Template:Harvtxt or Template:Harvtxt.

	Function	Fourier transform Template:Br unitary, ordinary frequency	Fourier transform Template:Br unitary, angular frequency	Fourier transform Template:Br non-unitary, angular frequency	Remarks
	<math> f(x)\,</math>	<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>	<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>	Definitions
301	<math> 1</math>	<math> \delta(\xi)</math>	<math> \sqrt{2\pi}\, \delta(\omega)</math>	<math> 2\pi\delta(\omega)</math>	The distribution Template:Math denotes the Dirac delta function.
302	<math> \delta(x)\,</math>	<math> 1</math>	<math> \frac{1}{\sqrt{2\pi}}\,</math>	<math> 1</math>	Dual of rule 301.
303	<math> e^{i a x}</math>	<math> \delta\left(\xi - \frac{a}{2\pi}\right)</math>	<math> \sqrt{2 \pi}\, \delta(\omega - a)</math>	<math> 2 \pi\delta(\omega - a)</math>	This follows from 103 and 301.
304	<math> \cos (a x)</math>	<math> \frac{ \delta\left(\xi - \frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2}</math>	<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)+\delta(\omega+a)}{2}</math>	<math> \pi\left(\delta(\omega-a)+\delta(\omega+a)\right)</math>	This follows from rules 101 and 303 using Euler's formula:Template:Br<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
305	<math> \sin( ax)</math>	<math> \frac{\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>	<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)-\delta(\omega+a)}{2i}</math>	<math> -i\pi\bigl(\delta(\omega-a)-\delta(\omega+a)\bigr)</math>	This follows from 101 and 303 usingTemplate:Br<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>
306	<math> \cos \left( a x^2 \right) </math>	<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>	<math> \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>	<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right) </math>	This follows from 101 and 207 usingTemplate:Br<math>\cos(a x^2) = \frac{e^{i a x^2} + e^{-i a x^2}}{2}.</math>
307	<math> \sin \left( a x^2 \right) </math>	<math> - \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>	<math> \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>	<math> -\sqrt{\frac{\pi}{a}}\sin \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right)</math>	This follows from 101 and 207 usingTemplate:Br<math>\sin(a x^2) = \frac{e^{i a x^2} - e^{-i a x^2}}{2i}.</math>
308	<math> e^{-\pi i\alpha x^2}\,</math>	<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\pi \xi^2}{\alpha}}</math>	<math> \frac{1}{\sqrt{2\pi \alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>	<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>	Here it is assumed <math>\alpha</math> is real. For the case that alpha is complex see table entry 206 above.
309	<math> x^n\,</math>	<math> \left(\frac{i}{2\pi}\right)^n \delta^{(n)} (\xi)</math>	<math> i^n \sqrt{2\pi} \delta^{(n)} (\omega)</math>	<math> 2\pi i^n\delta^{(n)} (\omega)</math>	Here, Template:Mvar is a natural number and Template:Math is the Template:Mvarth distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all polynomials.
310	<math> \delta^{(n)}(x)</math>	<math> (i 2\pi \xi)^n</math>	<math> \frac{(i\omega)^n}{\sqrt{2\pi}} </math>	<math> (i\omega)^n</math>	Dual of rule 309. Template:Math is the Template:Mvarth distribution derivative of the Dirac delta function. This rule follows from 106 and 302.
311	<math> \frac{1}{x}</math>	<math> -i\pi\sgn(\xi)</math>	<math> -i\sqrt{\frac{\pi}{2}}\sgn(\omega)</math>	<math> -i\pi\sgn(\omega)</math>	Here Template:Math is the sign function. Note that Template:Math is not a distribution. It is necessary to use the Cauchy principal value when testing against Schwartz functions. This rule is useful in studying the Hilbert transform.
312	<math>\begin{align} &\frac{1}{x^n} \\ &:= \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log |x| \end{align}</math>	<math> -i\pi \frac{(-i 2\pi \xi)^{n-1}}{(n-1)!} \sgn(\xi)</math>	<math> -i\sqrt{\frac{\pi}{2}}\, \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>	<math> -i\pi \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>	Template:Math is the homogeneous distribution defined by the distributional derivativeTemplate:Br<math>\frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log|x|</math>
313	x|^\alpha</math>	2\pi\xi|^{\alpha+1}}</math>	\omega|^{\alpha+1}} </math>	\omega|^{\alpha+1}} </math>	This formula is valid for Template:Math. For Template:Math some singular terms arise at the origin that can be found by differentiating 320. If Template:Math, then Template:Math is a locally integrable function, and so a tempered distribution. The function Template:Math 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 Template:Math for Template:Math (See homogeneous distribution.)
	x|}} </math>	\xi|}} </math>	\omega|}}</math>	\omega|}} </math>	Special case of 313.
314	<math> \sgn(x)</math>	<math> \frac{1}{i\pi \xi}</math>	<math> \sqrt{\frac{2}{\pi}} \frac{1}{i\omega } </math>	<math> \frac{2}{i\omega }</math>	The dual of rule 311. This time the Fourier transforms need to be considered as a Cauchy principal value.
315	<math> u(x)</math>	<math> \frac{1}{2}\left(\frac{1}{i \pi \xi} + \delta(\xi)\right)</math>	<math> \sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>	<math> \pi\left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>	The function Template:Math is the Heaviside unit step function; this follows from rules 101, 301, and 314.
316	<math> \sum_{n=-\infty}^{\infty} \delta (x - n T)</math>	<math> \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( \xi -\frac{k }{T}\right)</math>	<math> \frac{\sqrt{2\pi }}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>	<math> \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>	This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact thatTemplate:Br<math>\begin{align} & \sum_{n=-\infty}^{\infty} e^{inx} \\ = {}& 2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k) \end{align}</math>Template:Bras distributions.
317	<math> J_0 (x)</math>	<math> \frac{2\, \operatorname{rect}(\pi\xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>	<math> \sqrt{\frac{2}{\pi}} \, \frac{\operatorname{rect}\left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>	<math> \frac{2\,\operatorname{rect}\left(\frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}}</math>	The function Template:Math is the zeroth order Bessel function of first kind.
318	<math> J_n (x)</math>	<math> \frac{2 (-i)^n T_n (2 \pi \xi) \operatorname{rect}(\pi \xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>	<math> \sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>	<math> \frac{2(-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>	This is a generalization of 317. The function Template:Math is the Template:Mvarth order Bessel function of first kind. The function Template:Math is the Chebyshev polynomial of the first kind.
319	x \right|</math>	\xi \right|} - \gamma \delta \left( \xi \right) </math>	\omega \right|} - \sqrt{2 \pi} \gamma \delta \left( \omega \right) </math>	\omega \right|} - 2 \pi \gamma \delta \left( \omega \right) </math>	Template:Mvar is the Euler–Mascheroni constant. It is necessary to use a finite part integral when testing Template:Math or Template:Mathagainst Schwartz functions. The details of this might change the coefficient of the delta function.
320	<math> \left( \mp ix \right)^{-\alpha}</math>	<math> \frac{\left(2\pi\right)^\alpha}{\Gamma\left(\alpha\right)}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha-1} </math>	<math> \frac{\sqrt{2\pi}}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>	<math> \frac{2\pi}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>	This formula is valid for Template:Math. Use differentiation to derive formula for higher exponents. Template:Mvar is the Heaviside function.

Two-dimensional functions

	Function	Fourier transform Template:Br unitary, ordinary frequency	Fourier transform Template:Br unitary, angular frequency	Fourier transform Template:Br non-unitary, angular frequency	Remarks
400	<math> f(x,y)</math>	<math>\begin{align}& \hat{f}(\xi_x, \xi_y)\triangleq \\ & \iint f(x,y) e^{-i 2\pi(\xi_x x+\xi_y y)}\,dx\,dy \end{align}</math>	<math>\begin{align}& \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{align}</math>	<math>\begin{align}& \hat{f}(\omega_x,\omega_y)\triangleq \\ & \iint f(x,y) e^{-i(\omega_x x+\omega_y y)}\, dx\,dy \end{align}</math>	The variables Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar are real numbers. The integrals are taken over the entire plane.
401	<math> e^{-\pi\left(a^2x^2+b^2y^2\right)}</math>	ab|} e^{-\pi\left(\frac{\xi_x^2}{a^2} + \frac{\xi_y^2}{b^2}\right)}</math>	ab|} e^{-\frac{1}{4\pi}\left(\frac{\omega_x^2}{a^2} + \frac{\omega_y^2}{b^2}\right)}</math>	ab|} e^{-\frac{1}{4\pi}\left(\frac{\omega_x^2}{a^2} + \frac{\omega_y^2}{b^2}\right)}</math>	Both functions are Gaussians, which may not have unit volume.
402	<math> \operatorname{circ}\left(\sqrt{x^2+y^2}\right)</math>	<math> \frac{J_1\left(2 \pi \sqrt{\xi_x^2+\xi_y^2}\right)}{\sqrt{\xi_x^2+\xi_y^2}}</math>	<math> \frac{J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}</math>	<math> \frac{2\pi J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}</math>	The function is defined by Template:Math for Template:Math, and is 0 otherwise. The result is the amplitude distribution of the Airy disk, and is expressed using Template:Math (the order-1 Bessel function of the first kind).<ref>Template:Harvnb</ref>
403	<math> \frac{1}{\sqrt{x^2+y^2}}</math>	<math> \frac{1}{\sqrt{\xi_x^2+\xi_y^2}}</math>	<math> \frac{1}{\sqrt{\omega_x^2+\omega_y^2}}</math>	<math> \frac{2\pi}{\sqrt{\omega_x^2+\omega_y^2}}</math>	This is the Hankel transform of Template:Math, a 2-D Fourier "self-transform".<ref>Template:Harvnb</ref>
404	<math> \frac{i}{x+i y}</math>	<math> \frac{1}{\xi_x+i\xi_y}</math>	<math> \frac{1}{\omega_x+i\omega_y}</math>	<math> \frac{2\pi}{\omega_x+i\omega_y}</math>

Formulas for general Template:Math-dimensional functions

	Function	Fourier transform Template:Br unitary, ordinary frequency	Fourier transform Template:Br unitary, angular frequency	Fourier transform Template:Br non-unitary, angular frequency	Remarks
500	<math> f(\mathbf x)\,</math>	<math>\begin{align} &\hat{f_1}(\boldsymbol \xi) \triangleq \\ &\int_{\mathbb{R}^n}f(\mathbf x) e^{-i 2\pi \boldsymbol \xi \cdot \mathbf x }\, d \mathbf x \end{align}</math>	<math>\begin{align} &\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{align}</math>	<math>\begin{align} &\hat{f_3}(\boldsymbol \omega) \triangleq \\ &\int_{\mathbb{R}^n}f(\mathbf x) e^{-i \boldsymbol \omega \cdot \mathbf x}\, d \mathbf x \end{align}</math>
501	\mathbf x|)\left(1-|\mathbf x|^2\right)^\delta</math>	\boldsymbol \xi|^{\frac{n}{2} + \delta}} J_{\frac{n}{2}+\delta}(2\pi|\boldsymbol \xi|)</math>	\boldsymbol \omega\right|^{\frac{n}{2}+\delta}} J_{\frac{n}{2}+\delta}(|\boldsymbol \omega|)</math>	\frac{\boldsymbol \omega}{2\pi}\right|^{-\frac{n}{2}-\delta} J_{\frac{n}{2}+\delta}(\!|\boldsymbol \omega|\!)</math>	The function Template:Math is the indicator function of the interval Template:Math. The function Template:Math is the gamma function. The function Template:Math is a Bessel function of the first kind, with order Template:Math. Taking Template:Math and Template:Math produces 402.<ref>Template:Harvnb</ref>
502	\mathbf x|^{-\alpha}, \quad 0 < \operatorname{Re} \alpha < n.</math>	\boldsymbol \xi|^{-(n - \alpha)}</math>	\boldsymbol \omega|^{-(n - \alpha)}</math>	\boldsymbol \omega|^{-(n - \alpha)}</math>	See Riesz potential where the constant is given byTemplate:Br<math>c_{n, \alpha} = \pi^\frac{n}{2} 2^\alpha \frac{\Gamma\left(\frac{\alpha}{2}\right)}{\Gamma\left(\frac{n - \alpha}{2}\right)}.</math>Template:BrThe formula also holds for all Template:Math by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See homogeneous distribution.<ref group=note>In Template:Harvnb, with the non-unitary conventions of this table, the transform of <math>|\mathbf x|^\lambda</math> is given to beTemplate:Br <math>2^{\lambda+n}\pi^{\tfrac12 n}\frac{\Gamma\left(\frac{\lambda+n}{2}\right)}{\Gamma\left(-\frac{\lambda}{2}\right)}|\boldsymbol\omega|^{-\lambda-n}</math>Template:Brfrom which this follows, with <math>\lambda=-\alpha</math>.</ref>
503	\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}</math>	<math> e^{-2\pi^2 \boldsymbol \xi^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \xi} </math>	<math> (2\pi)^{-\frac{n}{2}} e^{-\frac{1}{2} \boldsymbol \omega^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \omega} </math>	<math> e^{-\frac{1}{2} \boldsymbol \omega^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \omega} </math>	This is the formula for a multivariate normal distribution normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page, Template:Math and Template:Math
504	\mathbf x|}</math>	\boldsymbol{\xi}|^2\right)^\frac{n+1}{2}}</math>	\boldsymbol{\omega}|^2\right)^\frac{n+1}{2}}</math>	\boldsymbol{\omega}|^2\right)^\frac{n+1}{2}}</math>	Here<ref>Template:Harvnb</ref>Template:Br<math>c_n=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\pi^\frac{n+1}{2}},</math> Template:Math

See also

• Analog signal processing
• Beevers–Lipson strip
• Constant-Q transform
• Discrete Fourier transform
• DFT matrix
• Fast Fourier transform
• Fourier integral operator
• Fourier inversion theorem
• Fourier multiplier
• Fourier series
• Fourier sine transform
• Fourier–Deligne transform
• Fourier–Mukai transform
• Fractional Fourier transform
• Indirect Fourier transform
• Integral transform
  • Hankel transform
  • Hartley transform
• Laplace transform
• Least-squares spectral analysis
• Linear canonical transform
• List of Fourier-related transforms
• Mellin transform
• Multidimensional transform
• NGC 4622, especially the image NGC 4622 Fourier transform Template:Math.
• Nonlocal operator
• Quantum Fourier transform
• Quadratic Fourier transform
• Short-time Fourier transform
• Spectral density
  • Spectral density estimation
• Symbolic integration
• Time stretch dispersive Fourier transform
• Transform (mathematics)

References

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External links

• Template:Commons category-inline
• Encyclopedia of Mathematics
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:FourierTransform%7CFourierTransform.html}} |title = Fourier Transform |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Fourier Transform in Crystallography