Fourier transform

thumb|The Fourier transform applied to the waveform of a [[C major piano chord (with logarithmic horizontal (frequency) axis). The first three peaks on the left correspond to the fundamental frequencies of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches.]]

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and 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 the mathematical operation and to this complex-valued function. 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.

class=skin-invert-image|thumb|right|The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. 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. 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.

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of "position space" to a function of 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. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on Real number#Arithmetic| or , notably includes the discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group = cyclic group|) and the Fourier series or circular Fourier transform (group = circle group|, 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 Lebesgue integrable complex-valued function <math>f(x)</math> on the real line, is the complex valued function , defined by the integral

When <math>f(x)</math> is (Lebesgue) integrable over the whole real line, the above integral converges for all <math>\xi\in\mathbb R</math>, and <math>\widehat{f}(\xi)</math> is a uniformly continuous function of <math>\xi</math> which decays to zero as .

However, the Fourier transform can also be defined for (generalized) functions for which the Lebesgue integral does not make sense. Interpreting the integral suitably (e.g. as an improper integral for locally integrable functions) extends the Fourier transform to functions that are not necessarily integrable over the whole real line. More generally, the Fourier transform also applies to generalized functions like the Dirac delta (and all other tempered distributions), in which case it is defined by duality rather than an integral.

First introduced in Fourier's Analytical Theory of Heat., the corresponding inversion formula for functions satisfying sufficient regularity and decay properties is given by the Fourier inversion theorem, i.e.,

The functions <math>f</math> and <math>\widehat{f}</math> are referred to as a Fourier transform pair.  A common notation for designating transform pairs is:

<math display="block">f(x)\ \stackrel{\mathcal{F{\longleftrightarrow}\ \widehat f(\xi).</math>

For example, the Fourier transform of the delta function is the constant function :

<math display="block">\delta(x)\ \stackrel{\mathcal{F{\longleftrightarrow}\ 1.</math>

Angular frequency (ω)

When the independent variable () represents time (often denoted by ), the transform variable () represents frequency (often denoted by ). For example, if time has the unit second, then frequency has the unit hertz. The transform variable can also be written in terms of angular frequency, , with the unit radian per second.

The substitution <math>\xi = \tfrac{\omega}{2 \pi}</math> into produces this convention, where function <math>\widehat f</math> is relabeled :

<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 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.

{| class="wikitable"

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

! ordinary frequency (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>

! rowspan="2" | angular frequency (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>

{| class="wikitable"

|+ Generalization for -dimensional functions

! ordinary frequency (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>

! rowspan="2" | angular frequency (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>

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 , is well-defined for all . Furthermore, <math>\widehat f\in L^\infty\cap C_0(\mathbb R)</math> is bounded, uniformly continuous and (by the Riemann–Lebesgue lemma) vanishing at infinity. Here <math>C_0(\mathbb R)</math> denotes the space of continuous functions on <math>\mathbb R</math> that approach 0 as x approaches positive or negative 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, is no longer valid, as it was stated only under the hypothesis that <math>f(x)</math> was "sufficiently nice" (e.g., <math>f(x)</math> decays with all derivatives).

While defines the Fourier transform for (complex-valued) functions in , it is not well-defined for other integrability classes, most importantly the space of square-integrable functions . 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 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 . This extension is important in part because, unlike the case of , the Fourier transform is an automorphism of the space .

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. and 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 finite linear combinations of Gaussians are dense in , 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:

that <math>e^{-\pi x^2}</math> is its own Fourier transform; and
that the Gaussian integral .

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, such that the Fourier transform is both unitary on and an algebra homomorphism from to , without renormalizing the Lebesgue measure.

Background

History

In 1822, Fourier claimed (see ') that any function, whether continuous or discontinuous, can be expanded into a series of sines. 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

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> () 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}\\[6pt]

&= \underbrace{A e^{i (2 \pi \xi x+\theta)_{\text{polar coordinate form\\[6pt]

&= \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 . Likewise, the intuitive interpretation of 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 . 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 ').

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 . is defined . Only certain complex-valued <math> f(x)</math> have transforms . (See Analytic signal; a simple example is .)  But negative frequency is necessary to characterize all other complex-valued , found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.

For a real-valued , has the symmetry property <math>\widehat f(-\xi) = \widehat {f}^* (\xi)</math> (see ' below). This redundancy enables to distinguish <math>f(x) = \cos(2 \pi \xi_0 x)</math> from . But it cannot determine the actual sign of , 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 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 , that has a convergent Fourier series, then:

\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 , 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

The Fourier transform of an integrable function <math>f</math> can be sampled at regular intervals of arbitrary length . These samples can be deduced from one cycle of a periodic function , 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(\tfrac{k}{P})</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 ). 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 ) 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 that is the range of <math>t</math> and measured in units of , and the other that is the range of <math>\xi</math> and measured in inverse units to the units of . 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 that 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 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 that 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 in the exponent instead of , and vice versa for the inversion formula. This convention is common in modern physics 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>\widehat f(\xi)</math> is the amplitude of the wave  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 have it replaced by . In electrical engineering the letter is typically used for the imaginary unit instead of because 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 of the probability density function of a random variable of continuous type is defined without a negative sign in the exponential, and since the units of are ignored, there is no either:

<math display="block">\varphi (\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 that 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>\widehat f(\xi)</math> and <math>\widehat h(\xi)</math> respectively.

Basic properties

The Fourier transform has the following basic properties:

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>

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:

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

<math display="block">\widehat{f}(-\xi)=\bigl(\widehat f(\xi)\bigr)^*.</math>

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 the total mass or DC bias of the function.

Uniform continuity and the Riemann–Lebesgue lemma

class=skin-invert-image|thumb|The [[rectangular function is Lebesgue integrable.]]

class=skin-invert-image|thumb|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>\widehat{f}</math> of any integrable function <math>f</math> is uniformly continuous and

<math display="block">\left\|\widehat{f}\right\|_\infty \leq \left\|f\right\|_1</math>

By the Riemann–Lebesgue lemma,

<math display="block">\widehat{f}(\xi) \to 0\text{ as }|\xi| \to \infty.</math>

However, <math>\widehat{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>\widehat{f}</math> are integrable, the inverse equality

<math display="block">f(x) = \int_{-\infty}^\infty \widehat f(\xi) e^{i 2\pi x \xi} \, d\xi</math> holds for almost every . As a result, the Fourier transform is injective on Lp space|.

Plancherel theorem and Parseval's theorem

Let and be integrable, and let and be their Fourier transforms. If and are also square-integrable, then the Parseval formula follows:

<math display="block">\langle f, g\rangle_{L^{2 = \int_{-\infty}^{\infty} f(x) \overline{g(x)} \,dx = \int_{-\infty}^\infty \widehat{f}(\xi) \overline{\widehat{g}(\xi)} \,d\xi,</math>

where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that

<math display="block">\|f\|^2_{L^{2 = \int_{-\infty}^\infty \left| f(x) \right|^2\,dx = \int_{-\infty}^\infty \left| \widehat{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 . On , this extension agrees with original Fourier transform defined on , thus enlarging the domain of the Fourier transform to (and consequently to for ). 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

The Fourier transform translates between convolution and multiplication of functions. If and are integrable functions with Fourier transforms and respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms and (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 denotes the convolution operation, then:

<math display="block">\widehat{h}(\xi) = \widehat{f}(\xi)\, \widehat{g}(\xi).</math>

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

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

Cross-correlation theorem

In an analogous manner, it can be shown that if is the cross-correlation of and :

<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 is:

<math display="block">\widehat{h}(\xi) = \overline{\widehat{f}(\xi)} \, \widehat{g}(\xi).</math>

As a special case, the autocorrelation of function 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">\widehat{h}(\xi) = \overline{\widehat{f}(\xi)}\widehat{f}(\xi) = \left|\widehat{f}(\xi)\right|^2.</math>

Differentiation

Suppose is differentiable almost everywhere, and both and its derivative are integrable (in ). 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\widehat{f}(\xi).</math>

More generally, the Fourier transformation of the th derivative 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\widehat{f}(\xi).</math>

Analogously, , so .

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 " is smooth if and only if quickly falls to for ". By using the analogous rules for the inverse Fourier transform, one can also say " quickly falls to for if and only if is smooth."

Eigenfunctions

The Fourier transform is a linear transform that has eigenfunctions obeying , with .

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. In other words, every solution <math>\psi(x)</math> and its Fourier transform <math>\widehat\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 , 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.

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 , which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator. The corresponding solutions provide an important choice of an orthonormal basis for Square-integrable function| 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 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">\widehat\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 . 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 and ) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose as a direct sum of four spaces , , , and where the Fourier transform acts on simply by multiplication by .

Since the complete set of Hermite functions 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. 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. In physics, this transform was introduced by Edward Condon. This change of basis 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

<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

<math display="block">N \equiv \frac{1}{4\pi}\left(2\pi x - \frac{\partial}{\partial x}\right)\left(2\pi x + \frac{\partial}{\partial x}\right) = -\frac{1}{4\pi}\frac{\partial^2}{\partial x^2} + \pi x^2 - \frac{1}{2}.</math>

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

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 , it can be recovered from its Fourier transform . Indeed, denoting the Fourier transform operator by , so , then for suitable functions, applying the Fourier transform twice simply flips the function: , which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields , so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: . In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining the parity operator <math>\mathcal{P}</math> such that , 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 -axis and frequency as the -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 SL2(R)| 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 of square integrable complex valued functions on the real line, generated by the translations and multiplication by , . 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 ) (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional Lie group of triples , 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^{-2 i \pi x_1 \xi_2}\right).</math>

Denote the Heisenberg group by . The above procedure describes not only the group structure, but also a standard unitary representation of on a Hilbert space, which we denote by . Define the linear automorphism of by

<math display="block">J \begin{pmatrix}

x \\

\xi

\end{pmatrix} = \begin{pmatrix}

-\xi \\

\end{pmatrix}</math>

so that . This can be extended to a unique automorphism of :

<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 and are unitarily equivalent, so there is a unique intertwiner such that

This operator is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. For example, the square of the Fourier transform, , is an intertwiner associated with , and so we have is the reflection of the original function .

Complex domain

The integral for the Fourier transform

<math display="block"> \widehat f (\xi) = \int _{-\infty}^\infty e^{-2i \pi t \xi} f(t) \, dt </math>

can be studied for complex values of its argument . Depending on the properties of , this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of , or something in between.

The Paley–Wiener theorem says that is smooth (i.e., -times differentiable for all positive integers ) and compactly supported if and only if is a holomorphic function for which there exists a constant such that for any integer ,

<math display="block"> \left\vert \xi ^n \widehat f(\xi) \right\vert \leq C_n e^{2\pi a\vert\tau\vert} </math>

for some constant . (In this case, is supported on .) This can be expressed by saying that is an entire function that is rapidly decreasing in (for fixed ) and of exponential growth in (uniformly in ).

(If is not smooth, but only , a corresponding version holds with the rapid-decrease condition replaced by the appropriate condition.) The space of such functions of a complex variable is called the Paley–Wiener space. This theorem has been generalised to semisimple Lie groups.

If is supported on the half-line , then 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, under suitable integrability hypotheses, extends to a holomorphic function on the complex lower half-plane that tends to zero as goes to . A simple converse in this form is false; precise converses require additional growth or Hardy-space hypotheses.

Laplace transform

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

It may happen that a function 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 is causal and of exponential growth, i.e.,

for some constants , then

<math display="block"> \widehat f (i\tau) = \int _0^\infty e^{ 2\pi \tau t} f(t) \, dt, </math>

convergent for all , is a one-sided Laplace transform of .

The usual one-sided version of the Laplace transform is

<math display="block"> F(s) = \int_0^\infty f(t) e^{-st} \, dt.</math>

If is causal and the integrals converge, then . 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 .

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term that lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for functions and integrals with at most exponential growth in the regulated direction, 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 that 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 compatible 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 of Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of harmonic analysis.

Inversion

Still with , if <math>\widehat f</math> is complex analytic for and has sufficient decay in horizontal strips, then

<math display="block"> \int _{-\infty}^\infty \widehat f (\sigma + ia) e^{ i 2\pi (\sigma+ia) t} \, d\sigma = \int _{-\infty}^\infty \widehat f (\sigma + ib) e^{ i 2\pi (\sigma+ib) 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.

Theorem: If for , and for some constants and , then

<math display="block"> f(t) = \int_{-\infty}^\infty \widehat f(\sigma + i\tau) e^{i 2 \pi (\sigma+i\tau) t} \, d\sigma,</math>

for any , under the usual hypotheses for Fourier inversion.

This theorem implies the Mellin inversion formula for the Laplace transformation,

versions of these inversion formulas are also available.

Fourier transform on Euclidean space

The Fourier transform can be defined in any arbitrary number of dimensions . As with the one-dimensional case, there are many conventions. For an integrable function , this article takes the definition:

<math display="block">\widehat{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 and are -dimensional vectors, and is the dot product of the vectors. Alternatively, can be viewed as belonging to the dual vector space , in which case the dot product becomes the contraction of and , usually written as .

All of the basic properties listed above hold for the -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.

<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 about zero.

The uncertainty principle states that, if is absolutely continuous and the functions and are square integrable, then

<math display="block">D_0(f)D_0(\widehat{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 \widehat{f}(\xi) &= \sigma C_1 \, e^{-\pi\sigma^2\xi^2} \end{align} </math>

where is arbitrary and so that is -normalized. In other words, where is a (normalized) Gaussian function with variance , centered at zero, and its Fourier transform is a Gaussian function with variance . 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|\widehat{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.

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|\widehat{f}\right|^2\right)\ge \log\left(\frac{e}{2}\right)</math>

where is the differential entropy of the probability density function :

<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

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 for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically) 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 and 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, , and the Fourier sine transform, .

The function 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.

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree on be denoted by . The set consists of the solid spherical harmonics of degree . The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if for some in , then . Let the set be the closure in of linear combinations of functions of the form where is in . The space is then a direct sum of the spaces and the Fourier transform maps each space to itself and it is possible to characterize the action of the Fourier transform on each space . This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases and 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 function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in for . It is possible in some cases to define the restriction of a Fourier transform to a set , provided has non-zero curvature. The case when is the unit sphere in 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 is a bounded operator on provided .

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 indexed by : such as balls of radius centered at the origin, or cubes of side . For a given integrable function , consider the function defined by:

<math display="block">f_R(x) = \int_{E_R}\widehat{f}(\xi) e^{i 2\pi x\cdot\xi}\, d\xi, \quad x \in \mathbb{R}^n.</math>

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

Fourier transform on function spaces

The definition of the Fourier transform naturally extends from <math>L^1(\mathbb R)</math> to . 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 \widehat{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\widehat{f}(\xi)\right\vert \leq \int_{\mathbb{R}^n} \vert f(x)\vert \,dx,</math>

which shows that its operator norm is bounded by . 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 that vanish at infinity, i.e., . Furthermore, the image of <math>L^1</math> under <math>\mathcal{F}</math> is a strict subset of .

Similarly to the case of one variable, the Fourier transform can be defined on . 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">\widehat{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 sense.

Furthermore, <math>\mathcal{F}:L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)</math> is a unitary operator. For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product. The Fourier inversion theorem implies that the transform is bijective. Also, for any 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>

\begin{align}

\int_{\mathbb{R}^n} \overline{\mathcal{F}f(x)}\mathcal{F}g(x)\,dx

&= \int_{\mathbb{R}^n} \mathcal{F}^{-1}\overline{f(x)}\mathcal{F}g(x)\,dx\\

&= \int_{\mathbb{R}^n} \mathcal{F}\mathcal{F}^{-1}\overline{f(x)}g(x)\,dx

= \int_{\mathbb{R}^n} \overline{f(x)}g(x)\,dx

\end{align}

</math>

So the transform preserves the inner product.

On other L<sup>p</sup>

For , the Fourier transform can be defined on <math>L^p(\mathbb R)</math> by Riesz–Thorin interpolation, which amounts to decomposing such functions into a fat tail part <math>|f|\le 1</math> in plus a fat body part <math>|f|>1</math> in . In each of these spaces, the Fourier transform of a function in is in , where is the Hölder conjugate of (by the Hausdorff–Young inequality). However, except for , the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in for the range requires the study of distributions. In fact, it can be shown that there are functions in with so that the Fourier transform is not defined as a function. If, in addition, the probability distribution has a probability density function, this definition is subject to the usual Fourier transform. Stated more generally, when <math>\mu</math> is absolutely continuous with respect to the Lebesgue measure, i.e.,

then

<math display="block">\widehat{\mu}(\xi)=\widehat{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.

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. Its Fourier transform is a constant function (whose value depends on the form of the Fourier transform used).

Locally compact abelian groups

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 is a locally compact abelian group, it has a translation invariant measure , called Haar measure. For a locally compact abelian group , 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 is itself a locally compact abelian group, called the Pontryagin dual of . For a function in , its Fourier transform is defined by

<math display="block">\widehat{f}(\xi) = \int_G \overline{\xi(x)}f(x)\,d\mu\quad \text{for any }\xi \in \widehat{G}.</math>

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

The Fourier transform on is an example; here is a locally compact abelian group, and the Haar measure on can be thought of as the Lebesgue measure on [0,1). Consider a representation of on the complex plane thought of as a 1-dimensional complex vector space. There is a group of such representations (which are irreducible since 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 .

The character of such representation, that is the trace of <math>e_{k}(x)</math> (thought of as a one-by-one matrix) for each <math>x\in T</math> and , is <math>e^{i 2\pi kx}</math> itself. Now, in the case of representations of finite groups, the character table of a group consists of rows of vectors such that each row is the character of one irreducible representation of , and these vectors form an orthonormal basis of the space of class (meaning conjugation-invariant) functions that map from to by Schur's lemma. The group is no longer finite but still compact, and it preserves the orthonormality of the character table. Each row of the table is the function <math>e_{k}(x)</math> of , and the inner product between two class functions (all functions being class functions since 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 . The sequence <math>\{e_{k}\mid k\in Z\}</math> is an orthonormal basis of the space of class functions .

For any representation of a finite group , <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 irreducible representations of ), such that . Similarly for <math>G = T</math> and , . The Pontriagin dual <math>\widehat{T}</math> is <math>\{e_{k}\}(k\in Z)</math> and for , <math display="inline">\widehat{f}(k) = \frac{1}{|T|}\int_{[0,1)}f(y)e^{-i 2\pi ky}dy</math> is its Fourier transform for .

Gelfand transform

The Fourier transform is also a special case of the 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 , as before we consider the space , defined using a Haar measure. With convolution as multiplication, 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 possible -norm gives its enveloping -algebra, called the group -algebra of . (Any -norm on is bounded by the norm, therefore their supremum exists.)

Given any abelian -algebra , the Gelfand transform gives an isomorphism between and , where is the multiplicative linear functionals, i.e. one-dimensional representations, on 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 , after suitable identification, are exactly the characters of , and the Gelfand transform, when restricted to the dense subset , 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. The Fourier transform on compact groups is a major tool in representation theory and non-commutative harmonic analysis.

Let be a compact Hausdorff topological group, and let be its normalized Haar measure. Let denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation on the Hilbert space of finite dimension for each .

For , the Fourier transform of at is the operator on defined by

\widehat f(\sigma)=\int_G f(g) U^{(\sigma)}_{g^{-1\,d\lambda(g).

</math>

Equivalently,

\left\langle \widehat f(\sigma)\xi,\eta\right\rangle_{H_\sigma}

\int_G f(g)\left\langle U^{(\sigma)}_{g^{-1\xi,\eta\right\rangle\,d\lambda(g).

</math>

Since is unitary, this may also be written using the adjoint <math>{U^{(\sigma)_g^*</math>.

If is a finite complex Borel measure on , then the Fourier–Stieltjes transform of is the operator on defined by

\widehat\mu(\sigma)=\int_G U^{(\sigma)}_{g^{-1\,d\mu(g),

</math>

or, weakly,

\left\langle \widehat{\mu}(\sigma)\xi,\eta\right\rangle_{H_\sigma}

\int_G \left\langle U^{(\sigma)}_{g^{-1\xi,\eta\right\rangle\,d\mu(g).

</math>

If is absolutely continuous with respect to , represented as

<math display="block">d\mu = f\,d\lambda</math>

for some , one identifies the Fourier transform of with the Fourier–Stieltjes transform of .

The mapping

<math display="block">\mu\mapsto\widehat{\mu}</math>

is injective and sends finite measures to bounded fields of operators , with

\sup_{\sigma\in\Sigma}\|\widehat\mu(\sigma)\|\leq \|\mu\|.

</math>

Thus it may be viewed as a representation of the Banach algebra of finite Borel measures, with multiplication given by convolution of measures. With the convention above, convolution corresponds to operator multiplication with the order reversed:

\widehat{\mu * \nu}(\sigma)=\widehat\nu(\sigma)\widehat\mu(\sigma).

</math>

Using the alternative convention reverses this order. The involution on is given, for absolutely continuous measures, by

f^*(g)=\overline{f(g^{-1})},

</math>

since compact groups are unimodular.

The Peter–Weyl theorem holds, and a version of the Fourier inversion formula follows: if , then

f(g)=\sum_{\sigma\in\Sigma} d_\sigma \operatorname{tr}\left(\widehat f(\sigma)U^{(\sigma)}_g\right),

</math>

where the summation is understood as convergent in the sense. The corresponding Plancherel formula is

\|f\|_2^2

\sum_{\sigma\in\Sigma} d_\sigma \|\widehat f(\sigma)\|_{\mathrm{HS^2,

</math>

where denotes the Hilbert–Schmidt norm.

The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry. 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 is no longer simply a transform of scalar-valued functions into scalar-valued 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, or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the Fourier transform being the continuous wavelet transform. 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 symbolically.

https://en.wikipedia.org/wiki/Help:Edit_summary

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. 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. 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 and denote their Fourier transforms by and . 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 or .

{| class="wikitable"

! !! Function !! Fourier transform unitary, ordinary frequency !! Fourier transform unitary, angular frequency !! Fourier transform 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>

|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>

|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>

|Shift in frequency domain, dual of 102

| 104

|<math> f(a x)</math>

|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\xi}{a} \right)</math>

|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)</math>

|Scaling in the time domain. If is large, then is concentrated around and <math display="inline"> \frac{1}{|a|}\widehat{f} \left( \frac{\omega}{a} \right)</math> spreads 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 replaces the frequency variable ( or ) 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>

|th-order derivative.

As 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>

|Integration. 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 display="inline">\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>

|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 denotes the convolution of and – 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 purely real

|<math> \widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}</math>

|<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}</math>

|Hermitian symmetry. indicates the complex conjugate.

<!-- A Symmetry section has been added instead of this.

| 111

|For purely real and even

| colspan=3 align=center |<math>\widehat f </math> is a purely real and even function.

| 112

|For purely real and odd

| colspan=3 align=center |<math>\widehat f </math> is a purely imaginary and odd function.

|-->

| 113

|For purely imaginary

|<math> \widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}</math>

|<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}</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>

|This follows from rules 101 and 103 using Euler's formula: .

|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>

|This follows from 101 and 103 using Euler's formula: .

Square-integrable functions, one-dimensional

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

{| class="wikitable"

! !! Function !! Fourier transform unitary, ordinary frequency !! Fourier transform unitary, angular frequency !! Fourier transform 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

| 201

|<math> \operatorname{rect}(a x) </math>

|<math> \frac{1}{|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>

|<math> \frac{1}{|a|}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>

|The rectangular pulse and the normalized sinc function, here defined as

| 202

|<math> \operatorname{sinc}(a x)</math>

|<math> \frac{1}{|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>

|<math> \frac{1}{|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 .

| 203

|<math> \operatorname{sinc}^2 (a x)</math>

|<math> \frac{1}{|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>

|<math> \frac{1}{|a|}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>

| The function is the triangular function

| 204

|<math> \operatorname{tri} (a x)</math>

|<math> \frac{1}{|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>

|<math> \frac{1}{|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 is the Heaviside unit step function and .

| 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 is its own Fourier transform for some choice of . For this to be integrable we must have .

| 208

|<math> e^{-a|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 . 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>

| is the th-order Hermite polynomial. If then the Gauss–Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see '. The formula reduces to 206 for .

Distributions, one-dimensional

The Fourier transforms in this table may be found in or .

{| class="wikitable"

! !! Function !! Fourier transform unitary, ordinary frequency !! Fourier transform unitary, angular frequency !! Fourier transform 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

| 301

|<math> 1</math>

|<math> \delta(\xi)</math>

|<math> \sqrt{2\pi}\, \delta(\omega)</math>

|<math> 2\pi\delta(\omega)</math>

|The distribution 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: .

| 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 using .

| 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 using .

| 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 using .

|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, is a natural number and is the th 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. is the th 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 is the sign function. Note that 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>

| is the homogeneous distribution defined by the distributional derivative<math>\tfrac{(-1)^{n-1{(n-1)!}\tfrac{d^n}{dx^n}\log|x|</math>

| 313

|<math> |x|^\alpha</math>

|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1</math>

|<math> \frac{-2}{\sqrt{2\pi\, \frac{\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1 </math>

|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1 </math>

|This formula is valid for . For some singular terms arise at the origin that can be found by differentiating 320. If , then is a locally integrable function, and so a tempered distribution. The function 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 for (see Homogeneous distribution).

|

|<math> \frac{1}{\sqrt{|x| </math>

|<math> \frac{1}{\sqrt{|\xi| </math>

|<math> \frac{1}{\sqrt{|\omega|</math>

|<math> \frac{\sqrt{2\pi{\sqrt{|\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 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 that <math display="inline">\sum_{n=-\infty}^{\infty} e^{inx}</math> <math display="inline">= 2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k) </math> as 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 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 is the th order Bessel function of first kind. The function is the Chebyshev polynomial of the first kind.

| 319

|<math> \log \left| x \right|</math>

|<math> -\frac{1}{2} \frac{1}{\left| \xi \right|} - \gamma \delta \left( \xi \right) </math>

|<math> -\frac{\sqrt\frac{\pi}{2{\left| \omega \right|} - \sqrt{2 \pi} \gamma \delta \left( \omega \right) </math>

|<math> -\frac{\pi}{\left| \omega \right|} - 2 \pi \gamma \delta \left( \omega \right) </math>

| is the Euler–Mascheroni constant. It is necessary to use a finite part integral when testing or against 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 . Use differentiation to derive formula for higher exponents. is the Heaviside function.

Two-dimensional functions

{| class="wikitable"

! !! Function !! Fourier transform unitary, ordinary frequency !! Fourier transform unitary, angular frequency !! Fourier transform non-unitary, angular frequency !! Remarks

|400

|<math> f(x,y)</math>

|<math>\begin{align}& \widehat{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}& \widehat{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}& \widehat{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 , , , are real numbers. The integrals are taken over the entire plane.

|401

|<math> e^{-\pi\left(a^2x^2+b^2y^2\right)}</math>

|<math> \frac{1}{|ab|} e^{-\pi\left(\frac{\xi_x^2}{a^2} + \frac{\xi_y^2}{b^2}\right)}</math>

|<math> \frac{1}{2\pi\,|ab|} e^{-\frac{1}{4\pi}\left(\frac{\omega_x^2}{a^2} + \frac{\omega_y^2}{b^2}\right)}</math>

|<math> \frac{1}{|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 for , and is 0 otherwise. The result is the amplitude distribution of the Airy disk, and is expressed using (the order-1 Bessel function of the first kind).

|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 , a 2-D Fourier "self-transform".

|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 <span class="texhtml">n</span>-dimensional functions

{| class="wikitable"

! !! Function !! Fourier transform unitary, ordinary frequency !! Fourier transform unitary, angular frequency !! Fourier transform non-unitary, angular frequency !! Remarks

|500

|<math> f(\mathbf x)</math>

|<math>\begin{align} &\widehat 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} &\widehat f_2 (\boldsymbol \omega) \triangleq \\ &\frac{1} \int_{\mathbb{R}^n} f(\mathbf x) e^{-i \boldsymbol \omega \cdot \mathbf x}\, d \mathbf x \end{align}</math>

|<math>\begin{align} &\widehat 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

|<math> \chi_{[0,1]}(|\mathbf x|)\left(1-|\mathbf x|^2\right)^\delta</math>

|<math> \frac{\Gamma(\delta+1)}{\pi^\delta\,|\boldsymbol \xi|^\gamma} J_{\gamma}(2\pi|\boldsymbol \xi|)</math>

|<math> 2^\delta \, \frac{\Gamma(\delta+1)}{\left|\boldsymbol \omega\right|^\gamma} J_{\gamma}(|\boldsymbol \omega|)</math>

|<math> \frac{\Gamma(\delta+1)}{\pi^\delta} \left|\frac{\boldsymbol \omega}{2\pi}\right|^{-\gamma} J_{\gamma}(\!|\boldsymbol \omega|\!)</math>

|Here, <math>\gamma = \tfrac{n}{2} + \delta</math>.

The function is the indicator function of the interval . The function is the gamma function. The function is a Bessel function of the first kind, with order . Taking and produces 402.

|502

|<math> |\mathbf x|^{-\alpha}, \quad 0 < \operatorname{Re} \alpha < n</math>

|<math> \frac{(2\pi)^{\alpha{c_{n, \alpha |\boldsymbol \xi|^{-(n - \alpha)}</math>

|<math> \frac{(2\pi)^{\frac{n}{2}{c_{n, \alpha |\boldsymbol \omega|^{-(n - \alpha)}</math>

|<math> \frac{(2\pi)^{n{c_{n, \alpha |\boldsymbol \omega|^{-(n - \alpha)}</math>

|See Riesz potential, where the constant is given by . The formula also holds for all by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions (see Homogeneous distribution).

|503

|<math> \frac{1}{\left|\boldsymbol \sigma\right|\left(2\pi\right)^\frac{n}{2 e^{-\frac{1}{2} \mathbf x^{\mathrm T} \boldsymbol \sigma^{-\mathrm T} \boldsymbol \sigma^{-1} \mathbf x}</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, and

|504

|<math> e^{-2\pi\alpha|\mathbf x|}</math>

|<math>\frac{c_n\alpha}{\left(\alpha^2+|\boldsymbol{\xi}|^2\right)^\frac{n+1}{2</math>

|<math>\frac{c_n (2\pi)^{\frac{n+2}{2 \alpha}{\left(4\pi^2\alpha^2+|\boldsymbol{\omega}|^2\right)^\frac{n+1}{2</math>

|<math>\frac{c_n (2\pi)^{n+1} \alpha}{\left(4\pi^2\alpha^2+|\boldsymbol{\omega}|^2\right)^\frac{n+1}{2</math>

|Here ,

Notes

Citations

References

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; also available at Fundamentals of Music Processing, Section 2.1, pages 40–56

External links

Encyclopedia of Mathematics
Fourier Transform in Crystallography

Definition

Angular frequency (&omega;)

Lebesgue integrable functions

Background

History

Complex sinusoids

Negative frequency

Fourier transform for periodic functions

Sampling the Fourier transform

Units

Properties

Basic properties

Linearity

Time shifting

Frequency shifting

Time scaling

Symmetry

Conjugation

Real and imaginary parts

Zero frequency component

Uniform continuity and the Riemann–Lebesgue lemma

Plancherel theorem and Parseval's theorem

Convolution theorem

Cross-correlation theorem

Differentiation

Eigenfunctions

Inversion and periodicity

Connection with the Heisenberg group

Complex domain

Laplace transform

Inversion

Fourier transform on Euclidean space

Sine and cosine transforms

Spherical harmonics

Restriction problems

Fourier transform on function spaces

On other L<sup>p</sup>

Locally compact abelian groups

Gelfand transform

Compact non-abelian groups

Alternatives

Numerical integration of closed-form continuous functions

Numerical integration of a series of ordered pairs

Tables of important Fourier transforms

Functional relationships, one-dimensional

Square-integrable functions, one-dimensional

Distributions, one-dimensional

Two-dimensional functions

Formulas for general <span class="texhtml">n</span>-dimensional functions

See also

Notes

Citations

References

External links

Angular frequency (ω)