fourier transform wiki

Fourier transform 1

Fourier transform

Fourier transforms

Continuous Fourier transform

Fourier series

Discrete-time Fourier transform

Discrete Fourier transform

Fourier analysis

Related transforms

The Fourier transform (English pronunciation: /frie/), named after Joseph Fourier, is a mathematical transformationemployed to transform signals between time (or spatial) domain and frequency domain, which has many applicationsin physics and engineering. It is reversible, being able to transform from either domain to the other. The term itselfrefers to both the transform operation and to the function it produces.In the case of a periodic function over time (for example, a continuous but not necessarily sinusoidal musical sound),the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourierseries coefficients. They represent the frequency spectrum of the original time-domain signal. Also, when atime-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a versionof the original Fourier transform according to the Poisson summation formula, also known as the discrete-timeFourier transform. See also Fourier analysis and List of Fourier-related transforms.

DefinitionThere are several common conventions for defining the Fourier transform of an integrable function

(Kaiser 1994, p.29), (Rahman 2011, p.11). This article will use the following definition:

, for any real number .

When the independent variable x represents time (with SI unit of seconds), the transform variable representsfrequency (in hertz). Under suitable conditions, is determined by via the inverse transform:

, for any real numberx.

The statement that can be reconstructed from is known as the Fourier inversion theorem, and was firstintroduced in Fourier's Analytical Theory of Heat (Fourier 1822, p.525), (Fourier & Freeman 1878, p.408), althoughwhat would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p.1). Thefunctions and often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p.10).For other common conventions and notations, including using the angular frequency instead of the frequency ,see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, inwhich the variable x often represents position and momentum.

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IntroductionSee also: Fourier analysis

In the first frames of the animation, a function f isresolved into Fourier series: a linear combination

of sines and cosines (in blue). The componentfrequencies of these sines and cosines spread

across the frequency spectrum, are represented aspeaks in the frequency domain (actually Diracdelta functions, shown in the last frames of the

animation). The frequency domain representationof the function, , is the collection of these

peaks at the frequencies that appear in thisresolution of the function.

The motivation for the Fourier transform comes from the study ofFourier series. In the study of Fourier series, complicated but periodicfunctions are written as the sum of simple waves mathematicallyrepresented by sines and cosines. The Fourier transform is an extensionof the Fourier series that results when the period of the representedfunction is lengthened and allowed to approach infinity (Taneja 2008,p.192).

Due to the properties of sine and cosine, it is possible to recover theamplitude of each wave in a Fourier series using an integral. In manycases it is desirable to use Euler's formula, which states that e2i =cos(2) + i sin(2), to write Fourier series in terms of the basicwaves e2i. This has the advantage of simplifying many of theformulas involved, and provides a formulation for Fourier series thatmore closely resembles the definition followed in this article.Re-writing sines and cosines as complex exponentials makes itnecessary for the Fourier coefficients to be complex valued. The usualinterpretation of this complex number is that it gives both theamplitude (or size) of the wave present in the function and the phase(or the initial angle) of the wave. These complex exponentialssometimes contain negative "frequencies". If is measured in seconds, then the waves e2i and e2i bothcomplete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequencyno longer measures the number of cycles per unit time, but is still closely related.

There is a close connection between the definition of Fourier series and the Fourier transform for functions f whichare zero outside of an interval. For such a function, we can calculate its Fourier series on any interval that includesthe points where f is not identically zero. The Fourier transform is also defined for such a function. As we increasethe length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to looklike the Fourier transform and the sum of the Fourier series of f begins to look like the inverse Fourier transform. Toexplain this more precisely, suppose that T is large enough so that the interval [T/2,T/2] contains the interval onwhich f is not identically zero. Then the n-th series coefficient cn is given by:

Comparing this to the definition of the Fourier transform, it follows that since f(x) is zerooutside [T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid ofwidth 1/T, multiplied by the grid width 1/T.Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:

where the last sum is simply the first sum rewritten using the definitions n = n/T, and = (n + 1)/T n/T = 1/T.This second sum is a Riemann sum, and so by letting T it will converge to the integral for the inverse Fouriertransform given in the definition section. Under suitable conditions this argument may be made precise (Stein &Shakarchi 2003).

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In the study of Fourier series the numbers cn could be thought of as the "amount" of the wave present in the Fourierseries of f. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how muchof each individual frequency is present in our function f, and we can recombine these waves by using an integral (or"continuous sum") to reproduce the original function.

ExampleThe following images provide a visual illustration of how the Fourier transform measures whether a frequency ispresent in a particular function. The function depicted f(t) = cos(6t) et2 oscillates at 3 hertz (if t measuresseconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes thecontinuous sinusoid into a short pulse. Its general form is a Gaussian function). This function was specially chosen tohave a real Fourier transform which can easily be plotted. The first image contains its graph. In order to calculate

we must integrate e2i(3t)f(t). The second image shows the plot of the real and imaginary parts of thisfunction. The real part of the integrand is almost always positive, because when f(t) is negative, the real part ofe2i(3t) is negative as well. Because they oscillate at the same rate, when f(t) is positive, so is the real part ofe2i(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in thiscase 0.5). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at

, the integrand oscillates enough so that the integral is very small. The general situation may be a bit morecomplicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequencyis present in a function f(t).

Original function showingoscillation 3 hertz.

Real and imaginary parts ofintegrand for Fourier transform

at 3 hertz

Real and imaginary parts ofintegrand for Fourier transform

at 5 hertz

Fourier transform with 3 and 5hertz labeled.

Properties of the Fourier transformHere we assume f(x), g(x) and h(x) are integrable functions, are Lebesgue-measurable on the real line, and satisfy:

We denote the Fourier transforms of these functions by , and respectively.

Basic propertiesThe Fourier transform has the following basic properties: (Pinsky 2002).Linearity

For any complex numbers a and b, if h(x) = af(x) + bg(x), then Translation

For any real number x0, if then Modulation

For any real number 0 if then

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Scaling

For a non-zero real number a, if h(x) = f(ax), then The case a = 1 leads to the

time-reversal property, which states: if h(x) = f(x), then Conjugation

If then

In particular, if f is real, then one has the reality condition , that is, is a Hermitian

function.And if f is purely imaginary, then

Integration

Substituting in the definition, we obtain

That is, the evaluation of the Fourier transform in the origin ( ) equals the integral of f all over its domain.

Invertibility and periodicityFurther information: Fourier inversion theorem and Fractional Fourier transform

Under suitable conditions on the function f, it can be recovered from its Fourier transform Indeed, denoting theFourier transform operator by so then for suitable functions, applying the Fourier transform twicesimply flips the function: which can be interpreted as "reversing time". Since reversingtime is two-periodic, applying this twice yields so the Fourier transform operator is four-periodic, andsimilarly 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 that inverts time, :

These equalities of operators require careful definition of the space of functions in question, defining equality offunctions (equality at every point? equality almost everywhere?) and defining equality of operators that is, definingthe topology on the function space and operator space in question. These are not true for all functions, but are trueunder various conditions, which are the content of the various forms of the Fourier inversion theorem.This four-fold periodicity of the Fourier transform is similar to a rotation of the plane by 90, particularly as thetwo-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform cansimply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transformswitching them back, more geometrically it can be interpreted as a rotation by 90 in the timefrequency domain(considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to thefractional Fourier transform, which involves rotations by other angles. This can be further generalized to linearcanonical transformations, which can be visualized as the action of the special linear group SL2(R) on thetimefrequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. Thisapproach is particularly studied in signal processing, under timefrequency analysis.

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Uniform continuity and the RiemannLebesgue lemma

The rectangular function is Lebesgue integrable.

The sinc function, which is the Fourier transformof the rectangular function, is bounded andcontinuous, but not Lebesgue integrable.

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

The Fourier transform, , of any integrable function f is uniformlycontinuous and (Katznelson 1976). By theRiemannLebesgue lemma (Stein & Weiss 1971),

However, need not be integrable. For example, the Fouriertransform of the rectangular function, which is integrable, is the sincfunction, which is not Lebesgue integrable, because its improperintegrals behave analogously to the alternating harmonic series, inconverging to a sum without being absolutely convergent.It is not generally possible to write the inverse transform as a Lebesgueintegral. However, when both f and are integrable, the inverseequality

holds almost everywhere. That is, the Fourier transform is injective onL1(R). (But if f is continuous, then equality holds for every x.)

Plancherel theorem and Parseval's theorem

Let f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are alsosquare-integrable, then we have Parseval's theorem (Rudin 1987, p. 187):

where the bar denotes complex conjugation.The Plancherel theorem, which is equivalent to Parseval's theorem, states (Rudin 1987, p. 186):

The Plancherel theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitaryoperator on L2(R). On L1(R)L2(R), this extension agrees with original Fourier transform defined on L1(R), thusenlarging the domain of the Fourier transform to L1(R) + L2(R) (and consequently to Lp(R) for 1 p 2). ThePlancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of theoriginal quantity. Depending on the author either of these theorems might be referred to as the Plancherel theorem oras Parseval's theorem.See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

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Poisson summation formulaMain article: Poisson summation formulaThe Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodicsummation of a function to values of the function's continuous Fourier transform. It has a variety of useful forms thatare derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. Thefrequency-domain dual of the standard PSF is also called discrete-time Fourier transform, which leads directly to: a popular, graphical, frequency-domain representation of the phenomenon of aliasing, and a proof of the Nyquist-Shannon sampling theorem.

Convolution theoremMain article: Convolution theoremThe Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) are integrablefunctions with Fourier transforms and respectively, then the Fourier transform of the convolution isgiven by the product of the Fourier transforms and (under other conventions for the definition of theFourier transform a constant factor may appear).This means that if:

where denotes the convolution operation, then:

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI systemwith input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case, represents the frequency response of the system.Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then theFourier transform of f(x) is given by the convolution of the respective Fourier transforms and .

Cross-correlation theoremMain article: Cross-correlationIn an analogous manner, it can be shown that if h(x) is the cross-correlation of f(x) and g(x):

then the Fourier transform of h(x) is:

As a special case, the autocorrelation of function f(x) is:

for which

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EigenfunctionsOne important choice of an orthonormal basis for L2(R) is given by the Hermite functions

where Hen(x) are the "probabilist's" Hermite polynomials, defined by

Under this convention for the Fourier transform, we have that

.In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fouriertransform on L2(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only fourdifferent eigenvalues of the Fourier transform (1 and i) and any linear combination of eigenfunctions with thesame eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L2(R) as a directsum of four spaces H0, H1, H2, and H3 where the Fourier transform acts on Hek simply by multiplication by i

k.Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can berepresented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed.This approach to define the Fourier transform was first done by Norbert Wiener(Duoandikoetxea 2001). Amongother properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they arethus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used intime-frequency analysis (Boashash 2003). In physics, this transform was introduced by Edward Condon(Condon1937).

Fourier transform on Euclidean spaceThe Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case,there are many conventions. For an integrable function f(x), this article takes the definition:

where x and are n-dimensional vectors, and x is the dot product of the vectors. The dot product is sometimeswritten as .All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's andParseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and theRiemannLebesgue lemma holds. (Stein & Weiss 1971)

Uncertainty principleFor more details on this topic, see Uncertainty principle.

Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. Inparticular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, itsFourier transform "stretches out" in . It is not possible to arbitrarily concentrate both a function and its Fouriertransform.The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of anuncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to thesymplectic form on the timefrequency domain: from the point of view of the linear canonical transformation, theFourier transform is rotation by 90 in the timefrequency domain, and preserves the symplectic form.

Fourier transform 8

Suppose f(x) is an integrable and square-integrable function. Without loss of generality, assume that f(x) isnormalized:

It follows from the Plancherel theorem that is also normalized.The spread around x= 0 may be measured by the dispersion about zero (Pinsky 2002, p.131) defined by

In probability terms, this is the second moment of |f(x)|2 about zero.The Uncertainty principle states that, if f(x) is absolutely continuous and the functions xf(x) and f(x) are squareintegrable, then

(Pinsky 2002).

The equality is attained only in the case (hence ) where > 0 isarbitrary and so that f is L2normalized (Pinsky 2002). In other words, where f is a (normalized)Gaussian function with variance 2, centered at zero, and its Fourier transform is a Gaussian function with variance2.In fact, this inequality implies that:

for any x0, 0 R (Stein & Shakarchi 2003, p.158).In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor ofPlanck's constant. With this constant properly taken into account, the inequality above becomes the statement of theHeisenberg uncertainty principle (Stein & Shakarchi 2003, p.158).A stronger uncertainty principle is the Hirschman uncertainty principle which is expressed as:

where H(p) is the differential entropy of the probability density function p(x):

where the logarithms may be in any base which is consistent. The equality is attained for a Gaussian, as in theprevious case.

Spherical harmonicsLet the set of homogeneous harmonic polynomials of degree k on Rn be denoted by Ak. The set Ak consists of thesolid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to theHermite polynomials in dimension one. Specifically, if f(x) = e|x|2P(x) for some P(x) in Ak, then

. Let the set Hk be the closure in L2(Rn) of linear combinations of functions of the form f(|x|)P(x)

where P(x) is in Ak. The space L2(Rn) is then a direct sum of the spaces Hk and the Fourier transform maps each

space Hk to itself and is possible to characterize the action of the Fourier transform on each space Hk (Stein & Weiss1971). Let f(x) = f0(|x|)P(x) (with P(x) in Ak), then where

Here J(n+2k2)/2 denotes the Bessel function of the first kind with order (n+2k2)/2. When k=0 this gives a useful formula for the Fourier transform of a radial function (Grafakos 2004). Note that this is essentially the Hankel

Fourier transform 9

transform. Moreover, there is a simple recursion relating the cases n+2 and n (Grafakos & Teschl 2013) allowing tocompute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.

Restriction problemsIn higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fouriertransform of an integrable function is continuous and the restriction of this function to any set is defined. But for asquare-integrable function the Fourier transform could be a general class of square integrable functions. As such, therestriction of the Fourier transform of an L2(Rn) function cannot be defined on sets of measure 0. It is still an activearea of study to understand restriction problems in Lp for 1

Fourier transform 10

where the limit is taken in the L2 sense. Many of the properties of the Fourier transform in L1 carry over to L2, by asuitable limiting argument.Furthermore : L2(Rn) L2(Rn) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). For an operator to beunitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from theFourier inversion theorem combined with the fact that for any f,gL2(Rn) we have

In particular, the image of L2(Rn) is itself under the Fourier transform.On other Lp

The definition of the Fourier transform can be extended to functions in Lp(Rn) for 1 p 2 by decomposing suchfunctions into a fat tail part in L2 plus a fat body part in L1. In each of these spaces, the Fourier transform of afunction in Lp(Rn) is in Lq(Rn), where is the Hlder conjugate of p. by the HausdorffYounginequality. However, except for p = 2, the image is not easily characterized. Further extensions become moretechnical. The Fourier transform of functions in Lp for the range 2 < p < requires the study of distributions(Katznelson 1976). In fact, it can be shown that there are functions in Lp with p > 2 so that the Fourier transform isnot defined as a function (Stein & Weiss 1971).

Tempered distributionsMain article: Tempered distributionsOne might consider enlarging the domain of the Fourier transform from L1+L2 by considering generalized functions,or distributions. A distribution on Rn is a continuous linear functional on the space Cc(R

n) of compactly supportedsmooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fouriertransform on Cc(R

n) and pass to distributions by duality. The obstruction to do this is that the Fourier transform doesnot map Cc(R

n) to Cc(Rn). In fact the Fourier transform of an element in Cc(R

n) can not vanish on an open set; seethe above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartzfunctions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thusinduces an automorphism on its dual, the space of tempered distributions(Stein & Weiss 1971). The tempereddistribution include all the integrable functions mentioned above, as well as well-behaved functions of polynomialgrowth and distributions of compact support.

For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let and be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplicationformula (Stein & Weiss 1971),

Every integrable function f defines (induces) a distribution Tf by the relation

for all Schwartz functions .

So it makes sense to define Fourier transform of Tf by

for all Schwartz functions . Extending this to all tempered distributions T gives the general definition of the Fouriertransform.


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

Generalizations

FourierStieltjes transformThe Fourier transform of a finite Borel measure on Rn is given by (Pinsky 2002, p.256):

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. Onenotable difference is that the RiemannLebesgue lemma fails for measures (Katznelson 1976). In the case that d =f(x)dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that is theprobability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to thecharacteristic function, but the typical conventions in probability theory take eix instead of e2ix (Pinsky 2002).In the case when the distribution has a probability density function this definition reduces to the Fourier transformapplied to the probability density function, again with a different choice of constants.The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes whichfunctions may arise as the FourierStieltjes transform of a positive measure on the circle (Katznelson 1976).Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is aconstant function (whose specific value depends upon the form of the Fourier transform used).

Locally compact abelian groupsMain article: Pontryagin dualityThe Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group isan abelian group which is at the same time a locally compact Hausdorff topological space so that the group operationis continuous. If G is a locally compact abelian group, it has a translation invariant measure , called Haar measure.For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are calledits characters. With its natural group structure and the topology of pointwise convergence, the set of characters isitself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L1(G), its Fouriertransform is defined by (Katznelson 1976):

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

Gelfand transformMain article: Gelfand representationThe Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to thePontryagin duality map defined above.Given an abelian locally compact Hausdorff topological group G, as before we consider space L1(G), defined using aHaar measure. With convolution as multiplication, L1(G) is an abelian Banach algebra. It also has an involution *given by

Taking the completion with respect to the largest possibly C*-norm gives its enveloping C*-algebra, called the groupC*-algebra C*(G) of G. (Any C*-norm on L1(G) is bounded by the L1 norm, therefore their supremum exists.)


Given any abelian C*-algebra A, the Gelfand transform gives an isomorphism between A and C0(A^), where A^ isthe multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology. The mapis simply given by

It turns out that the multiplicative linear functionals of C*(G), after suitable identification, are exactly the charactersof G, and the Gelfand transform, when restricted to the dense subset L1(G) is the Fourier-Pontryagin transform.

Non-abelian groupsThe 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 alwaysbe one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators(Hewitt & Ross 1970, Chapter 8). The Fourier transform on compact groups is a major tool in representation theory(Knapp 2001) and non-commutative harmonic analysis.Let G be a compact Hausdorff topological group. Let denote the collection of all isomorphism classes offinite-dimensional irreducible unitary representations, along with a definite choice of representation U() on theHilbert space H of finite dimension d for each . If is a finite Borel measure on G, then the FourierStieltjestransform of is the operator on H defined by

where is the complex-conjugate representation of U() acting on H. If is absolutely continuous with respectto the left-invariant probability measure on G, represented as

for some f L1(), one identifies the Fourier transform of f with the FourierStieltjes transform of .The mapping defines an isomorphism between the Banach space M(G) of finite Borel measures (see rcaspace) and a closed subspace of the Banach space C() consisting of all sequences E = (E) indexed by of(bounded) linear operators E: H H for which the norm

is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact anisometric isomorphism of C* algebras into a subspace of C(). Multiplication on M(G) is given by convolution ofmeasures and the involution * defined by

and C() has a natural C*-algebra structure as Hilbert space operators.The PeterWeyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f L2(G), then

where the summation is understood as convergent in the L2 sense.The generalization of the Fourier transform to the noncommutative situation has also in part contributed to thedevelopment of noncommutative geometry.Wikipedia:Citation needed In this context, a categorical generalization ofthe Fourier transform to noncommutative groups is TannakaKrein duality, which replaces the group of characterswith the category of representations. However, this loses the connection with harmonic functions.


AlternativesIn signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but nofrequency information, while the Fourier transform has perfect frequency resolution, but no time information: themagnitude of the Fourier transform at a point is how much frequency content there is, but location is only given byphase (argument of the Fourier transform at a point), and standing waves are not localized in time a sine wavecontinues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signalsthat 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 ortime-frequency distributions to represent signals in a form that has some time information and some frequencyinformation by the uncertainty principle, there is a trade-off between these. These can be generalizations of theFourier transform, such as the short-time Fourier transform or fractional Fourier transform, or other functions torepresent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous)Fourier transform being the continuous wavelet transform. (Boashash 2003).

Applications

Some problems, such as certain differential equations, become easier to solve when theFourier transform is applied. In that case the solution to the original problem is recovered

using the inverse Fourier transform.

Analysis of differentialequations

Fourier transforms and the closelyrelated Laplace transforms are widelyused in solving differential equations.The Fourier transform is compatiblewith differentiation in the followingsense: if f(x) is a differentiable functionwith Fourier transform , then the Fourier transform of its derivative is given by . This can be usedto transform differential equations into algebraic equations. This technique only applies to problems whose domainis the whole set of real numbers. By extending the Fourier transform to functions of several variables partialdifferential equations with domain Rn can also be translated into algebraic equations.

Fourier transform spectroscopyMain article: Fourier transform spectroscopyThe 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 domainand Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used inmagnetic resonance imaging (MRI) and mass spectrometry.


Quantum mechanics and signal processingIn quantum mechanics, Fourier transforms of solutions to the Schrdinger equation are known as momentum space(or k space) wave functions. They display the amplitudes for momenta. Their absolute square is the probabilities ofmomenta. This is valid also for classical waves treated in signal processing, such as in swept frequency radar wheredata is taken in frequency domain and transformed to time domain, yielding range. The absolute square is then thepower.

Other notationsOther common notations for include:

Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such asf(x) and F()) is especially common in the sciences and engineering. In electronics, the omega () is often usedinstead of due to its interpretation as angular frequency, sometimes it is written as F(j), where j is the imaginaryunit, to indicate its relationship with the Laplace transform, and sometimes it is written informally as F(2f) in orderto use ordinary frequency.

The interpretation of the complex function may be aided by expressing it in polar coordinate form

in terms of the two real functions A() and () where:

is the amplitude and

is the phase (see arg function).Then the inverse transform can be written:

which is a recombination of all the frequency components of f(x). Each component is a complex sinusoid of theform e2ix whose amplitude is A() and whose initial phase angle (at x=0) is ().The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted and

is used to denote the Fourier transform of the function f. This mapping is linear, which means that canalso be seen as a linear transformation on the function space and implies that the standard notation in linear algebraof applying a linear transformation to a vector (here the function f) can be used to write instead of .Since the result of applying the Fourier transform is again a function, we can be interested in the value of thisfunction evaluated at the value for its variable, and this is denoted either as or as . Notice thatin the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluatedat , not the other way around.In mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value off when its variable equals x, denoted f(x). This means that a notation like formally can be interpreted asthe Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when aparticular function or a function of a particular variable is to be transformed.For example, is sometimes used to express that the Fourier transform of a rectangularfunction is a sinc function,or 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 x, notof x0.

Other conventionsThe Fourier transform can also be written in terms of angular frequency: = 2 whose units are radians per second.The substitution = /(2) into the formulas above produces this convention:

Under this convention, the inverse transform becomes:

Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer aunitary transformation on L2(Rn). There is also less symmetry between the formulas for the Fourier transform and itsinverse.Another convention is to split the factor of (2)n evenly between the Fourier transform and its inverse, which leadsto definitions:

Under this convention, the Fourier transform is again a unitary transformation on L2(Rn). It also restores thesymmetry between the Fourier transform and its inverse.Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forwardand the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.

Summary of popular forms of the Fourier transform

ordinary frequency (hertz) unitary

angular frequency (rad/s) non-unitary

unitary

As discussed above, the characteristic function of a random variable is the same as the FourierStieltjes transform ofits distribution measure, but in this context it is typical to take a different convention for the constants. Typicallycharacteristic function is defined .

As in the case of the "non-unitary angular frequency" convention above, there is no factor of 2 appearing in eitherof the integral, or in the exponential. Unlike any of the conventions appearing above, this convention takes theopposite sign in the exponential.


Tables of important Fourier transformsThe following tables record some closed-form Fourier transforms. For functions f(x), g(x) and h(x) denote theirFourier transforms by , , and respectively. Only the three most common conventions are included. It maybe useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the originalfunction, which can be seen as relating the Fourier transform and its inverse.

Functional relationshipsThe Fourier transforms in this table may be found in Erdlyi (1954) or Kammler (2000, appendix).

Function Fourier transformunitary, ordinary

frequency

Fourier transformunitary, angular frequency

Fourier transformnon-unitary, angular

frequency

Remarks

Definition

101 Linearity

102 Shift in time domain

103 Shift in frequency domain, dualof 102

104 Scaling in the time domain. Ifis large, then is

concentrated around 0 and

spreads out and

flattens.

105 Duality. Here needs to becalculated using the samemethod as Fourier transformcolumn. Results from swapping"dummy" variables of and or or .

106

107 This is the dual of 106

108 The notation denotes theconvolution of and thisrule is the convolution theorem

109 This is the dual of 108

110 For purely real Hermitian symmetry. indicates the complexconjugate.

111 For a purely realeven function

, and are purely real even functions.

112 For a purely realodd function

, and are purely imaginary odd functions.

113 Complex conjugation,generalization of 110


Square-integrable functionsThe Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdlyi 1954), or the appendix of(Kammler 2000).

Function Fourier transformunitary, ordinary

frequency



frequency

Remarks

201 The rectangular pulse and thenormalized sinc function, here definedas sinc(x) = sin(x)/(x)

202 Dual of rule 201. The rectangularfunction is an ideal low-pass filter, andthe sinc function is the non-causalimpulse response of such a filter. Thesinc function is defined here as sinc(x)= sin(x)/(x)

203 The function tri(x) is the triangularfunction

204 Dual of rule 203.

205 The function u(x) is the Heaviside unitstep function and a>0.

206 This shows that, for the unitary Fouriertransforms, the Gaussian functionexp(x2) is its own Fourier transformfor some choice of . For this to beintegrable we must have Re()>0.

207 For a>0. That is, the Fourier transformof a decaying exponential function is aLorentzian function.

208 Hyperbolic secant is its own Fouriertransform

209 is the Hermite's polynomial. If a =1 then the Gauss-Hermite functions areeigenfunctions of the Fourier transformoperator. For a derivation, see Hermitepolynomial. The formula reduces to206 for n = 0.


DistributionsThe Fourier transforms in this table may be found in (Erdlyi 1954) or the appendix of (Kammler 2000).

Function Fourier transformunitary, ordinary frequency



frequency

Remarks

301 The distribution ()denotes the Dirac deltafunction.

302 Dual of rule 301.

303 This follows from 103and 301.

304 This follows from rules101 and 303 using Euler'sformula:

305 This follows from 101and 303 using

306

307

308 Here, n is a naturalnumber and isthe n-th distributionderivative of the Diracdelta function. This rulefollows from rules 107and 301. Combining thisrule with 101, we cantransform allpolynomials.

309 Here sgn() is the signfunction. Note that 1/x isnot a distribution. It isnecessary to use theCauchy principal valuewhen testing againstSchwartz functions. Thisrule is useful in studyingthe Hilbert transform.

310 1/xn is the homogeneousdistribution defined bythe distributionalderivative


311 This formula is valid for0 > > 1. For > 0some singular terms ariseat the origin that can befound by differentiating318. If Re > 1, then

is a locallyintegrable function, andso a tempereddistribution. The function

is aholomorphic functionfrom the right half-planeto the space of tempereddistributions. It admits aunique meromorphicextension to a tempereddistribution, also denoted

for 2, 4,...(See homogeneousdistribution.)

312 The dual of rule 309. Thistime the Fouriertransforms need to beconsidered as Cauchyprincipal value.

313 The function u(x) is theHeaviside unit stepfunction; this followsfrom rules 101, 301, and312.

314 This function is known asthe Dirac comb function.This result can be derivedfrom 302 and 102,together with the fact that

as

distributions.

315 The function J0(x) is thezeroth order Besselfunction of first kind.

316 This is a generalization of315. The function Jn(x) isthe n-th order Besselfunction of first kind. Thefunction Tn(x) is theChebyshev polynomial ofthe first kind.

317 is theEulerMascheroniconstant.


318 This formula is valid for1 > > 0. Usedifferentiation to deriveformula for higherexponents. u is theHeaviside function.

Two-dimensional functions



Fourier transformnon-unitary, angular frequency

400

401

402

RemarksTo 400: The variables x, y, x, y, x and y are real numbers. The integrals are taken over the entire plane.To 401: Both functions are Gaussians, which may not have unit volume.To 402: The function is defined by circ(r)=1 0r1, and is 0 otherwise. This is the Airy distribution, and isexpressed using J1 (the order 1 Bessel function of the first kind). (Stein & Weiss 1971, Thm. IV.3.3)

Formulas for general n-dimensional functions



Fourier transformnon-unitary, angular frequency

500

501

502

503

504

RemarksTo 501: The function [0, 1] is the indicator function of the interval [0, 1]. The function (x) is the gamma function.The function Jn/2 + is a Bessel function of the first kind, with order n/2 + . Taking n = 2 and = 0 produces 402.(Stein & Weiss 1971, Thm. 4.15)To 502: See Riesz potential. The formula also holds for all n, n 1, by analytic continuation, but then thefunction and its Fourier transforms need to be understood as suitably regularized tempered distributions. Seehomogeneous distribution.


To 503: This is the formula for a multivariate normal distribution normalized to 1 with a mean of 0. Bold variablesare vectors or matrices. Following the notation of the aforementioned page, and

To 504: Here . See (Stein & Weiss 1971, p.6).

References Boashash, B., ed. (2003), Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Oxford:

Elsevier Science, ISBN0-08-044335-4 Bochner S., Chandrasekharan K. (1949), Fourier Transforms, Princeton University Press Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill,

ISBN0-07-116043-4. Campbell, George; Foster, Ronald (1948), Fourier Integrals for Practical Applications, New York: D. Van

Nostrand Company, Inc.. Condon, E. U. (1937), "Immersion of the Fourier transform in a continuous group of functional transformations",

Proc. Nat. Acad. Sci. USA 23: 158164. Duoandikoetxea, Javier (2001), Fourier Analysis, American Mathematical Society, ISBN0-8218-2172-5. Dym, H; McKean, H (1985), Fourier Series and Integrals, Academic Press, ISBN978-0-12-226451-1. Erdlyi, Arthur, ed. (1954), Tables of Integral Transforms 1, New Your: McGraw-Hill Fourier, J. B. Joseph (1822), Thorie Analytique de la Chaleur [1], Paris: Chez Firmin Didot, pre et fils Fourier, J. B. Joseph; Freeman, Alexander, translator (1878), The Analytical Theory of Heat [2], The University

Press Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Prentice-Hall, ISBN0-13-035399-X. Grafakos, Loukas; Teschl, Gerald (2013), "On Fourier transforms of radial functions and distributions", J. Fourier

Anal. Appl. 19: 167179, doi:10.1007/s00041-012-9242-5 [3]. Hewitt, Edwin; Ross, Kenneth A. (1970), Abstract harmonic analysis. Vol. II: Structure and analysis for compact

groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band152, Berlin, New York: Springer-Verlag, MR0262773 [4].

Hrmander, L. (1976), Linear Partial Differential Operators, Volume 1, Springer-Verlag,ISBN978-3-540-00662-6.

James, J.F. (2011), A Student's Guide to Fourier Transforms (3rd ed.), New York: Cambridge University Press,ISBN978-0-521-17683-5.

Kaiser, Gerald (1994), A Friendly Guide to Wavelets [5], Birkhuser, ISBN0-8176-3711-7 Kammler, David (2000), A First Course in Fourier Analysis, Prentice Hall, ISBN0-13-578782-3 Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN0-486-63331-4 Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples [6],

Princeton University Press, ISBN978-0-691-09089-4 Pinsky, Mark (2002), Introduction to Fourier Analysis and Wavelets [7], Brooks/Cole, ISBN0-534-37660-6 Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: CRC Press,

ISBN0-8493-2876-4. Rudin, Walter (1987), Real and Complex Analysis (Third ed.), Singapore: McGraw Hill, ISBN0-07-100276-6. Rahman, Matiur (2011), Applications of Fourier Transforms to Generalized Functions [8], WIT Press,

ISBN1845645642. Stein, Elias; Shakarchi, Rami (2003), Fourier Analysis: An introduction [9], Princeton University Press,

ISBN0-691-11384-X. Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces [10], Princeton, N.J.:

Princeton University Press, ISBN978-0-691-08078-9.


Taneja, HC (2008), "Chapter 18: Fourier integrals and Fourier transforms" [11], Advanced EngineeringMathematics:, Volume 2, New Delhi, India: I. K. International Pvt Ltd, ISBN8189866567.

Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), Oxford University: ClarendonPress (published 1986), ISBN978-0-8284-0324-5.

Wilson, R. G. (1995), Fourier Series and Optical Transform Techniques in Contemporary Optics, New York:Wiley, ISBN0-471-30357-7.

Yosida, K. (1968), Functional Analysis, Springer-Verlag, ISBN3-540-58654-7.

External links The Discrete Fourier Transformation (DFT): Definition and numerical examples [12] A Matlab tutorial The Fourier Transform Tutorial Site [13] (thefouriertransform.com) Fourier Series Applet [14] (Tip: drag magnitude or phase dots up or down to change the wave form). Stephan Bernsee's FFTlab [15] (Java Applet) Stanford Video Course on the Fourier Transform [16]

Hazewinkel, Michiel, ed. (2001), "Fourier transform" [17], Encyclopedia of Mathematics, Springer,ISBN978-1-55608-010-4

Weisstein, Eric W., "Fourier Transform" [18], MathWorld. The DFT Pied: Mastering The Fourier Transform in One Day [19] at The DSP Dimension An Interactive Flash Tutorial for the Fourier Transform [20]

Java Library for DFT [21]

Fourier Transforms [22] Gary D. Knott Fourier Transforms [23] on www.continuummechanics.org [24]

References[1] http:/ / books. google. com/ books?id=TDQJAAAAIAAJ& pg=PA525& dq=%22c%27est-%C3%A0-dire+ qu%27on+ a+

l%27%C3%A9quation%22& hl=en& sa=X& ei=SrC7T9yKBorYiALVnc2oDg& sqi=2& ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22& f=false

[2] http:/ / books. google. com/ books?id=-N8EAAAAYAAJ& pg=PA408& dq=%22that+ is+ to+ say,+ that+ we+ have+ the+ equation%22&hl=en& sa=X& ei=F667T-u5I4WeiALEwpHXDQ& ved=0CDgQ6AEwAA#v=onepage&q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22& f=false

[3] http:/ / dx. doi. org/ 10. 1007%2Fs00041-012-9242-5[4] http:/ / www. ams. org/ mathscinet-getitem?mr=0262773[5] http:/ / books. google. com/ books?id=rfRnrhJwoloC& pg=PA29& dq=%22becomes+ the+ Fourier+ %28integral%29+ transform%22&

hl=en& sa=X& ei=osO7T7eFOqqliQK3goXoDQ& ved=0CDQQ6AEwAA#v=onepage&q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22& f=false

[6] http:/ / books. google. com/ ?id=QCcW1h835pwC[7] http:/ / books. google. com/ books?id=tlLE4KUkk1gC& pg=PA256& dq=%22The+ Fourier+ transform+ of+ the+ measure%22& hl=en&

sa=X& ei=w8e7T43XJsiPiAKZztnRDQ& ved=0CEUQ6AEwAg#v=onepage&q=%22The%20Fourier%20transform%20of%20the%20measure%22& f=false

[8] http:/ / books. google. com/ books?id=k_rdcKaUdr4C& pg=PA10[9] http:/ / books. google. com/ books?id=FAOc24bTfGkC& pg=PA158& dq=%22The+ mathematical+ thrust+ of+ the+ principle%22& hl=en&

sa=X& ei=Esa7T5PZIsqriQKluNjPDQ& ved=0CDQQ6AEwAA#v=onepage&q=%22The%20mathematical%20thrust%20of%20the%20principle%22& f=false

[10] http:/ / books. google. com/ books?id=YUCV678MNAIC& dq=editions:xbArf-TFDSEC& source=gbs_navlinks_s[11] http:/ / books. google. com/ books?id=X-RFRHxMzvYC& pg=PA192& dq=%22The+ Fourier+ integral+ can+ be+ regarded+ as+ an+

extension+ of+ the+ concept+ of+ Fourier+ series%22& hl=en& sa=X& ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=false

[12] http:/ / www. nbtwiki. net/ doku. php?id=tutorial:the_discrete_fourier_transformation_dft[13] http:/ / www. thefouriertransform. com[14] http:/ / www. westga. edu/ ~jhasbun/ osp/ Fourier. htm


[15] http:/ / www. dspdimension. com/ fftlab/[16] http:/ / www. academicearth. org/ courses/ the-fourier-transform-and-its-applications[17] http:/ / www. encyclopediaofmath. org/ index. php?title=p/ f041150[18] http:/ / mathworld. wolfram. com/ FourierTransform. html[19] http:/ / www. dspdimension. com/ admin/ dft-a-pied/[20] http:/ / www. fourier-series. com/ f-transform/ index. html[21] http:/ / www. patternizando. com. br/ 2013/ 05/ transformadas-discretas-wavelet-e-fourier-em-java/[22] http:/ / www. civilized. com/ files/ newfourier. pdf[23] http:/ / www. continuummechanics. org/ cm/ fourierxforms. html[24] http:/ / www. continuummechanics. org/

Article Sources and Contributors 24

Article Sources and ContributorsFourier transform Source: http://en.wikipedia.org/w/index.php?oldid=624967828 Contributors: 9258fahsflkh917fas, A Doon, A. Pichler, Abecedare, Adam.stinchcombe, Admartch,Adoniscik, Ahoerstemeier, Akbg, Albandil, Alejo2083, Ali salsa, AliceNovak, Alipson, Amaher, AnAj, Andrei Polyanin, Andres, Angalla, Anna Lincoln, Anna Roy, Anoko moonlight, Ap,Army1987, Arondals, Asmeurer, Astronautameya, Avicennasis, Avoided, AxelBoldt, BD2412, Barak Sh, Bci2, Bdmy, BehzadAhmadi, BenFrantzDale, BigJohnHenry, Bmcginty2, Bo Jacoby,Bob K, Bobblewik, Bobo192, BorisG, Breno, Brews ohare, Bugnot, Bumm13, Burhem, Butala, Bwb1729, CSTAR, Caio2112, Cassandra B, Catslash, Cburnett, CecilWard, Ch mad, CharlesMatthews, Chris the speller, ChrisGualtieri, ClickRick, Cmghim925, Colonies Chris, Complexica, Compsonheir, Coppertwig, CrisKatz, Crisfilax, Cuzkatzimhut, Cyrapas, DVdm, DX-MON, Danuke, DabMachine, Daqu, David R. Ingham, DavidCBryant, Davyzhu, Dcirovic, Demosta, Dhabih, Discospinster, DmitTrix, Dmmaus, Dougweller, Download, Dr.enh, DrBob, Drew335,Drilnoth, Dysprosia, EconoPhysicist, Ed g2s, Eliyak, Elkman, Enochlau, Epzcaw, Favonian, Feline Hymnic, Feraudyh, Fgnievinski, Fizyxnrd, Forbes72, Formula255, Foxj, Fr33kman,Frappyjohn, Fred Bradstadt, Freiddie, Frietjes, Fropuff, Futurebird, Gaidheal1, Gaius Cornelius, Gareth Owen, Geekdiva, Giftlite, Giovannidimauro, Glenn, Graphium, GuidoGer, GyroMagician,H2g2bob, HappyCamper, Heimstern, HenningThielemann, Herr Lip, Hesam7, HirsuteSimia, Hrafeiro, Ht686rg90, I am a mushroom, Ianweiner, Igny, Iihki, Imroy, Ivan Shmakov, Iwfyita,Jaakobou, Jdorje, Jfmantis, Jhealy, Jko, Joerite, John Cline, JohnBlackburne, JohnOFL, JohnQPedia, Joriki, Jose Brox, Jpbowen, Justwantedtofixonething, KHamsun, KYN, Keenan Pepper,Kevmitch, Klallas, Kostmo, Kri, Kunaporn, Larsobrien, Linas, LokiClock, Loodog, Looxix, Lovibond, LucasVB, Luciopaiva, Lupin, M1ss1ontomars2k4, Maine12329, Manik762007, Maschen,MathKnight, Maxim, Mckee, Mct mht, Mecanismo, Metacomet, Michael Hardy, Mikeblas, Mikiemike, Millerdl, Moxfyre, Mr. PIM, NTUDISP, Naddy, NameIsRon, NathanHagen, Nbarth,NickGarvey, Nihil, Nishantjr, Njerseyguy, Njm7203, Nk, Nmnogueira, NokMok, NotWith, Nscozzaro, Od Mishehu, Offsure, Oleg Alexandrov, Oli Filth, Omegatron, Oreo Priest, Ouzel Ring,PAR, Pak21, Papa November, Paul August, Pbubenik, Pedrito, Pete463251, Petergans, Phasmatisnox, Phils, PhotoBox, PigFlu Oink, Poincarecon, Pol098, Policron, PsiEpsilon, PtDw832,Publichealthguru, Quercus solaris, Quietbritishjim, Quintote, Qwfp, R.e.b., Raffamaiden, Rainwarrior, Rbj, Red Winged Duck, Riesz, Rifleman 82, Rijkbenik, Rjwilmsi, RobertHannah89, Rror,Rs2, Rurz2007, SKvalen, Safenner1, Sai2020, Sandb, Sbyrnes321, SchreiberBike, SebastianHelm, Sepia tone, Sgoder, Sgreddin, Shreevatsa, Silly rabbit, Slawekb, SlimDeli, Smibu, Snigbrook,Snoyes, Sohale, Soulkeeper, SpaceFlight89, Sprocedato, Stausifr, Stevan White, Stevenj, Stpasha, StradivariusTV, Sun Creator, Sunev, Sverdrup, Sylvestersteele, Sawomir Biay, THEN WHOWAS PHONE?, TYelliot, Tabletop, Tahome, TakuyaMurata, TarryWorst, Tedickey, Tetracube, The Thing That Should Not Be, Thennicke, Thenub314, Thermochap, Thinking of England, TimGoodwyn, Tim Starling, Tinos, Tobias Bergemann, Tobych, TranceThrust, Transistortron, Trovatore, Tunabex, Ujjalpatra, User A1, Vadik wiki, Vasi, Verdy p, VeryNewToThis, VictorAnyakin,Vidalian Tears, Vnb61, Voronwae, WLior, Waldir, Wavelength, Whikie, Wiki Edit Testing, WikiDao, Wile E. Heresiarch, Writer130, Wwheaton, Ybhatti, YouRang?, Zoz, Zvika, 668anonymous edits

Image Sources, Licenses and ContributorsFile:Fourier transform time and frequency domains (small).gif Source: http://en.wikipedia.org/w/index.php?title=File:Fourier_transform_time_and_frequency_domains_(small).gif License:Public Domain Contributors: Lucas V. BarbosaFile:Function ocsillating at 3 hertz.svg Source: http://en.wikipedia.org/w/index.php?title=File:Function_ocsillating_at_3_hertz.svg License: Creative Commons Attribution-Sharealike 3.0Contributors: Thenub314File:Onfreq.svg Source: http://en.wikipedia.org/w/index.php?title=File:Onfreq.svg License: GNU Free Documentation License Contributors: Original: Nicholas Longo, SVG conversion:DX-MON (Richard Mant)File:Offfreq.svg Source: http://en.wikipedia.org/w/index.php?title=File:Offfreq.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Thenub314File:Fourier transform of oscillating function.svg Source: http://en.wikipedia.org/w/index.php?title=File:Fourier_transform_of_oscillating_function.svg License: Creative CommonsAttribution-Sharealike 3.0 Contributors: Thenub314File:Rectangular function.svg Source: http://en.wikipedia.org/w/index.php?title=File:Rectangular_function.svg License: GNU Free Documentation License Contributors: Aflafla1, Axxgreazz,Bender235, Darapti, Jochen Burghardt, Omegatron, Perhelion, 1 anonymous editsFile:Sinc function (normalized).svg Source: http://en.wikipedia.org/w/index.php?title=File:Sinc_function_(normalized).svg License: GNU Free Documentation License Contributors: Aflafla1,Bender235, Jochen Burghardt, Juiced lemon, Krishnavedala, Omegatron, Perhelion, Pieter Kuiper, SarangFile:Commutative diagram illustrating problem solving via the Fourier transform.svg Source:http://en.wikipedia.org/w/index.php?title=File:Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg License: Creative Commons Attribution-Sharealike 3.0Contributors: User:Quietbritishjim

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Fourier transformDefinitionIntroductionExample

Properties of the Fourier transformBasic propertiesInvertibility and periodicityUniform continuity and the RiemannLebesgue lemmaPlancherel theorem and Parseval's theoremPoisson summation formulaConvolution theoremCross-correlation theoremEigenfunctions

Fourier transform on Euclidean spaceUncertainty principleSpherical harmonicsRestriction problems

Fourier transform on function spacesOn Lp spacesTempered distributions

GeneralizationsFourierStieltjes transformLocally compact abelian groupsGelfand transformNon-abelian groups

AlternativesApplicationsAnalysis of differential equationsFourier transform spectroscopyQuantum mechanics and signal processing

Other notationsOther conventionsTables of important Fourier transformsFunctional relationshipsSquare-integrable functionsDistributionsTwo-dimensional functionsFormulas for general n-dimensional functions

ReferencesExternal links

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