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Fourier transforms 

Continuous Fourier transform 
Fourier series 
Discretetime Fourier transform 
Discrete Fourier transform 
Fourier transforms 

Continuous Fourier transform 
Fourier series 
Discretetime Fourier transform 
Discrete Fourier transform 
The Fourier transform, named for Joseph Fourier, is a mathematical transform that expresses a mathematical function of time as a function of frequency. For instance, the transform of a musical chord made up of pure notes without overtones, expressed as loudness as a function of time, is a mathematical representation of the amplitudes and phases of the individual notes that make it up. The function of time is often called the time domain representation, and the function of frequency is called the frequency domain representation. The inverse Fourier transform expresses a frequency domain function in the time domain. Each value of the function is usually expressed as a complex number (called complex amplitude) that can be interpreted as an absolute value and a phase component. In the case of a periodic function, such as a musical tone (possibly with overtones), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients.
The Fourier transform has many applications in physics and engineering. Fourier transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. This often results in simplification of needed mathematical manipulations. The Fourier transform is reversible, being able to transform from either domain to the other. The term itself refers to both the transform operation and to the function it produces.
There 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:
When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, is determined by via the inverse transform:
The statement that can be reconstructed from is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions 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 ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.
The motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity (Taneja 2008, p. 192).
Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral. In many cases it is desirable to use Euler's formula, which states that e^{2πiθ} = cos(2πθ) + i sin(2πθ), to write Fourier series in terms of the basic waves e^{2πiθ}. This has the advantage of simplifying many of the formulas involved, and provides a formulation for Fourier series that more closely resembles the definition followed in this article. Rewriting sines and cosines as complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. These complex exponentials sometimes contain negative "frequencies". If θ is measured in seconds, then the waves e^{2πiθ} and e^{−2πiθ} both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no 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 which are zero outside of an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where f is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of f begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [−T/2, T/2] contains the interval on which f is not identically zero. Then the nth series coefficient c_{n} is given by:
Comparing this to the definition of the Fourier transform, it follows that since f(x) is zero outside [−T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 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 ξ_{n} = n/T, and Δξ = (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 Fourier transform given in the definition section. Under suitable conditions this argument may be made precise (Stein & Shakarchi 2003).
In the study of Fourier series the numbers c_{n} could be thought of as the "amount" of the wave present in the Fourier series of f. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of 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.
The following images provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The function depicted f(t) = cos(6πt) e^{−πt2} oscillates at 3 hertz (if t measures seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse. Its general form is a Gaussian function). This function was specially chosen to have a real Fourier transform which can easily be plotted. The first image contains its graph. In order to calculate we must integrate e^{−2πi(3t)}f(t). The second image shows the plot of the real and imaginary parts of this function. The real part of the integrand is almost always positive, because when f(t) is negative, the real part of e^{−2πi(3t)} is negative as well. Because they oscillate at the same rate, when f(t) is positive, so is the real part of e^{−2πi(3t)}. The result is that when you integrate the real part of the integrand you get a relatively large number (in this case 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 more complicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequency is present in a function f(t).
Here we assume f(x), g(x) and h(x) are integrable functions, are Lebesguemeasurable on the real line, and satisfy:
We denote the Fourier transforms of these functions by , and respectively.
The Fourier transform has the following basic properties: (Pinsky 2002).
That is, the evaluation of the Fourier transform in the origin () equals the integral of f all over its domain.
Under suitable conditions on the function f, 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 twoperiodic, applying this twice yields so the Fourier transform operator is fourperiodic, 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 that inverts time, :
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 twofold 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 xaxis and frequency as the yaxis), 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 SL_{2}(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.
The Fourier transform may be defined in some cases for nonintegrable functions, but the Fourier transforms of integrable functions have several strong properties.
The Fourier transform, , of any integrable function f is uniformly continuous and (Katznelson 1976). By the Riemann–Lebesgue lemma (Stein & Weiss 1971),
However, 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 f and are integrable, the inverse equality
holds almost everywhere. That is, the Fourier transform is injective on L^{1}(R). (But if f is continuous, then equality holds for every x.)
Let f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are also squareintegrable, 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 unitary operator on L^{2}(R). On L^{1}(R)∩L^{2}(R), this extension agrees with original Fourier transform defined on L^{1}(R), thus enlarging the domain of the Fourier transform to L^{1}(R) + L^{2}(R) (and consequently to L^{p}(R) for 1 ≤ p ≤ 2). The Plancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. Depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.
See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and timeshifting properties. The frequencydomain dual of the standard PSF is also called discretetime Fourier transform, which leads directly to:
The Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) 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:
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 system with 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 the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms and .
In an analogous manner, it can be shown that if h(x) is the crosscorrelation 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
One important choice of an orthonormal basis for L^{2}(R) is given by the Hermite functions
where He_{n}(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 Fourier transform on L^{2}(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L^{2}(R) as a direct sum of four spaces H_{0}, H_{1}, H_{2}, and H_{3} where the Fourier transform acts on He_{k} simply by multiplication by i^{k}.
Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be represented 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). 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 timefrequency analysis (Boashash 2003). In physics, this transform was introduced by Edward Condon (Condon 1937).
The Fourier transform can be defined in any arbitrary number of dimensions n. As with the onedimensional case, there are many conventions. For an integrable function f(x), this article takes the definition:
where x and ξ are ndimensional vectors, and x · ξ is the dot product of the vectors. The dot product is sometimes written as .
All of the basic properties listed above hold for the ndimensional 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. (Stein & Weiss 1971)
Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its Fourier transform "stretches out" in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.
The tradeoff between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.
Suppose f(x) is an integrable and squareintegrable function. Without loss of generality, assume that f(x) is normalized:
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 x·f(x) and f′(x) are square integrable, then
The equality is attained only in the case (hence ) where σ > 0 is arbitrary and so that f is L^{2}–normalized (Pinsky 2002). In other words, where f is a (normalized) Gaussian function with variance σ^{2}, centered at zero, and its Fourier transform is a Gaussian function with variance σ^{−2}.
In fact, this inequality implies that:
for any x_{0}, ξ_{0} ∈ R (Stein & Shakarchi 2003, p. 158).
In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg 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 the previous case.
Let the set of homogeneous harmonic polynomials of degree k on R^{n} be denoted by A_{k}. The set A_{k} consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e^{−πx2}P(x) for some P(x) in A_{k}, then . Let the set H_{k} be the closure in L^{2}(R^{n}) of linear combinations of functions of the form f(x)P(x) where P(x) is in A_{k}. The space L^{2}(R^{n}) is then a direct sum of the spaces H_{k} and the Fourier transform maps each space H_{k} to itself and is possible to characterize the action of the Fourier transform on each space H_{k} (Stein & Weiss 1971). Let f(x) = f_{0}(x)P(x) (with P(x) in A_{k}), then where
Here J_{(n + 2k − 2)/2} denotes the Bessel function of the first kind with order (n + 2k − 2)/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 transform. Moreover, there is a simple recursion relating the cases n+2 and n (Grafakos & Teschl 2013) allowing to compute, e.g., the threedimensional Fourier transform of a radial function from the onedimensional one.
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 squareintegrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L^{2}(R^{n}) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in L^{p} for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has nonzero curvature. The case when S is the unit sphere in R^{n} is of particular interest. In this case the TomasStein restriction theorem states that the restriction of the Fourier transform to the unit sphere in R^{n} is a bounded operator on L^{p} provided 1 ≤ p ≤ (2n + 2) / (n + 3).
One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets E_{R} indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function f, consider the function f_{R} defined by:
Suppose in addition that f ∈ L^{p}(R^{n}). For n = 1 and 1 < p < ∞, if one takes E_{R} = (−R, R), then f_{R} converges to f in L^{p} as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that E_{R} is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball E_{R} = {ξ : ξ < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in L^{p}(R^{n}). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2 (Duoandikoetxea 2001). In fact, when p ≠ 2, this shows that not only may f_{R} fail to converge to f in L^{p}, but for some functions f ∈ L^{p}(R^{n}), f_{R} is not even an element of L^{p}.
The definition of the Fourier transform by the integral formula
is valid for Lebesgue integrable functions f; that is, f ∈ L^{1}(R^{n}).
The Fourier transform : L^{1}(R^{n}) → L^{∞}(R^{n}) is a bounded operator. This follows from the observation that
which shows that its operator norm is bounded by 1. Indeed it equals 1, which can be seen, for example, from the transform of the rect function. The image of L^{1} is a subset of the space C_{0}(R^{n}) of continuous functions that tend to zero at infinity (the Riemann–Lebesgue lemma), although it is not the entire space. Indeed, there is no simple characterization of the image.
Since compactly supported smooth functions are integrable and dense in L^{2}(R^{n}), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L^{2}(R^{n}) by continuity arguments. The Fourier transform in L^{2}(R^{n}) is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an L^{2} function f,
where the limit is taken in the L^{2} sense. Many of the properties of the Fourier transform in L^{1} carry over to L^{2}, by a suitable limiting argument.
Furthermore : L^{2}(R^{n}) → L^{2}(R^{n}) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any f,g∈L^{2}(R^{n}) we have
In particular, the image of L^{2}(R^{n}) is itself under the Fourier transform.
The definition of the Fourier transform can be extended to functions in L^{p}(R^{n}) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L^{2} plus a fat body part in L^{1}. In each of these spaces, the Fourier transform of a function in L^{p}(R^{n}) is in L^{q}(R^{n}), where is the Hölder conjugate of p. by the Hausdorff–Young inequality. However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in L^{p} for the range 2 < p < ∞ requires the study of distributions (Katznelson 1976). In fact, it can be shown that there are functions in L^{p} with p > 2 so that the Fourier transform is not defined as a function (Stein & Weiss 1971).
One might consider enlarging the domain of the Fourier transform from L^{1}+L^{2} by considering generalized functions, or distributions. A distribution on R^{n} is a continuous linear functional on the space C_{c}(R^{n}) of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on C_{c}(R^{n}) and pass to distributions by duality. The obstruction to do this is that the Fourier transform does not map C_{c}(R^{n}) to C_{c}(R^{n}). In fact the Fourier transform of an element in C_{c}(R^{n}) can not vanish on an open set; see the above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartz functions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions (Stein & Weiss 1971). The tempered distributions include all the integrable functions mentioned above, as well as wellbehaved functions of polynomial growth 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 multiplication formula (Stein & Weiss 1971),
Every integrable function f defines (induces) a distribution T_{f} by the relation
So it makes sense to define Fourier transform of T_{f} by
for all Schwartz functions φ. Extending this to all tempered distributions T gives the general definition of the Fourier transform.
Distributions can be differentiated and the abovementioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
The Fourier transform of a finite Borel measure μ on R^{n} is given by (Pinsky 2002, p. 256):
This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures (Katznelson 1976). In the case that dμ = f(x)dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the FourierStieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take e^{ix·ξ} instead of e^{−2πix·ξ} (Pinsky 2002). In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied 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 which functions may arise as the Fourier–Stieltjes 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 a constant function (whose specific value depends upon the form of the Fourier transform used).
The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group which is at the same time a locally compact Hausdorff topological space so that the group operation is continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G, the set of irreducible, i.e. onedimensional, unitary representations are called its characters. With its natural group structure and the topology of pointwise convergence, the set of characters is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L^{1}(G), its Fourier transform is defined by (Katznelson 1976):
The RiemannLebesgue lemma holds in this case; is a function vanishing at infinity on .
The Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to the Pontryagin duality map defined above.
Given an abelian locally compact Hausdorff topological group G, as before we consider space L^{1}(G), defined using a Haar measure. With convolution as multiplication, L^{1}(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 group C*algebra C*(G) of G. (Any C*norm on L^{1}(G) is bounded by the L^{1} norm, therefore their supremum exists.)
Given any abelian C*algebra A, the Gelfand transform gives an isomorphism between A and C_{0}(A^), where A^ is the multiplicative linear functionals, i.e. onedimensional representations, on A with the weak* topology. The map is simply given by
It turns out that the multiplicative linear functionals of C*(G), after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset L^{1}(G) is the FourierPontryagin transform.
The Fourier transform can also be defined for functions on a nonabelian group, provided that the group is compact. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be onedimensional. This means the Fourier transform on a nonabelian 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 noncommutative harmonic analysis.
Let G be a compact Hausdorff topological group. Let Σ denote the collection of all isomorphism classes of finitedimensional irreducible unitary representations, along with a definite choice of representation U^{(σ)} on the Hilbert space H_{σ} of finite dimension d_{σ} for each σ ∈ Σ. If μ is a finite Borel measure on G, then the Fourier–Stieltjes transform of μ is the operator on H_{σ} defined by
where is the complexconjugate representation of U^{(σ)} acting on H_{σ}. If μ is absolutely continuous with respect to the leftinvariant probability measure λ on G, represented as
for some f ∈ L^{1}(λ), one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.
The mapping defines an isomorphism between the Banach space M(G) of finite Borel measures (see rca space) and a closed subspace of the Banach space C_{∞}(Σ) consisting of all sequences E = (E_{σ}) indexed by Σ of (bounded) linear operators E_{σ}: H_{σ} → H_{σ} for which the norm
is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of C* algebras into a subspace of C_{∞}(Σ). Multiplication on M(G) is given by convolution of measures and the involution * defined by
and C_{∞}(Σ) has a natural C*algebra structure as Hilbert space operators.
The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f ∈ L^{2}(G), then
where the summation is understood as convergent in the L^{2} sense.
The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry.^{[citation needed]} In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.
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 timefrequency analysis, one uses timefrequency transforms or timefrequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a tradeoff between these. These can be generalizations of the Fourier transform, such as the shorttime Fourier transform or fractional Fourier transform, or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. (Boashash 2003).
Fourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(x) is a differentiable function with Fourier transform , then the Fourier transform of its derivative is given by . This can be used to transform differential equations into algebraic equations. This technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables partial differential equations with domain R^{n} can also be translated into algebraic equations.
The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fouriertransformed to a Lorentzian lineshape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry.
In quantum mechanics, Fourier transforms of solutions to the Schrödinger equation are known as momentum space (or k space) wave functions. They display the amplitudes for momenta. Their absolute squares are the probabilities of momenta. This is valid also for classical waves treated in signal processing, such as in swept frequency radar where data is taken in frequency domain and transformed to time domain, yielding range. The absolute square is then the power.
Other common notations for include:
Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such as f(x) and F(ξ)) is especially common in the sciences and engineering. In electronics, the omega (ω) is often used instead of ξ due to its interpretation as angular frequency, sometimes it is written as F(jω), where j is the imaginary unit, to indicate its relationship with the Laplace transform, and sometimes it is written informally as F(2πf) in order to 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(ξ) and φ(ξ) 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 the form e^{2πixξ} whose amplitude is A(ξ) and whose initial phase angle (at x = 0) is φ(ξ).
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 can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function 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 this function evaluated at the value ξ for its variable, and this is denoted either as or as . Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around.
In mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted f(x). This means that a notation like formally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.
For example, is sometimes used to express that the Fourier transform of a rectangular function 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, not of x_{0}.
The Fourier transform can also be written in terms of angular frequency: ω = 2πξ whose units are radians per second.
The substitution ξ = ω/(2π) 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 a unitary transformation on L^{2}(R^{n}). There is also less symmetry between the formulas for the Fourier transform and its inverse.
Another convention is to split the factor of (2π)^{n} evenly between the Fourier transform and its inverse, which leads to definitions:
Under this convention, the Fourier transform is again a unitary transformation on L^{2}(R^{n}). It also restores the symmetry between the Fourier transform and its inverse.
Variations of all three conventions can be created by conjugating the complexexponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.
ordinary frequency ξ (hertz)  unitary  

angular frequency ω (rad/s)  nonunitary  
unitary 
As discussed above, the characteristic function of a random variable is the same as the Fourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined .
As in the case of the "nonunitary angular frequency" convention above, there is no factor of 2π appearing in either of the integral, or in the exponential. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponential.
It has been suggested that this section be split into a new article. (Discuss) Proposed since October 2014. 
The following tables record some closedform Fourier transforms. For functions f(x), g(x) and h(x) denote their Fourier transforms by , , and respectively. 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.
The Fourier transforms in this table may be found in Erdélyi (1954) or Kammler (2000, appendix).
Function  Fourier transform unitary, ordinary frequency  Fourier transform unitary, angular frequency  Fourier transform nonunitary, angular frequency  Remarks  

Definition  
101  Linearity  
102  Shift in time domain  
103  Shift in frequency domain, dual of 102  
104  Scaling in the time domain. If is large, then is concentrated around 0 and spreads out and flattens.  
105  Duality. Here needs to be calculated using the same method as Fourier transform column. Results from swapping "dummy" variables of and or or .  
106  
107  This is the dual of 106  
108  The notation denotes the convolution of and — this rule is the convolution theorem  
109  This is the dual of 108  
110  For purely real  Hermitian symmetry. indicates the complex conjugate.  
111  For a purely real even function  , and are purely real even functions.  
112  For a purely real odd function  , and are purely imaginary odd functions.  
113  Complex conjugation, generalization of 110 
The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdélyi 1954), or the appendix of (Kammler 2000).
Function  Fourier transform unitary, ordinary frequency  Fourier transform unitary, angular frequency  Fourier transform nonunitary, angular frequency  Remarks  

201  The rectangular pulse and the normalized sinc function, here defined as sinc(x) = sin(πx)/(πx)  
202  Dual of rule 201. The rectangular function is an ideal lowpass filter, and the sinc function is the noncausal impulse response of such a filter. The sinc function is defined here as sinc(x) = sin(πx)/(πx)  
203  The function tri(x) is the triangular function  
204  Dual of rule 203.  
205  The function u(x) is the Heaviside unit step function and a>0.  
206  This shows that, for the unitary Fourier transforms, the Gaussian function exp(−αx^{2}) is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α)>0.  
207  For a>0. That is, the Fourier transform of a twosided decaying exponential function is a Lorentzian function.  
208  Hyperbolic secant is its own Fourier transform  
209 


 is the Hermite's polynomial. If a = 1 then the GaussHermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for n = 0. 
The Fourier transforms in this table may be found in (Erdélyi 1954) or the appendix of (Kammler 2000).
Function  Fourier transform unitary, ordinary frequency  Fourier transform unitary, angular frequency  Fourier transform nonunitary, angular frequency  Remarks  

301  The distribution δ(ξ) denotes the Dirac delta function.  
302  Dual of rule 301.  
303  This follows from 103 and 301.  
304  This follows from rules 101 and 303 using Euler's formula:  
305  This follows from 101 and 303 using  
306  
307  
308  Here, n is a natural number and is the nth 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.  
309  Here sgn(ξ) is the sign function. Note that 1/x 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.  
310  1/x^{n} is the homogeneous distribution defined by the distributional derivative  
311  This formula is valid for 0 > α > −1. For α > 0 some singular terms arise at the origin that can be found by differentiating 318. If Re α > −1, then is a locally integrable function, and so a tempered distribution. The function is a holomorphic function from the right halfplane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted for α ≠ −2, −4, ... (See homogeneous distribution.)  
Special case of 311.  
312  The dual of rule 309. This time the Fourier transforms need to be considered as Cauchy principal value.  
313  The function u(x) is the Heaviside unit step function; this follows from rules 101, 301, and 312.  
314  This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact that as distributions.  
315  The function J_{0}(x) is the zeroth order Bessel function of first kind.  
316  This is a generalization of 315. The function J_{n}(x) is the nth order Bessel function of first kind. The function T_{n}(x) is the Chebyshev polynomial of the first kind.  
317  is the Euler–Mascheroni constant.  
318  This formula is valid for 1 > α > 0. Use differentiation to derive formula for higher exponents. u is the Heaviside function. 
Function  Fourier transform unitary, ordinary frequency  Fourier transform unitary, angular frequency  Fourier transform nonunitary, angular frequency  

400  
401  
402 
To 400: The variables ξ_{x}, ξ_{y}, ω_{x}, ω_{y}, ν_{x} and ν_{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 0≤r≤1, and is 0 otherwise. This is the Airy distribution, and is expressed using J_{1} (the order 1 Bessel function of the first kind). (Stein & Weiss 1971, Thm. IV.3.3)
Function  Fourier transform unitary, ordinary frequency  Fourier transform unitary, angular frequency  Fourier transform nonunitary, angular frequency  

500  
501  
502  
503  
504 
To 501: The function χ_{[0, 1]} is the indicator function of the interval [0, 1]. The function Γ(x) is the gamma function. The function J_{n/2 + δ} is a Bessel function of the first kind, with order n/2 + δ. Taking n = 2 and δ = 0 produces 402. (Stein & Weiss 1971, Thm. 4.15)
To 502: See Riesz potential. The formula also holds for all α ≠ −n, −n − 1, … by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See homogeneous distribution.
To 503: 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
To 504: Here . See (Stein & Weiss 1971, p. 6).