The
Fourier transform is a mathematical operation that decomposes a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
into its constituent frequencies, known as a
frequency spectrumThe frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...
. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes (and
phase-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
) of the individual notes that make it up. The composite waveform depends on
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, and therefore is called the
time domainTime domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
representation. The frequency spectrum is a function of frequency and is called the
frequency domainIn electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
representation. Each value of the function is a complex number (called
complex amplitude) that encodes both a magnitude and phase component. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces.
In the case of a
periodic functionIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
, like the musical chord, the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called
Fourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
coefficients. Also, when a time-domain function is
sampledIn signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
to facilitate storage and/or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the
Poisson summation formulaIn mathematics, the Poisson summation formula 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. Consequently, the periodic summation of a function is completely defined by discrete samples...
, also known as
discrete-time Fourier transformIn mathematics, the discrete-time Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function . But the DTFT requires an input function...
. These topics are addressed in separate articles. For an overview of those and other related operations, refer to
Fourier analysis or
List of Fourier-related transforms.
Definition
There are several common conventions for defining the Fourier transform
of an
integrableIn mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
function . This article will use the definition:
for every
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
ξ.
When the independent variable
x represents
time (with
SISi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of
secondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
s), the transform variable
ξ represents
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
(in
hertzThe hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
). Under suitable conditions,
ƒ can be reconstructed from
by the
inverse transform:
for every real number
x.
For other common conventions and notations, including using the
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
ωOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
instead of the
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal 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.
Introduction
The motivation for the Fourier transform comes from the study of
Fourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
. In the study of Fourier series, complicated functions are written as the sum of simple waves mathematically represented by
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s and cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
, which states that
e2πiθ = cos 2
πθ +
i sin 2
πθ, to write Fourier series in terms of the basic waves
e2πiθ. This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed in this article. This passage from sines and cosines to 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
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
(or size) of the wave present in the function and the
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
(or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". If
θ were measured in seconds then the waves
e2πiθ and
e−2πiθ would 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 closely related.
There is a close connection between the definition of Fourier series and the Fourier transform for functions
ƒ which are zero outside of an interval. For such a function we can calculate its Fourier series on any interval that includes the interval where
ƒ 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
ƒ 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
ƒ 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
ƒ(
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. As
T increases the Fourier coefficients more closely represent the Fourier transform of the function.
Under appropriate conditions the sum of the Fourier series of
ƒ will equal the function
ƒ. In other words
ƒ can be written
:
where the last sum is simply the first sum rewritten using the definitions
ξn =
n/
T, and Δ
ξ = (
n + 1)/
T −
n/
T = 1/
T.
This second sum is a
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
, 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 .
In the study of Fourier series the numbers
cn could be thought of as the "amount" of the wave in the Fourier series of
ƒ. 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
ƒ, 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
oscillates at 3 hertz (if
t measures seconds) and tends quickly to 0. 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)ƒ(
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, this is because when
ƒ(
t) is negative, then the real part of
e−2πi(3t) is negative as well. Because they oscillate at the same rate, when
ƒ(
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
ƒ(
t).
Properties of the Fourier transform
An
integrable function is a function
ƒ on the real line that is Lebesgue-measurable and satisfies
Basic properties
Given integrable functions
f(
x),
g(
x), and
h(
x), their Fourier transforms are denoted by
,
, and
respectively. The Fourier transform has the following basic properties .
Linearity
- For any complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s a and b, if h(x) = aƒ(x) + bg(x), then 
Translation
- For any real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
x0, if h(x) = ƒ(x − x0), then 
Modulation
- For any real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
ξ0, if h(x) = e2πixξ0ƒ(x), then 
.
Scaling
- For a non-zero real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
a, if h(x) = ƒ(ax), then 
. The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then 
.
Conjugation
- If
, then 
- In particular, if ƒ is real, then one has the reality condition
- And if ƒ is purely imaginary, then
Duality
- If
then 
Convolution
- If
, then 
Uniform continuity and the Riemann–Lebesgue lemma
The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties.
The Fourier transform
of any integrable function
ƒ is uniformly continuous and
. By the
Riemann–Lebesgue lemma ,
Furthermore,
is bounded and continuous, but 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 integralIn calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
s 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
ƒ and
are integrable, the inverse equality
holds
almost everywhereIn measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
. That is, the Fourier transform is injective on
L1(R)In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
.
(But if
ƒ is continuous, then equality holds for every
x.)
The 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 also square-integrable, then we have
Parseval's theoremIn mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
:
-
where the bar denotes complex conjugation.
The
Plancherel theoremIn mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
, which is equivalent to
Parseval's theoremIn mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
, states
:
The Plancherel theorem makes it possible to define the Fourier transform for functions in
L2(
R), as described in Generalizations below. The Plancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.
See
Pontryagin dualityIn mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
for a general formulation of this concept in the context of locally compact abelian groups.
Poisson summation formula
The Poisson summation formula 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 time-shifting properties. One such form leads directly to a proof of the Nyquist-Shannon sampling theorem.
Convolution theorem
The Fourier transform translates between
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
and multiplication of functions. If
ƒ(
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 theoryLinear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
, it is common to interpret
g(
x) as the
impulse responseIn signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
of an LTI system with input
ƒ(
x) and output
h(
x), since substituting the
unit impulseThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
for
ƒ(
x) yields
h(
x) =
g(
x). In this case,
represents the
frequency responseFrequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
of the system.
Conversely, if
ƒ(
x) can be decomposed as the product of two square integrable functions
p(
x) and
q(
x), then the Fourier transform of
ƒ(
x) is given by the convolution of the respective Fourier transforms
and
.
Cross-correlation theorem
In an analogous manner, it can be shown that if
h(
x) is the
cross-correlationIn signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
of
ƒ(
x) and
g(
x):
then the Fourier transform of
h(
x) is:
As a special case, the
autocorrelationAutocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
of function
ƒ(
x) is:
for which
Eigenfunctions
One important choice of an orthonormal basis for
L2(R)In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
is given by the Hermite functions
-
where
are the "probabilist's" Hermite polynomials, defined by
Hen(
x) = (−1)
nexp(
x2/2) D
n exp(−
x2/2). 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
L2(
R) . 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
L2(
R) as a direct sum of four spaces
H0,
H1,
H2, and
H3 where the Fourier transform acts on
Hek simply by multiplication by
ik. This approach to define the Fourier transform is due to N. Wiener . The choice of Hermite functions is convenient because they are exponentially localized in both frequency and time domains, and thus give rise to the
fractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...
used in time-frequency analysis .
Fourier transform on Euclidean space
The Fourier transform can be in any arbitrary number of dimensions
n. As with the one-dimensional case there are many conventions, for an integrable function
ƒ(
x) this article takes the definition
:
where
x and
ξ are
n-dimensional vectors, and is the
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of the vectors. The dot product is sometimes written as
.
All of the basic properties listed above hold for the
n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.
Uncertainty principle
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 trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
by viewing a function and its Fourier transform as
conjugate variablesConjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals of one-another, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty...
with respect to the symplectic form on the time–frequency domain: from the point of view of the
linear canonical transformationIn Hamiltonian mechanics, the linear canonical transformation is a family of integral transforms that generalizes many classical transforms...
, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the
symplectic formIn mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...
.
Suppose
ƒ(
x) is an integrable and square-integrable function. Without loss of generality, assume that
ƒ(
x) is normalized:
It follows from the
Plancherel theoremIn mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
that
is also normalized.
The spread around
x = 0 may be measured by the
dispersion about zero defined by
In probability terms, this is the
second momentIn mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
of
about zero.
The Uncertainty principle states that, if
ƒ(
x) is absolutely continuous and the functions
x·
ƒ(
x) and
ƒ′(
x) are square integrable, then
.
The equality is attained only in the case
(hence
) where
σ > 0 is arbitrary and
C1 is such that
ƒ is
L2–normalized . In other words, where
ƒ is a (normalized)
Gaussian function with variance σ
2, centered at zero, and its Fourier transform is a Gaussian function with variance 1/σ
2.
In fact, this inequality implies that:
-
for any
in
R .
In
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
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 .
A stronger uncertainty principle is the
Hirschman uncertainty principleIn quantum mechanics, information theory, and Fourier analysis, the Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies...
which is expressed as:
where
H(p) is the
differential entropyDifferential entropy is a concept in information theory that extends the idea of entropy, a measure of average surprisal of a random variable, to continuous probability distributions.-Definition:...
of the
probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
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.
Spherical harmonics
Let the set of
homogeneousIn mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
harmonicIn mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s of degree
k on
Rn be denoted by
Ak. The set
Ak 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−π|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 . Let
ƒ(
x) =
ƒ0(|
x|)
P(
x) (with
P(
x) in
Ak), then
where
Here
J(n + 2k − 2)/2 denotes the
Bessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first kind with order (
n + 2
k − 2)/2. When
k = 0 this gives a useful formula for the Fourier transform of a radial function .
Restriction problems
In higher dimensions it becomes interesting to study
restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general
class of square integrable functions. As such, the restriction of the Fourier transform of an
L2(
Rn) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in
Lp 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 non-zero curvature. The case when
S is the unit sphere in
Rn is of particular interest. In this case the Tomas-Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in
Rn is a bounded operator on
Lp provided 1 ≤
p ≤ .
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
ER indexed by
R ∈ (0,∞): such as balls of radius
R centered at the origin, or cubes of side 2
R. For a given integrable function
ƒ, consider the function
ƒR defined by:
Suppose in addition that
ƒ is in
Lp(
Rn). For
n = 1 and , if one takes
ER = (−R, R), then
ƒR converges to
ƒ in
Lp as
R tends to infinity, by the boundedness of the
Hilbert transformIn mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
. Naively one may hope the same holds true for
n > 1. In the case that
ER is taken to be a cube with side length
R, then convergence still holds. Another natural candidate is the Euclidean ball
ER = {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
Lp(
Rn). For
n ≥ 2 it is a celebrated theorem of
Charles FeffermanCharles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis....
that the multiplier for the unit ball is never bounded unless
p = 2 . In fact, when , this shows that not only may
ƒR fail to converge to
ƒ in
Lp, but for some functions
ƒ ∈
Lp(
Rn),
ƒR is not even an element of
Lp.
Fourier transform on other function spaces
It is possible to extend the definition of the Fourier transform to other spaces of functions. Since compactly supported smooth functions are integrable and dense in
L2(
R), the
Plancherel theoremIn mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
allows us to extend the definition of the Fourier transform to general functions in
L2(
R) by continuity arguments. Further
:
L2(
R) →
L2(
R) is a
unitary operatorIn functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
. Many of the properties remain the same for the Fourier transform. The
Hausdorff–Young inequalityIn mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. proved the inequality for some special values of q, and proved it in general...
can be used to extend the definition of the Fourier transform to include functions in
Lp(
R) for 1 ≤
p ≤ 2.
Unfortunately, further extensions become more technical. The Fourier transform of functions in
Lp for the range 2 <
p < ∞ requires the study of distributions . In fact, it can be shown that there are functions in
Lp with
p>2 so that the Fourier transform is not defined as a function .
Fourier–Stieltjes transform
The Fourier transform of a finite Borel measure
μ on
Rn is given by :
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 . In the case that
dμ =
ƒ(
x)
dx, then the formula above reduces to the usual definition for the Fourier transform of
ƒ. In the case that
μ is the probability distribution associated to a random variable
X, the Fourier-Stieltjes transform is closely related to the
characteristic functionIn probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
, but the typical conventions in probability theory take
eix·ξ instead of
e−2πix·ξ . In the case when the distribution has a
probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
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 continuous measures.
Bochner's theoremIn mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line.- Background :...
characterizes which functions may arise as the Fourier–Stieltjes transform of a measure .
Furthermore, the
Dirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
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).
Tempered distributions
The Fourier transform maps the space of
Schwartz functions to itself, and gives a
homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
of the space to itself . Because of this it is possible to define the Fourier transform of tempered distributions. These include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support, and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution.
The following two facts provide some motivation for the definition of the Fourier transform of a distribution. First let
ƒ and
g be integrable functions, and let
and
be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula ,
Secondly, every integrable function
ƒ defines a distribution
Tƒ by the relation
for all Schwartz functions
φ.
In fact, given a distribution
T, we define the Fourier transform by the relation
for all Schwartz functions
φ.
It follows that
Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
Locally compact abelian groups
The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
which is at the same time a locally compact
Hausdorff topological spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
so that the group operations are continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called
Haar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
. For a locally compact abelian group G it is possible to place a topology on the set of
charactersIn mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters which arises in the related context of character theory...
so that
is also a locally compact abelian group. For a function
ƒ in
L1(
G) it is possible to define the Fourier transform by :
Locally compact Hausdorff space
The Fourier transform may be generalized to any locally compact Hausdorff space, which recovers the topology but loses the group structure.
Given a
locally compactIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
HausdorffIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X, the space
A=
C0(
X) of continuous complex-valued functions on
X which
vanish at infinity is in a natural way a commutative C*-algebra, via pointwise addition, multiplication, complex conjugation, and with norm as the
uniform norm. Conversely, the characters of this algebra
A, denoted
are naturally a topological space, and can be identified with evaluation at a point of
x, and one has an isometric isomorphism
In the case where
X=
R is the real line, this is exactly the Fourier transform.
Non-abelian groups
The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Unlike the Fourier transform on an abelian group, which is scalar-valued, the Fourier transform on a non-abelian group is operator-valued . The Fourier transform on compact groups is a major tool in
representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
and non-commutative harmonic analysis.
Let
G be a compact
HausdorffIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible
unitary representationIn mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
s, along with a definite choice of representation
U(σ) on the
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
Hσ of finite dimension
dσ for each σ ∈ Σ. If μ is a finite
Borel measure on
G, then the Fourier–Stieltjes transform of μ is the operator on
Hσ defined by
where
is the complex-conjugate representation of
U(σ) acting on
Hσ. As in the abelian case, if μ is absolutely continuous with respect to the
left-invariant probability measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
λ on
G, then it is
representedIn mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
as
for some
ƒ ∈
L1(λ)In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. In this case, one identifies the Fourier transform of
ƒ with the Fourier–Stieltjes transform of μ.
The mapping
defines an isomorphism between the
Banach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
M(
G) of finite Borel measures (see rca space) and a closed subspace of the Banach space
C∞(Σ) consisting of all sequences
E = (
Eσ) indexed by Σ of (bounded) linear operators
Eσ :
Hσ →
Hσ for which the norm
is finite. The "
convolution theoremIn mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isomorphism of C
* algebras into a subspace of
C∞(Σ), in which
M(
G) is equipped with the product given by
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of measures and
C∞(Σ) the product given by multiplication of operators in each index σ.
The Peter-Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if
ƒ ∈ 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 geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
. In this context, a categorical generalization of the Fourier transform to noncommutative groups is
Tannaka-Krein dualityIn mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. Its natural extension to the non-Abelian case is the Grothendieck duality theory....
, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.
Alternatives
In
signal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
terms, a function (of time) is a representation of a signal with perfect
time resolution, but no frequency information, while the Fourier transform has perfect
frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.
As alternatives to the Fourier transform, in
time-frequency analysisIn signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations...
, one uses time-frequency transforms or time-frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the
short-time Fourier transformThe short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
or
fractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...
, or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the
continuous wavelet transformA continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization...
. .
Analysis of differential equations
Fourier transforms and the closely related
Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with
differentiationIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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. Note that 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 equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s with domain
Rn can also be translated into algebraic equations.
Fourier transform spectroscopy
The Fourier transform is also used in
nuclear magnetic resonanceNuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
(NMR) and in other kinds of
spectroscopySpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
, e.g. infrared (
FTIRFourier transform infrared spectroscopy is a technique which is used to obtain an infrared spectrum of absorption, emission, photoconductivity or Raman scattering of a solid, liquid or gas. An FTIR spectrometer simultaneously collects spectral data in a wide spectral range...
). In NMR an exponentially-shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in
magnetic resonance imagingMagnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...
(MRI) and
mass spectrometryMass spectrometry is an analytical technique that measures the mass-to-charge ratio of charged particles.It is used for determining masses of particles, for determining the elemental composition of a sample or molecule, and for elucidating the chemical structures of molecules, such as peptides and...
.
Domain and range of the Fourier transform
It is often desirable to have the most general domain for the Fourier transform possible. The definition of Fourier transform as an integral naturally restricts the domain to the space of integrable functions. Unfortunately, there is no simple characterization of which functions are Fourier transforms of integrable functions . It is possible to extend the domain of the Fourier transform in various ways, as discussed in the generalizations above. The following list details some of the more common domains and ranges on which the Fourier transform is defined.
- The space of Schwartz functions is closed under the Fourier transform. Schwartz functions are rapidly decaying functions and do not include all functions which are relevant for the Fourier transform. More details may be found in .
- The space Lp maps into the space Lq, where 1/p + 1/q = 1 and 1 ≤ p ≤ 2 (Hausdorff–Young inequality
In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. proved the inequality for some special values of q, and proved it in general...
).
- In particular, the space L2 is closed under the Fourier transform, but here the Fourier transform is no longer defined by integration.
- The space L1 of Lebesgue integrable functions maps into C0, the space of continuous functions that tend to zero at infinity – not just into the space
of bounded functions (the Riemann–Lebesgue lemma).
- The set of tempered distributions is closed under the Fourier transform. Tempered distributions are also a form of generalization of functions. It is in this generality that one can define the Fourier transform of objects like the Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...
.
Other notations
Other common notations for
are these:
Though less commonly other notations are used. 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 unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, to indicate its relationship with the
Laplace transform, and sometimes it is written informally as
F(2
πf) 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
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
and
is the
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
(see
arg functionIn mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...
).
Then the inverse transform can be written
:
which is a recombination of all the
frequency components of
ƒ(
x). Each component is a complex sinusoid of the form
e2πixξ whose
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
is
A(
ξ) and whose initial
phase angleIn the context of vectors and phasors, the term phase angle refers to the angular component of the polar coordinate representation. The notation A\ang \!\ \theta, for a vector with magnitude A and phase angle θ, is called angle notation.In the context of periodic phenomena, such as a wave,...
(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
x0.
Other conventions
The Fourier transform can also be written in terms of
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
: 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 transformationIn mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....
on
L2(
Rn). There is also less symmetry between the formulas for the Fourier transform and its inverse.
Another convention is to split the factor of (2
π)
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
L2(
Rn). It also restores the symmetry between the Fourier transform and its inverse.
Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.
Summary of popular forms of the Fourier transform
| ordinary frequency ξ (hertz) |
unitary |
 |
| angular frequency ω (rad/s) |
non-unitary |
 |
| unitary |
 |
As discussed above, the
characteristic functionIn probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
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 "non-unitary angular frequency" convention above, there is no factor of 2
π
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.
Tables of important Fourier transforms
The following tables record some closed form Fourier transforms. For functions
ƒ(
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.
Functional relationships
The Fourier transforms in this table may be found in or the appendix of .
| | Function | Fourier transform
unitary, ordinary frequency | Fourier transform
unitary, angular frequency | Fourier transform
non-unitary, angular frequency | Remarks |
|
 |
 |
 |
 |
Definition |
| 101 |
 |
 |
 |
 |
Linearity |
| 102 |
 |
 |
 |
 |
Shift in time domain |
| 103 |
 |
 |
 |
 |
Shift in frequency domain, dual of 102 |
| 104 |
 |
|
|
|
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 convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of  and  — this rule is the convolution theoremIn mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
|
| 109 |
 |
 |
 |
 |
This is the dual of 108 |
| 110 |
For  a purely real |
 |
 |
 |
Hermitian symmetry.  indicates the complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
. |
| 111 |
For  a purely real even function |
 ,  and  are purely real even functions. |
|
| 112 |
For  a purely real odd function |
 ,  and  are purely imaginaryAn imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
odd functions. |
|
Square-integrable functions
The Fourier transforms in this table may be found in , , or the appendix of .
| | Function | Fourier transform
unitary, ordinary frequency | Fourier transform
unitary, angular frequency | Fourier transform
non-unitary, 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 low-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
, and the sinc function is the non-causal impulse response of such a filter. |
| 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 The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
and a>0. |
| 206 |
 |
 |
 |
 |
This shows that, for the unitary Fourier transforms, the Gaussian function exp(−αx2) 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 decaying
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
is a Lorentzian function.
|-
| 208
|
|
|
|
|
Hyperbolic secantIn mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
is its own Fourier transform
|-
| 209
|
|
|
|
|
is the Hermite's polynomial. If
then the Gauss-Hermite functions are
eigenfunctionsIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for
.
|}
Distributions
The Fourier transforms in this table may be found in or the appendix of .
| | Function | Fourier transform
unitary, ordinary frequency | Fourier transform
unitary, angular frequency | Fourier transform
non-unitary, angular frequency | Remarks |
|
 |
 |
 |
 |
|
| 301 |
 |
 |
 |
 |
The distribution δ(ξ) denotes the Dirac delta function The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
. |
| 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 In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
and  is the n-th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s. |
| 309 |
 |
 |
 |
 |
Here sgn(ξ) is the sign functionIn mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
. Note that 1/x is not a distribution. It is necessary to use the Cauchy principal valueIn mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...
when testing against Schwartz functions. This rule is useful in studying the Hilbert transformIn mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
. |
| 310 |
 |
 |
 |
 |
xn is the homogeneous distribution defined by the distributional derivative  |
| 311 |
|
|
|
|
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 half-plane 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.) |
| 312 |
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The dual of rule 309. This time the Fourier transforms need to be considered as Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...
. |
| 313 |
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The function u(x) is the Heaviside unit step function The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
; this follows from rules 101, 301, and 312. |
| 314 |
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This function is known as the Dirac comb In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...
function. This result can be derived from 302 and 102, together with the fact that 
 as distributions. |
| 315 |
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The function J0(x) is the zeroth order Bessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of first kind. |
| 316 |
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This is a generalization of 315. The function Jn(x) is the n-th order Bessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of first kind. The function Tn(x) is the Chebyshev polynomial of the first kindIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
. |
| 317 |
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 is the Euler–Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
. |
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| 318 |
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This formula is valid for 1 > α > 0. Use differentiation to drive formula for higher exponents.  is the Heaviside function. |
Two-dimensional functions
| | Function | Fourier transform
unitary, ordinary frequency | Fourier transform
unitary, angular frequency | Fourier transform
non-unitary, angular frequency |
| 400 |
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| 401 |
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e^{\frac{-\left(\omega_x^2/a^2 + \omega_y^2/b^2\right)}{4\pi}}
|align="center"|
|-
|402
|
|align="center"|
|align="center"|
|align="center"|
|}
Remarks
To 400: The variables
ξx,
ξy,
ωx,
ωy,
νx and
νy 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 functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first kind).
Formulas for general n-dimensional functions
| | Function | Fourier transform
unitary, ordinary frequency | Fourier transform
unitary, angular frequency | Fourier transform
non-unitary, angular frequency |
| 500 |
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| 501 |
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| 502 |
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Remarks
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
Jn/2 + δ is a Bessel function of the first kind, with order
n/2 +
δ. Taking
n = 2 and
δ = 0 produces 402.
To 502:
See
Riesz potentialIn mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
. 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.
External links