Fourier transform

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term "Fourier transform" refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the "time domain". There is also an "inverse Fourier transform" that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 'position space' to a function of momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on or (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group = ) and the Fourier series or circular Fourier transform (group = , the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.
Definition.
The Fourier transform of a function is traditionally denoted formula_1, by adding a circumflex to the symbol of the function. There are several common conventions for defining the Fourier transform of an integrable function formula_2. One of them is
for any real number .
A reason for the negative sign in the exponent is that it is common in electrical engineering to represent
by formula_3 a signal with zero initial phase and frequency formula_4 The negative sign convention causes the product formula_5 to be 1 (frequency zero) when formula_6 causing the integral to diverge. The result is a Dirac delta function at formula_7, which is the only frequency component of the sinusoidal signal formula_8
When the independent variable represents "time", the transform variable represents frequency (e.g. if time is measured in seconds, then frequency is in hertz). Under suitable conditions, is determined by formula_9 via the inverse transform:
for any real number .
The statement that can be reconstructed from formula_9 is known as the Fourier inversion theorem, and was first introduced in Fourier's "Analytical Theory of Heat", although what would be considered a proof by modern standards was not given until much later. The functions and formula_9 often are referred to as a "Fourier integral pair" or "Fourier transform pair".  A common notation for designating transform pairs is:
For other common conventions and notations, including using the angular frequency instead of the frequency , see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable often represents position and momentum. The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on and an algebra homomorphism from to , without renormalizing the Lebesgue measure.
Many other characterizations of the Fourier transform exist. For example, one uses the Stone–von Neumann theorem: the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group.
History.
In 1822, Joseph Fourier showed that some functions could be written as an infinite sum of harmonics.
Introduction.
One 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.
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 , to write Fourier series in terms of the basic waves . 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. Re-writing 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 is measured in seconds, then the waves and 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 that are zero outside an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval in which we calculate the Fourier series, then the Fourier series coefficients begin to resemble the Fourier transform and the sum of the Fourier series of begins to resemble the inverse Fourier transform. More precisely, suppose is large enough that the interval contains the interval in which is not identically zero. Then, the th series coefficient is given by:
Comparing this to the definition of the Fourier transform, it follows that:
since is zero outside . Thus, the Fourier coefficients are equal to the values of the Fourier transform sampled on a grid of width , multiplied by the grid width .
Under appropriate conditions, the Fourier series of will equal the function . In other words, can be written:
where the last sum is simply the first sum rewritten using the definitions , and .
This second sum is a Riemann sum. By letting it will converge to the integral for the inverse Fourier transform as expressed above. Under suitable conditions, this argument may be made precise.
In the study of Fourier series the numbers could be thought of as the "amount" of the wave present in the Fourier series of . 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 , and we can recombine these waves by using an integral (or "continuous sum") to reproduce the original function.
Example.
The following figures provide a visual illustration how the Fourier transform measures whether a frequency is present in a particular function. The depicted function oscillates at 3 Hz (if 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 that can be easily plotted. The first image contains its graph. In order to calculate formula_16 we must integrate . 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 is negative, the real part of is negative as well. Because they oscillate at the same rate, when is positive, so is the real part of . The result is that when you integrate the real part of the integrand you get a relatively large number (in this case ). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at formula_17, you see that both real and imaginary component of this function vary rapidly between positive and negative values, as plotted in the third image. Therefore, in this case, the integrand oscillates fast enough so that the integral is very small and the value for the Fourier transform for that frequency is nearly zero.
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 .
Properties of the Fourier transform.
Here we assume , and are "integrable functions": Lebesgue-measurable on the real line satisfying:
We denote the Fourier transforms of these functions as , and respectively.
Basic properties.
The Fourier transform has the following basic properties:
Invertibility and periodicity.
Under suitable conditions on the function , it can be recovered from its Fourier transform formula_9. Indeed, denoting the Fourier transform operator by , so , then for suitable functions, applying the Fourier transform twice simply flips the function: , which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields , so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: . In particular the Fourier transform is invertible (under suitable conditions).
More precisely, defining the "parity operator" 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 two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the time–frequency domain (considering time as the -axis and frequency as the -axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves rotations by other angles. This can be further generalized to linear canonical transformations, which can be visualized as the action of the special linear group 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.
Units and duality.
In mathematics, one often does not think of any units as being attached to the two variables and . But in physical applications, must have inverse units to the units of . For example, if is measured in seconds, 
should be in cycles per second for the formulas here to be valid. If the scale of is changed and is measured in units of 2 seconds, then either must be in the so-called "angular frequency", or one must insert some constant scale factor into some of the formulas. If is measured in units of length, then must be in inverse length, e.g., wavenumbers. That is to say, there are two copies of the real line: one measured in one set of units, where ranges, and the other in inverse units to the units of , and which is the range of . So these are two distinct copies of the real line, and cannot be identified with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.
In general, must always be taken to be a linear form on the space of s, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalisations of the Fourier transform to general symmetry groups, including the case of Fourier series.
That there is no one preferred way (often, one says "no canonical way") to compare the two copies of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants. If the units of are in seconds but the units of are in angular frequency, then the angular frequency variable is often denoted by one or another Greek letter, for example, is quite common. Thus (writing for the alternative definition and for the definition adopted in this article)
The corresponding alternative inversion formula is
Another alternative definition of the Fourier transform, with a factor of is
and the corresponding inversion formula is
In other conventions, the Fourier transform has in the exponent instead of , and vice versa for the inversion formula. Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve have it replaced by .
For example, in probability theory, the characteristic function of the probability density function of a random variable of continuous type is defined without a negative sign in the exponential, and since the units of are ignored, there is no 2 either:
From the higher point of view of group characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a locally compact Abelian group.
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 is uniformly continuous and
By the "Riemann–Lebesgue lemma",
However, formula_9 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 and formula_9 are integrable, the inverse equality
holds almost everywhere. That is, the Fourier transform is injective on . (But if is continuous, then equality holds for every .)
Plancherel theorem and Parseval's theorem.
Let and be integrable, and let and be their Fourier transforms. If "f" ("x") and "g"("x") are also square-integrable, then the Parseval formula follows:
where the bar denotes complex conjugation.
The Plancherel theorem, which follows from the above, states that
Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on . On , this extension agrees with original Fourier transform defined on , thus enlarging the domain of the Fourier transform to (and consequently to for ). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.
See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
Poisson summation formula.
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. The Poisson summation formula says that for sufficiently regular functions ,
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. The formula has applications in engineering, physics, and number theory. The frequency-domain dual of the standard Poisson summation formula is also called the discrete-time Fourier transform.
Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a theta function. It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form, and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula.
Differentiation.
Suppose is an absolutely continuous differentiable function, and both and its derivative are integrable. Then the Fourier transform of the derivative is given by
More generally, the Fourier transformation of the th derivative is given by
By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb " is smooth if and only if quickly falls to 0 for ." By using the analogous rules for the inverse Fourier transform, one can also say " quickly falls to 0 for if and only if is smooth."
Convolution theorem.
The Fourier transform translates between convolution and multiplication of functions. If and are integrable functions with Fourier transforms and respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms and (under other conventions for the definition of the Fourier transform a constant factor may appear).
This means that if:
where denotes the convolution operation, then:
In linear time invariant (LTI) system theory, it is common to interpret as the impulse response of an LTI system with input and output , since substituting the unit impulse for yields . In this case, represents the frequency response of the system.
Conversely, if can be decomposed as the product of two square integrable functions and , then the Fourier transform of is given by the convolution of the respective Fourier transforms and .
Cross-correlation theorem.
In an analogous manner, it can be shown that if is the cross-correlation of and :
then the Fourier transform of is:
As a special case, the autocorrelation of function is:
for which
Eigenfunctions.
One important choice of an orthonormal basis for is given by the Hermite functions
where are the "probabilist's" Hermite polynomials, defined as
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 . However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose as a direct sum of four spaces , , , and where the Fourier transform acts on simply by multiplication by .
Since the complete set of Hermite functions provides a resolution of the identity, 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. Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time-frequency analysis. In physics, this transform was introduced by Edward Condon.
Connection with the Heisenberg group.
The Heisenberg group is a certain group of unitary operators on the Hilbert space of square integrable complex valued functions on the real line, generated by the translations and multiplication by , . These operators do not commute, as their (group) commutator is
which is multiplication by the constant (independent of ) (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional Lie group of triples , with the group law
Denote the Heisenberg group by . The above procedure describes not only the group structure, but also a standard unitary representation of on a Hilbert space, which we denote by . Define the linear automorphism of by
so that . This can be extended to a unique automorphism of :
According to the Stone–von Neumann theorem, the unitary representations and are unitarily equivalent, so there is a unique intertwiner such that
This operator is the Fourier transform.
Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. For example, the square of the Fourier transform, , is an intertwiner associated with , and so we have is the reflection of the original function .
Complex domain.
The integral for the Fourier transform
can be studied for complex values of its argument . Depending on the properties of , this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of , or something in between.
The Paley–Wiener theorem says that is smooth (i.e., -times differentiable for all positive integers ) and compactly supported if and only if is a holomorphic function for which there exists a constant such that for any integer ,
for some constant . (In this case, is supported on .) This can be expressed by saying that is an entire function which is rapidly decreasing in (for fixed ) and of exponential growth in (uniformly in ).
(If is not smooth, but only , the statement still holds provided .) The space of such functions of a complex variable is called the Paley—Wiener space. This theorem has been generalised to semisimple Lie groups.
If is supported on the half-line , then is said to be "causal" because the impulse response function of a physically realisable filter must have this property, as no effect can precede its cause. Paley and Wiener showed that then extends to a holomorphic function on the complex lower half-plane which tends to zero as goes to infinity. The converse is false and it is not known how to characterise the Fourier transform of a causal function.
Laplace transform.
The Fourier transform is related to the Laplace transform , which is also used for the solution of differential equations and the analysis of filters.
It may happen that a function for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the complex plane.
For example, if is of exponential growth, i.e.,
for some constants , then
convergent for all , is the two-sided Laplace transform of .
The more usual version ("one-sided") of the Laplace transform is
If is also causal, then
Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case—the case of causal functions—but with the change of variable .
Inversion.
If is complex analytic for , then
by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.
Theorem: If for , and for some constants , then
for any .
This theorem implies the Mellin inversion formula for the Laplace transformation,
for any , where is the Laplace transform of .
The hypotheses can be weakened, as in the results of Carleman and Hunt, to being , provided that is of bounded variation in a closed neighborhood of (cf. Dirichlet-Dini theorem), the value of at is taken to be the arithmetic mean of the left and right limits, and provided that the integrals are taken in the sense of Cauchy principal values.
 versions of these inversion formulas are also available.
Fourier transform on Euclidean space.
The Fourier transform can be defined in any arbitrary number of dimensions . As with the one-dimensional case, there are many conventions. For an integrable function , this article takes the definition:
where and are -dimensional vectors, and is the dot product of the vectors. Alternatively, can be viewed as belonging to the dual vector space formula_69, in which case the dot product becomes the contraction of and , usually written as .
All of the basic properties listed above hold for the -dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.
Uncertainty principle.
Generally speaking, the more concentrated 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 , its Fourier transform stretches out in . It is not possible to arbitrarily concentrate both a function and its Fourier transform.
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.
Suppose is an integrable and square-integrable function. Without loss of generality, assume that is normalized:
It follows from the Plancherel theorem that is also normalized.
The spread around may be measured by the "dispersion about zero" defined by
In probability terms, this is the second moment of about zero.
The uncertainty principle states that, if is absolutely continuous and the functions and are square integrable, then
The equality is attained only in the case
where is arbitrary and so that is -normalized. In other words, where is a (normalized) Gaussian function with variance , centered at zero, and its Fourier transform is a Gaussian function with variance .
In fact, this inequality implies that:
for any , .
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.
A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:
where is the differential entropy of the probability density function :
where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.
Sine and cosine transforms.
Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically) by
This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions and can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):
and
Older literature refers to the two transform functions, the Fourier cosine transform, , and the Fourier sine transform, .
The function can be recovered from the sine and cosine transform using
together with trigonometric identities. This is referred to as Fourier's integral formula.
Spherical harmonics.
Let the set of homogeneous harmonic polynomials of degree on be denoted by . The set consists of the solid spherical harmonics of degree . The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if for some in , then . Let the set be the closure in of linear combinations of functions of the form where is in . The space is then a direct sum of the spaces and the Fourier transform maps each space to itself and is possible to characterize the action of the Fourier transform on each space .
Let (with in ), then
where
Here denotes the Bessel function of the first kind with order . When this gives a useful formula for the Fourier transform of a radial function. This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases and allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.
Restriction problems.
In higher dimensions it becomes interesting to study "restriction problems" for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general "class" of square integrable functions. As such, the restriction of the Fourier transform of an function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in for . Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set , provided has non-zero curvature. The case when is the unit sphere in is of particular interest. In this case the Tomas–Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in is a bounded operator on provided .
One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets indexed by : such as balls of radius centered at the origin, or cubes of side . For a given integrable function , consider the function defined by:
Suppose in addition that . For and , if one takes , then converges to in as tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for . In the case that is taken to be a cube with side length , then convergence still holds. Another natural candidate is the Euclidean ball . In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in . For it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless . In fact, when , this shows that not only may fail to converge to in , but for some functions , is not even an element of .
Fourier transform on function spaces.
On spaces.
On.
The definition of the Fourier transform by the integral formula
is valid for Lebesgue integrable functions ; that is, .
The Fourier transform 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 is a subset of the space 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.
On.
Since compactly supported smooth functions are integrable and dense in , the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in by continuity arguments. The Fourier transform in is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an function ,
Formulas for general -dimensional functions.
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