Fourier transform explained

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

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.^[1] 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.^[2]

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.^[3] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on or, 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 is an analysis process, decomposing a complex-valued function

style f(x) into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates style f(x) from its transform.

We can start with an analogy, the Fourier series, which analyzes

style f(x) on a bounded interval style x \in [ - P/ 2 , P/ 2 ], for some positive real number P. The constituent frequencies are a discrete set of harmonics at frequencies \tfrac{n}{P}, n \in Z, whose amplitude and phase are given by the analysis formula:$$c\_n\; =\; \backslash tfrac\; \backslash int\_^\; f(x)\; \backslash ,\; e^\; \backslash ,\; dx.$$The actual Fourier series is the synthesis formula:$$f(x)\; =\; \backslash sum\_^\backslash infty\; c\_n\backslash ,\; e^,\backslash quad\; \backslash textstyle\; x\; \backslash in\; [-P/2,\; P/2].$$

The analogy for a function

style f(x) can be obtained formally from the analysis formula by taking the limit as P\to inf ty , while at the same time taking n so that \tfrac{n}{P} \to \xi \in R. Formally carrying this out, we obtain, for rapidly decreasing f :^[4]

It is easy to see, assuming the hypothesis of rapid decreasing, that the integral converges for all real

\xi , and (using the Riemann–Lebesgue lemma) that the transformed function \widehat f is also rapidly decreasing. The validity of this definition for classes of functions f that are not necessarily rapidly decreasing is discussed later in this section.

Evaluating for all values of

\xi produces the frequency-domain function. The complex number \widehat{f}(\xi) , in polar coordinates, conveys both amplitude and phase of frequency \xi. The intuitive interpretation of is that the effect of multiplying f(x) by e ^{-i} is to subtract \xi from every frequency component of function f(x). ^[5] Only the component that was at frequency \xi can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see)

The corresponding synthesis formula for such a function is:

is a representation of

f(x) as a weighted summation of complex exponential functions.

This is also known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat.^[6]

The functions

f and \widehat{f} are referred to as a Fourier transform pair.^[7]   A common notation for designating transform pairs is:$$f(x)\backslash \; \backslash stackrel\backslash \; \backslash widehat\; f(\backslash xi),$$   for example   \operatorname{rect}(x) \stackrel{ l{F}}{\longleftrightarrow} \operatorname{sinc}(\xi).

Definition for Lebesgue integrable functions

f: R\to C is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite:Two measurable functions are equivalent if they are equal except on a set of measure zero. The set of all equivalence classes of integrable functions is denoted L ^{1(R)} . Then:The integral is well-defined for all \xi\in R, because of the assumption \|f\| _{1<infty} . (It can be shown that the function \widehat f\in L ^{infty\cap} C( R) is bounded and uniformly continuous in the frequency domain, and moreover, by the Riemann–Lebesgue lemma, it is zero at infinity.)

However, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.

Unitarity and definition for square integrable functions

While defines the Fourier transform for (complex-valued) functions in

L ^{1(R)} , it is easy to see that it is not well-defined for other integrability classes, most importantly L ^{2(R)} . For functions in L ^{1\cap} L ^{2(R)} , and with the conventions of, the Fourier transform is a unitary operator with respect to the Hilbert inner product on L ^{2(R)} , restricted to the dense subspace of integrable functions. Therefore, it admits a unique continuous extension to a unitary operator on L ^{2(R)} , also called the Fourier transform. This extension is important in part because the Fourier transform preserves the space L ^{2(R)} so that, unlike the case of L ^{1} , the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself.

Importantly, for functions in

L ^{2} , the Fourier transform is no longer given by (interpreted as a Lebesgue integral). For example, the function f(x) = ( 1+ x ^{2)} ^{-1/2} is in L ^{2} but not L ^{1} , so the integral diverges. In such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit. In practice, the integral is often regarded as an improper integral instead of a proper Lebesgue integral, but sometimes for convergence one needs to use weak limit or principal value instead of the (pointwise) limits implicit in an improper integral. and each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.

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.

Angular frequency (ω)

When the independent variable (

x ) represents time (often denoted by t ), the transform variable (\xi ) represents frequency (often denoted by f ). For example, if time is measured in seconds, then frequency is in hertz. The Fourier transform can also be written in terms of angular frequency, \omega = 2 \pi \xi, whose units are radians per second.

The substitution

\xi = \tfrac{\omega}{ 2 \pi} into produces this convention, where function \widehat f is relabeled \widehat {f _{1}:} Unlike the definition, the Fourier transform is no longer a unitary transformation, and there is less symmetry between the formulas for the transform and its inverse. Those properties are restored by splitting the 2 \pi factor evenly between the transform and its inverse, which leads to another convention: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.

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

ordinary frequency (Hz)	unitary	\begin{align} \widehat{f _{1}(\xi) &\triangleq \int} ^{inf ty} ^{-inf ty} f(x) e ^{-i} dx = \sqrt{ 2 \pi} \widehat{f _{2}(2} \pi \xi) = \widehat{f _{3}(2} \pi \xi) \\ f(x) & = \int ^{inf ty} _{-inf ty} \widehat{f _{1}(\xi)} e ^{i} d\xi \end{align}
angular frequency (rad/s)	unitary	\begin{align} \widehat{f _{2 }(\omega) &\triangleq (1)/(\sqrt{ 2 \pi)} }\ \int_^ f(x)\, e^\, dx = \frac\ \ \widehat \! \left(\frac\right) = \frac\ \ \widehat(\omega) \\f(x) &= \frac\ \int_^ \widehat(\omega)\, e^\, d\omega \end
	non-unitary	\begin{align} \widehat{f _{3}(\omega)} &\triangleq \int ^{inf ty} _{-inf ty} f(x) e ^{-i\omega} dx = \widehat{f _{1}} \left( (\omega)/(2 \pi) \right) = \sqrt{ 2 \pi} \widehat{f _{2}(\omega)} \\ f(x) & = (1)/(2 \pi) \int ^{inf ty} _{-inf ty} \widehat{f _{3}(\omega)} e ^{i} d\omega \end{align}

Generalization for -dimensional functions

ordinary frequency (Hz)	unitary	\begin{align} \widehat{f _{1}(\xi) &\triangleq \int} ^{R ^{n}} f(x) e ^{-i} dx = ( 2 \pi) ^{(n)/(2)} \widehat{f _{2}(2\pi} \xi) = \widehat{f _{3}(2\pi} \xi) \\ f(x) & = \int _{R ^{n}} \widehat{f _{1}(\xi)} e ^{i} d\xi \end{align}
angular frequency (rad/s)	unitary	\begin{align} \widehat{f _{2 }(\omega) &\triangleq (1)/(( 2 \pi) ^{(n)/(2)})} \int _{R ^{n}} f(x) e ^{-i} dx = (1)/(( 2 \pi) ^{(n)/(2)}) \widehat{f _{1}} \left( (\omega)/(2 \pi) \right) = (1)/(( 2 \pi) ^{(n)/(2)}) \widehat{f _{3}(\omega)} \\ f(x) & = (1)/(( 2 \pi) ^{(n)/(2)}) \int _{R ^{n}} \widehat{f ^{i \omega ⋅ x} _{2 }(\omega)e} d\omega \end{align}
	non-unitary	\begin{align} \widehat{f _{3}(\omega)} &\triangleq \int _{R ^{n}} f(x) e ^{-i\omega ⋅} dx = \widehat{f _{1}} \left( (\omega)/(2 \pi) \right) = ( 2 \pi) ^{(n)/(2)} \widehat{f _{2}(\omega)} \\ f(x) & = (1)/(( 2 \pi) ^{n}) \int _{R ^{n}} \widehat{f _{3}(\omega)} e ^{i} d\omega \end{align}

Extension of the definition

For

1 <p< 2 , the Fourier transform can be defined on L ^{p(R)} by Marcinkiewicz interpolation.

The Fourier transform can be defined on domains other than the real line. The Fourier transform on Euclidean space and the Fourier transform on locally abelian groups are discussed later in the article.

The Fourier transform can also be defined for tempered distributions, dual to the space of rapidly decreasing functions (Schwartz functions). A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives. The space of Schwartz functions is denoted by

l S( R) , and its dual l S'( R) is the space of tempered distributions. It is easy to see, by differentiating under the integral and applying the Riemann-Lebesgue lemma, that the Fourier transform of a Schwartz function (defined by the formula) is again a Schwartz function. The Fourier transform of a tempered distribution T\in l S'( R) is defined by duality:$$\backslash langle\; \backslash widehat\; T,\; \backslash phi\backslash rangle\; =\; \backslash langle\; T,\backslash widehat\; \backslash phi\backslash rangle;\backslash quad\; \backslash forall\; \backslash phi\backslash in\backslash mathcal\; S(\backslash mathbb\; R).$$

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.

Background

History

In 1822, Fourier claimed (see) that any function, whether continuous or discontinuous, can be expanded into a series of sines. That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

Complex sinusoids

In general, the coefficients

\widehat f(\xi) are complex numbers, which have two equivalent forms (see Euler's formula):$$\backslash widehat\; f(\backslash xi)\; =\; \backslash underbrace\_=\; \backslash underbrace\_.$$

The product with

e ^{i} has these forms:

It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.

Negative frequency

Euler's formula introduces the possibility of negative

\xi.   And is defined \forall \xi \in R . Only certain complex-valued f(x) have transforms \widehat f =0 , \forall \xi < 0 (See Analytic signal. A simple example is e ^{i 2 \pi \xi _{0} x} (\xi _{0} > 0 ). )  But negative frequency is necessary to characterize all other complex-valued f(x), found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.

For a real-valued

f(x), has the symmetry property \widehat f( - \xi) = \widehat {f} ^{*} (\xi) (see below). This redundancy enables to distinguish f(x) = \cos( 2 \pi \xi _{0} x) from e ^{i 2 \pi \xi _{0} x} .   But of course it cannot tell us the actual sign of \xi _{0,} because \cos( 2 \pi \xi _{0} x) and \cos( 2 \pi ( - \xi _{0)} x) are indistinguishable on just the real numbers line.

Fourier transform for periodic functions

The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.

This makes it possible to see a connection between the Fourier series and the Fourier transform for periodic functions that have a convergent Fourier series. If

f(x) is a periodic function, with period P , that has a convergent Fourier series, then:$$\backslash widehat(\backslash xi)\; =\; \backslash sum\_^\backslash infty\; c\_n\; \backslash cdot\; \backslash delta\; \backslash left(\backslash xi\; -\; \backslash tfrac\backslash right),$$where c _{n} are the Fourier series coefficients of f , and \delta is the Dirac delta function. In other words, the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.

Sampling the Fourier transform

The Fourier transform of an integrable function

f can be sampled at regular intervals of arbitrary length \tfrac{ 1 }{P}. These samples can be deduced from one cycle of a periodic function f _{P} which has Fourier series coefficients proportional to those samples by the Poisson summation formula:$$f\_P(x)\; \backslash triangleq\; \backslash sum\_^\; f(x+nP)\; =\; \backslash frac\backslash sum\_^\; \backslash widehat\; f\backslash left(\backslash tfrac\backslash right)\; e^,\; \backslash quad\; \backslash forall\; k\; \backslash in\; \backslash mathbb$$

The integrability of

f ensures the periodic summation converges. Therefore, the samples \widehat f\left(\tfrac{k}{P}\right) can be determined by Fourier series analysis:$$\backslash widehat\; f\backslash left(\backslash tfrac\backslash right)\; =\; \backslash int\_\; f\_P(x)\; \backslash cdot\; e^\; \backslash ,dx.$$

When

f(x) has compact support, f _{P(x)} has a finite number of terms within the interval of integration. When f(x) does not have compact support, numerical evaluation of f _{P(x)} requires an approximation, such as tapering f(x) or truncating the number of terms.

Example

The following figures provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The depicted function

f(t) = \cos( 2 \pi 3 t) e ^{- \pi t ^{2}} oscillates at 3 Hz (if t measures seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse.). f(t) was specially chosen to have a real Fourier transform that can be easily plotted. The first image is its graph. In order to calculate \widehat{f}( 3 ) we must integrate the product f(t) e ^{-i} . The next 2 images are the real and imaginary parts of that product. The real part of the integrand has a non-negative average value, because the alternating signs of f(t) and \operatorname{Re}(e ^{-i} ) oscillate at the same rate and same phase, whereas f(t) and \operatorname{Im} (e ^{-i} ) are same rate but orthogonal phase. The result is that when you integrate the real part of the integrand you get a relatively large number (in this case \tfrac{ 1 }{ 2 } ). Also, when you try to measure a frequency that is not present, as in the case when we look at \widehat{f}( 5 ), both real and imaginary component of the product vary rapidly between positive and negative values. Therefore, the integral is very small and the value for the Fourier transform for that frequency is nearly zero. The general situation is usually more complicated than this, but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a function f(t).

To re-enforce an earlier point, the reason for the response at  

\xi =-3 Hz  is because  \cos( 2 \pi 3 t)   and  \cos( 2 \pi( -3 )t)   are indistinguishable. The transform of  e ^{i2\pi} ⋅ e ^{- \pi t ^{2}}   would have just one response, whose amplitude is the integral of the smooth envelope: e ^{- \pi t ^{2}} ,   whereas  \operatorname{Re}(f(t) ⋅ e ^{-i2\pi} ) (second graph above) is  e ^{- \pi t ^{2}} ( 1 + \cos( 2 \pi 6 t))/ 2 .

Properties of the Fourier transform

Let

f(x) and h(x) represent integrable functions Lebesgue-measurable on the real line satisfying:We denote the Fourier transforms of these functions as \hat f(\xi) and \hat h(\xi) respectively.

Basic properties

The Fourier transform has the following basic properties:

Linearity

$$a\backslash \; f(x)\; +\; b\backslash \; h(x)\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; a\backslash \; \backslash widehat\; f(\backslash xi)\; +\; b\backslash \; \backslash widehat\; h(\backslash xi);\backslash quad\; \backslash \; a,b\; \backslash in\; \backslash mathbb\; C$$

Time shifting

$$f(x-x\_0)\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; e^\backslash \; \backslash widehat\; f(\backslash xi);\backslash quad\; \backslash \; x\_0\; \backslash in\; \backslash mathbb\; R$$

Frequency shifting

$$e^\; f(x)\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash widehat\; f(\backslash xi\; -\; \backslash xi\_0);\backslash quad\; \backslash \; \backslash xi\_0\; \backslash in\; \backslash mathbb\; R$$

Time scaling

$$f(ax)\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash frac$$

\widehat\left(\frac\right);\quad \ a \ne 0 The case a =-1 leads to the time-reversal property:$$f(-x)\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash widehat\; f\; (-\backslash xi)$$

Symmetry

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:

From this, various relationships are apparent, for example:

• The transform of a real-valued function

(f _{_{RE}} + f _{_{RO}} ) is the conjugate symmetric function \hat f _{RE} + i \hat f _{IO} . Conversely, a conjugate symmetric transform implies a real-valued time-domain.

• The transform of an imaginary-valued function

(i f _{_{IE}} + i f _{_{IO}} ) is the conjugate antisymmetric function \hat f _{RO} + i \hat f _{IE} , and the converse is true.

• The transform of an even-symmetric function

(f _{_{RE}} + i f _{_{IO}} ) is the real-valued function \hat f _{RE} + \hat f _{RO} , and the converse is true.

• The transform of an odd-symmetric function

(f _{_{RO}} + i f _{_{IE}} ) is the imaginary-valued function i \hat f _{IE} + i\hat f _{IO} , and the converse is true.

Conjugation

$$\backslash bigl(f(x)\backslash bigr)^*\backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash left(\backslash widehat(-\backslash xi)\backslash right)^*$$(Note: the ∗ denotes complex conjugation.)

In particular, if

f is real, then \widehat f is even symmetric (aka Hermitian function):$$\backslash widehat(-\backslash xi)=\backslash bigl(\backslash widehat\; f(\backslash xi)\backslash bigr)^*.$$

And if

f is purely imaginary, then \widehat f is odd symmetric:$$\backslash widehat\; f(-\backslash xi)\; =\; -(\backslash widehat\; f(\backslash xi))^*.$$

Real and imaginary part in time

$$\backslash operatorname\backslash \backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash tfrac\; \backslash left(\backslash widehat\; f(\backslash xi)\; +\; \backslash bigl(\backslash widehat\; f\; (-\backslash xi)\; \backslash bigr)^*\; \backslash right)$$$$\backslash operatorname\backslash \backslash \; \backslash \; \backslash stackrel\backslash \; \backslash \; \backslash tfrac\; \backslash left(\backslash widehat\; f(\backslash xi)\; -\; \backslash bigl(\backslash widehat\; f\; (-\backslash xi)\; \backslash bigr)^*\; \backslash right)$$

Zero frequency component

Substituting

\xi = 0 in the definition, we obtain:$$\backslash widehat(0)\; =\; \backslash int\_^\; f(x)\backslash ,dx.$$

The integral of

f over its domain is known as the average value or DC bias of the function.

Invertibility and periodicity

Under suitable conditions on the function

f , it can be recovered from its Fourier transform \hat{f} . Indeed, denoting the Fourier transform operator by l{F} , so l{F} f : = \hat{f} , then for suitable functions, applying the Fourier transform twice simply flips the function: \left( l{F} ^{2} f\right)(x) = f( - x) , which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields l{F} ^{4(f)} = f , so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: l{F} ^{3\left(\hat{f}\right)} = f . In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining the parity operator

l{P} such that ( l{P} f)(x) = f( - x) , we have: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

The frequency variable must have inverse units to the units of the original function's domain (typically named or). For example, if is measured in seconds, should be in cycles per second or hertz. If the scale of time is in units of 2 seconds, then another Greek letter typically is used instead to represent angular frequency (where) in units of radians per second. If using for units of length, then must be in inverse length, e.g., wavenumbers. That is to say, there are two versions of the real line: one which is the range of and measured in units of, and the other which is the range of and measured in inverse units to the units of . These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In general, must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to general symmetry groups, including the case of Fourier series.

That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line 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.

In other conventions, the Fourier transform has in the exponent instead of, and vice versa for the inversion formula. This convention is common in modern physics and is the default for Wolfram Alpha, and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means that

\hat f(\xi) is the amplitude of the wave  e ^{-i}   instead of the wave  e ^{i} (the former, with its minus sign, is often seen in the time dependence for Sinusoidal plane-wave solutions of the electromagnetic wave equation, or in the time dependence for quantum wave functions). Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve have it replaced by . In Electrical engineering the letter is typically used for the imaginary unit instead of because is used for current.

When using dimensionless units, the constant factors might not even be written in the transform definition. For instance, in probability theory, the characteristic function of the probability density function of a random variable of continuous type is defined without a negative sign in the exponential, and since the units of are ignored, there is no 2 either:$$\backslash phi\; (\backslash lambda)\; =\; \backslash int\_^\backslash infty\; f(x)\; e^\; \backslash ,dx.$$

(In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but distributions, i.e., measures which possess "atoms".)

From the higher point of view of group characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a locally compact Abelian group.

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$$\backslash left\backslash |\backslash hat\backslash right\backslash |\_\backslash infty\; \backslash leq\; \backslash left\backslash |f\backslash right\backslash |\_1$$

By the Riemann–Lebesgue lemma,$$\backslash hat(\backslash xi)\; \backslash to\; 0\backslash text|\backslash xi|\; \backslash to\; \backslash infty.$$

However,

\hat{f} 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

\hat{f} are integrable, the inverse equality$$f(x)\; =\; \backslash int\_^\backslash infty\; \backslash hat\; f(\backslash xi)\; e^\; \backslash ,\; d\backslash xi$$holds holds for almost every . As a result, the Fourier transform is injective on .

Plancherel theorem and Parseval's theorem

Let and be integrable, and let and be their Fourier transforms. If and are also square-integrable, then the Parseval formula follows:$$\backslash langle\; f,\; g\backslash rangle\_\; =\; \backslash int\_^\; f(x)\; \backslash overline\; \backslash ,dx\; =\; \backslash int\_^\backslash infty\; \backslash hat(\backslash xi)\; \backslash overline\; \backslash ,d\backslash xi,$$where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that$$\backslash |f\backslash |^2\_\; =\; \backslash int\_^\backslash infty\; \backslash left|\; f(x)\; \backslash right|^2\backslash ,dx\; =\; \backslash int\_^\backslash infty\; \backslash left|\; \backslash hat(\backslash xi)\; \backslash right|^2\backslash ,d\backslash xi.$$

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

See main article: 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,$$\backslash sum\_n\; \backslash hat\; f(n)\; =\; \backslash sum\_n\; f\; (n).$$

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$$\backslash widehat(\backslash xi)\; =\; \backslash mathcal\backslash left\backslash \; =\; i\; 2\backslash pi\; \backslash xi\backslash hat(\backslash xi).$$More generally, the Fourier transformation of the th derivative is given by$$\backslash widehat(\backslash xi)\; =\; \backslash mathcal\backslash left\backslash \; =\; (i\; 2\backslash pi\; \backslash xi)^n\backslash hat(\backslash xi).$$

Analogically,

l{F}\left\{ (d ^{n})/(d\xi ^{n}) \hat{f}(\xi)\right\} = (i 2 \pi x) ^{n} f(x) , so l{F}\left\{ x ^{n} f(x)\right\} = \left( (i)/(2 \pi) \right) ^{n} (d ^{n})/(d\xi ^{n}) \hat{f}(\xi).

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

See main article: 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:$$h(x)\; =\; (f*g)(x)\; =\; \backslash int\_^\backslash infty\; f(y)g(x\; -\; y)\backslash ,dy,$$where denotes the convolution operation, then:$$\backslash hat(\backslash xi)\; =\; \backslash hat(\backslash xi)\backslash ,\; \backslash hat(\backslash xi).$$

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

See main article: Cross-correlation.

In an analogous manner, it can be shown that if is the cross-correlation of and :$$h(x)\; =\; (f\; \backslash star\; g)(x)\; =\; \backslash int\_^\backslash infty\; \backslash overlineg(x\; +\; y)\backslash ,dy$$then the Fourier transform of is:$$\backslash hat(\backslash xi)\; =\; \backslash overline\; \backslash ,\; \backslash hat(\backslash xi).$$

As a special case, the autocorrelation of function is:$$h(x)\; =\; (f\; \backslash star\; f)(x)\; =\; \backslash int\_^\backslash infty\; \backslash overlinef(x\; +\; y)\backslash ,dy$$for which$$\backslash hat(\backslash xi)\; =\; \backslash overline\backslash hat(\backslash xi)\; =\; \backslash left|\backslash hat(\backslash xi)\backslash right|^2.$$

Eigenfunctions

See also: Mehler kernel. The Fourier transform is a linear transform which has eigenfunctions obeying

l{F}[\psi] = λ \psi, with λ \in C .

A set of eigenfunctions is found by noting that the homogeneous differential equation $$\backslash left[U\backslash left(\backslash frac\{1\}\{2\backslash pi\}\backslash frac\{d\}\{dx\}\; \backslash right)\; +\; U(x)\; \backslash right]\; \backslash psi(x)\; =\; 0$$ leads to eigenfunctions

\psi(x) of the Fourier transform l{F} as long as the form of the equation remains invariant under Fourier transform.^[8] In other words, every solution \psi(x) and its Fourier transform \hat\psi(\xi) obey the same equation. Assuming uniqueness of the solutions, every solution \psi(x) must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if U(x) can be expanded in a power series in which for all terms the same factor of either one of \pm 1 , \pm i arises from the factors i ^{n} introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable U(x) = x leads to the standard normal distribution.

More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation $$\backslash left[W\backslash left(\backslash frac\{i\}\{2\backslash pi\}\backslash frac\{d\}\{dx\}\; \backslash right)\; +\; W(x)\; \backslash right]\; \backslash psi(x)\; =\; C\; \backslash psi(x)$$with

C constant and W(x) being a non-constant even function remains invariant in form when applying the Fourier transform l{F} to both sides of the equation. The simplest example is provided by W(x) = x ^{2} which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator. The corresponding solutions provide an important choice of an orthonormal basis for and are given by the "physicist's" Hermite functions. Equivalently one may use$$\backslash psi\_n(x)\; =\; \backslash frac\; e^\backslash mathrm\_n\backslash left(2x\backslash sqrt\backslash right),$$where are the "probabilist's" Hermite polynomials, defined as$$\backslash mathrm\_n(x)\; =\; (-1)^n\; e^\backslash left(\backslash frac\backslash right)^n\; e^.$$

Under this convention for the Fourier transform, we have that$$\backslash hat\backslash psi\_n(\backslash xi)\; =\; (-i)^n\; \backslash psi\_n(\backslash xi).$$

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on . However, this choice of eigenfunctions is not unique. Because of

l{F} ^{4} = id there are only four different eigenvalues of the Fourier transform (the fourth roots of unity ±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 they diagonalize the Fourier operator, i.e. the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed:$$\backslash mathcal[f](\backslash xi)\; =\; \backslash int\; dx\; f(x)\; \backslash sum\_\; (-i)^n\; \backslash psi\_n(x)\; \backslash psi\_n(\backslash xi)\; ~.$$

This approach to define the Fourier transform was first proposed by Norbert Wiener. Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time–frequency analysis. In physics, this transform was introduced by Edward Condon. This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right conventions. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator

N via$$\backslash mathcal[\backslash psi]\; =\; e^\; \backslash psi.$$

The operator

N is the number operator of the quantum harmonic oscillator written as$$N\; \backslash equiv\; \backslash frac\backslash left(x\; -\; \backslash frac\backslash right)\backslash left(x\; +\; \backslash frac\backslash right)\; =\; \backslash frac\backslash left(-\backslash frac\; +\; x^2\; -\; 1\backslash right).$$

It can be interpreted as the generator of fractional Fourier transforms for arbitrary values of, and of the conventional continuous Fourier transform

l{F} for the particular value t = \pi/ 2 , with the Mehler kernel implementing the corresponding active transform. The eigenfunctions of N are the Hermite functions \psi _{n(x)} which are therefore also eigenfunctions of l{F}.

Upon extending the Fourier transform to distributions the Dirac comb is also an eigenfunction of the Fourier transform.

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$$\backslash left(M^\_\backslash xi\; T^\_y\; M\_\backslash xi\; T\_yf\backslash right)(x)\; =\; e^f(x)$$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$$\backslash left(x\_1,\; \backslash xi\_1,\; t\_1\backslash right)\; \backslash cdot\; \backslash left(x\_2,\; \backslash xi\_2,\; t\_2\backslash right)\; =\; \backslash left(x\_1\; +\; x\_2,\; \backslash xi\_1\; +\; \backslash xi\_2,\; t\_1\; t\_2\; e^\backslash right).$$

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$$J\; \backslash begin\; x\; \backslash \backslash \; \backslash xi\backslash end\; =\; \backslash begin\; -\backslash xi\; \backslash \backslash \; x\backslash end$$so that . This can be extended to a unique automorphism of :$$j\backslash left(x,\; \backslash xi,\; t\backslash right)\; =\; \backslash left(-\backslash xi,\; x,\; te^\backslash right).$$

According to the Stone–von Neumann theorem, the unitary representations and are unitarily equivalent, so there is a unique intertwiner such that$$\backslash rho\; \backslash circ\; j\; =\; W\; \backslash rho\; W^*.$$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$$\backslash hat\; f\; (\backslash xi)\; =\; \backslash int\; \_^\backslash infty\; e^\; f(t)\; \backslash ,\; dt$$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,$$\backslash left\backslash vert\; \backslash xi\; ^n\; \backslash hat\; f(\backslash xi)\; \backslash right\backslash vert\; \backslash leq\; C\; e^$$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$$\backslash hat\; f\; (i\backslash tau)\; =\; \backslash int\; \_^\backslash infty\; e^\; f(t)\; \backslash ,\; dt,$$convergent for all, is the two-sided Laplace transform of .

The more usual version ("one-sided") of the Laplace transform is$$F(s)\; =\; \backslash int\_0^\backslash infty\; f(t)\; e^\; \backslash ,\; dt.$$

If is also causal, and analytical, then:

\hat f(i\tau) = F( -2 \pi\tau). Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable .

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements. Two particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle. Such designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.

Furthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response. There, only the delay circuit in-between admits a classical Fourier description, which is critical. Both the circuits to the side are unstable, and do not admit a convergent Fourier decomposition. However, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.

In modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of harmonic analysis.

Inversion

Still with

\xi = \sigma + i\tau , if \widehat f is complex analytic for, then

$$\backslash int\; \_^\backslash infty\; \backslash hat\; f\; (\backslash sigma\; +\; ia)\; e^\; \backslash ,\; d\backslash sigma\; =\; \backslash int\; \_^\backslash infty\; \backslash hat\; f\; (\backslash sigma\; +\; ib)\; e^\; \backslash ,\; d\backslash sigma$$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$$f(t)\; =\; \backslash int\_^\backslash infty\; \backslash hat\; f(\backslash sigma\; +\; i\backslash tau)\; e^\; \backslash ,\; d\backslash sigma,$$for any .

This theorem implies the Mellin inversion formula for the Laplace transformation,$$f(t)\; =\; \backslash frac\; 1\; \backslash int\_^\; F(s)\; e^\backslash ,\; ds$$for any, where is the Laplace transform of .

The hypotheses can be weakened, as in the results of Carleson 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

\R ^{n\star} , 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:$$\backslash int\_^\backslash infty\; |f(x)|^2\; \backslash ,dx=1.$$

It follows from the Plancherel theorem that is also normalized.

The spread around may be measured by the dispersion about zero defined by$$D\_0(f)=\backslash int\_^\backslash infty\; x^2|f(x)|^2\backslash ,dx.$$

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$$D\_0(f)D\_0(\backslash hat)\; \backslash geq\; \backslash frac.$$

The equality is attained only in the casewhere 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:$$\backslash left(\backslash int\_^\backslash infty\; (x-x\_0)^2|f(x)|^2\backslash ,dx\backslash right)\backslash left(\backslash int\_^\backslash infty(\backslash xi-\backslash xi\_0)^2\backslash left|\backslash hat(\backslash xi)\backslash right|^2\backslash ,d\backslash xi\backslash right)\backslash geq\; \backslash frac$$for any, .

In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, up to a factor of the Planck constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle.

A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:$$H\backslash left(\backslash left|f\backslash right|^2\backslash right)+H\backslash left(\backslash left|\backslash hat\backslash right|^2\backslash right)\backslash ge\; \backslash log\backslash left(\backslash frac\backslash right)$$where is the differential entropy of the probability density function :$$H(p)\; =\; -\backslash int\_^\backslash infty\; p(x)\backslash log\backslash bigl(p(x)\backslash bigr)\; \backslash ,\; dx$$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

See main article: 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$$f(t)\; =\; \backslash int\_0^\backslash infty\; \backslash bigl(a(\backslash lambda)\; \backslash cos(2\backslash pi\; \backslash lambda\; t)\; +\; b(\backslash lambda)\; \backslash sin(2\backslash pi\; \backslash lambda\; t)\backslash bigr)\; \backslash ,\; d\backslash lambda.$$

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):$$a\; (\backslash lambda)\; =\; 2\backslash int\_^\backslash infty\; f(t)\; \backslash cos(2\backslash pi\backslash lambda\; t)\; \backslash ,\; dt$$and$$b\; (\backslash lambda)\; =\; 2\backslash int\_^\backslash infty\; f(t)\; \backslash sin(2\backslash pi\backslash lambda\; t)\; \backslash ,\; dt.$$

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$$f(t)\; =\; 2\backslash int\_0\; ^\; \backslash int\_^\; f(\backslash tau)\; \backslash cos\backslash bigl(2\backslash pi\; \backslash lambda(\backslash tau-t)\backslash bigr)\; \backslash ,\; d\backslash tau\; \backslash ,\; d\backslash lambda.$$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$$\backslash hat(\backslash xi)=F\_0(|\backslash xi|)P(\backslash xi)$$where$$F\_0(r)\; =\; 2\backslash pi\; i^r^\; \backslash int\_0^\backslash infty\; f\_0(s)J\_\backslash frac(2\backslash pi\; rs)s^\backslash frac\backslash ,ds.$$

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 . 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:$$f\_R(x)\; =\; \backslash int\_\backslash hat(\backslash xi)\; e^\backslash ,\; d\backslash xi,\; \backslash quad\; x\; \backslash in\; \backslash mathbb^n.$$

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

Notes and References

1. Depending on the application a Lebesgue integral, distributional, or other approach may be most appropriate.
2. provides solid justification for these formal procedures without going too deeply into functional analysis or the theory of distributions.
3. In relativistic quantum mechanics one encounters vector-valued Fourier transforms of multi-component wave functions. In quantum field theory, operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for example .
4. For this article, a rapidly decreasing function is a function f(x) on the reals that tends to zero together with all derivatives as x\to\pm inf ty : \lim _{x\toinfty} f ^{(n)} (x) = 0 , n =0 , 1 , 2 , ... . See Schwartz function.
5. A possible source of confusion is the frequency-shifting property; i.e. the transform of function f(x)e ^{- i 2 \pi \xi _{0} x} is \widehat{f}(\xi + \xi _{0).}   The value of this function at  \xi =0   is \widehat{f}(\xi _{0),} meaning that a frequency \xi _{0} has been shifted to zero (also see Negative frequency).
6. proves on pp. 216–226 the Fourier integral theorem before studying Fourier series.
7. .
8. The operator U\left( (1)/(2 \pi) (d)/(dx) \right) is defined by replacing x by (1)/(2 \pi) (d)/(dx) in the Taylor expansion of U(x).