Fourier transform

In mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies 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 Fourier transform 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 terms of the intensity of its constituent pitches.

An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.

The red sinusoid can be described by peak amplitude (1), peak-to-peak (2), RMS (3), and wavelength (4). The red and blue sinusoids have a phase difference of θ.

${\displaystyle \scriptstyle f(t)}$

${\displaystyle \scriptstyle {\hat {f}}(\omega )}$

${\displaystyle \scriptstyle g(t)}$

${\displaystyle \scriptstyle {\hat {g}}(\omega )}$

${\displaystyle \scriptstyle t}$

${\displaystyle \scriptstyle \omega }$

${\displaystyle \scriptstyle t}$

${\displaystyle \scriptstyle \omega }$

The top row shows a unit pulse as a function of time (f(t)) and its Fourier transform as a function of frequency (f̂(ω)). The bottom row shows a delayed unit pulse as a function of time (g(t)) and its Fourier transform as a function of frequency (ĝ(ω)). Translation (i.e. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. The Fourier transform decomposes a function into eigenfunctions for the group of translations. The imaginary part of ĝ(ω) is negated because a negative sign exponent has been used in the Fourier transform, which is the default as derived from the Fourier series, but the sign does not matter for a transform that is not going to be reversed.

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.^{[note 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.^{[note 2]}

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional 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.^{[note 3]} Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R^{[clarification needed]}or Rⁿ (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S¹, 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.

Definitions

The Fourier transform on R

The Fourier transform is an extension of the Fourier series, which in its most general form introduces the use of complex exponential functions. For example, for a function ${\displaystyle f(x)}$ , the amplitude and phase of a frequency component at frequency ${\displaystyle n/P,n\in \mathbb {Z} }$ , is given by this complex number:

${\displaystyle c_{n}={\tfrac {1}{P}}\int _{P}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.}$ 

The extension provides a frequency continuum of components ${\displaystyle \left(\xi \in \mathbb {R} \right),}$  using an infinite integral of integration:

Fourier transform

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx.}$

(Eq.1)

Here, the transform of function ${\displaystyle f(x)}$  at frequency ${\displaystyle \xi }$  is denoted by the complex number ${\displaystyle {\hat {f}}(\xi )}$ , which is just one of several common conventions. Evaluating Eq.1 for all values of ${\displaystyle \xi }$  produces the frequency-domain function. When the independent variable (${\displaystyle x}$ ) represents time (often denoted by ${\displaystyle t}$ ), the transform variable (${\displaystyle \xi }$ ) represents frequency (often denoted by ${\displaystyle f}$ ). For example, if time is measured in seconds, then frequency is in hertz.

For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that frequency integrated over the domain, and the argument of the complex value represents that complex sinusoid's phase offset. If a frequency is not present, the transform has a value of 0 for 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. The Fourier inversion theorem provides a synthesis process that recreates the original function from its frequency domain representation.

A key to interpreting Eq.1 is that the effect of multiplying ${\displaystyle f(x)}$  by ${\displaystyle e^{-i2\pi \xi x}}$  is to subtract ${\displaystyle \xi }$  from every frequency component of function ${\displaystyle f(x).}$ ^{[note 4]} (also see Negative frequency) So the component that was at ${\displaystyle \xi }$  ends up at zero hertz, and the integral produces its amplitude, because all the other components are oscillatory and integrate to zero over an infinite interval.

The functions ${\displaystyle f}$  and ${\displaystyle {\hat {f}}}$  are often referred to as a Fourier transform pair.^[1]  A common notation for designating transform pairs is:^[2]

${\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\hat {f}}(\xi )\quad {\text{and therefore}}\quad \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi )}$ 

A function can be recovered from its Fourier series, under suitable conditions. When this is possible, the Fourier series provides the inversion formula:

${\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x}.}$ 

Similarly, under suitable conditions on ${\displaystyle f}$ , the Fourier inversion formula on ${\displaystyle \mathbb {R} }$  is:

Fourier inversion integral

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} ,}$

(Eq.2)

The complex number, ${\displaystyle {\hat {f}}(\xi )}$ , conveys both amplitude and phase of frequency ${\displaystyle \xi }$ . So Eq.2 is a representation of ${\displaystyle f(x)}$  as a weighted summation of complex exponential functions. This is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat,^[3][4] although a proof by modern standards was not given until much later.^[5][6]

Other notational conventions

For other common conventions and notations, including using the angular frequency ω instead of the ordinary frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. The 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 L² and an algebra homomorphism from L¹ to L^∞, without renormalizing the Lebesgue measure.^[7]

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

Main articles: Fourier analysis § History, and Fourier series § History

In 1821, Fourier claimed (see Joseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be expanded into a series of sines.^[8] 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 ${\displaystyle {\hat {f}}(\xi )}$  are complex numbers, which have two equivalent forms (see Euler's formula):

${\displaystyle {\hat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}$ 

The product with ${\displaystyle e^{i2\pi \xi x}}$  (Eq.2) has these forms:

${\displaystyle {\hat {f}}(\xi )\cdot e^{i2\pi \xi x}=Ae^{i\theta }\cdot e^{i2\pi \xi x}=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.}$ 

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

See also: Negative frequency § Simplifying the Fourier transform

One aspect of the Fourier transform that is often confusing is its use of negative frequency. When frequency is thought of as the rate how often something happens, or in physics, as 1/T for some period of time T. Those notions of frequency are inherently positive. But the value of ${\displaystyle \xi }$  in Eq.1 is allow to take on any real number.

For real-valued functions, there is a simple relationship between the values of the Fourier transform for positive and negative ${\displaystyle \xi }$  (see conjugation below). This makes it possible to avoid the subject of negative frequencies by using the sine and cosine transforms. But most authors prefer using Eq.1 rather than using two transforms. One reason for this is that many applications have to take the Fourier transform of complex-valued functions, such as partial differential equations, radar, nonlinear optics, quantum mechanics, and others. In these cases, the value of the Fourier transform at negative frequencies is distinct from the value at real frequencies, and they are important. In these situations, the concept of what is a frequency is defined by the Fourier transform rather than appealing to a rate or period.

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 Eq.1 to be defined the function must be absolutely integrable. Instead it is common to instead 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 which have a convergent Fourier series. If ${\displaystyle f(x)}$  periodic function, with period ${\displaystyle P}$ , that has a convergent Fourier series, then:

${\displaystyle {\hat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}$ 

where ${\displaystyle c_{n}}$  are the Fourier series coefficients of ${\displaystyle f}$ . 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 ${\displaystyle f}$  can be sampled at regular intervals of ${\displaystyle {\tfrac {1}{P}}.}$  These samples can be deduced from one cycle of a periodic function ${\displaystyle f_{P}}$  which has Fourier series coefficients proportional to those samples by the Poisson summation formula:

${\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\hat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }$ 

The integrability of ${\displaystyle f}$  ensures the periodic summation converges. Therefore, the samples ${\displaystyle {\hat {f}}\left({\tfrac {k}{P}}\right)}$  can be determined by:

${\displaystyle {\hat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}$ 

When ${\displaystyle f(x)}$  has compact support, ${\displaystyle f_{P}(x)}$  has a finite number of terms within the interval of integration. When ${\displaystyle f(x)}$  does not have compact support, numerical evaluation of ${\displaystyle f_{P}(x)}$  requires an approximation, such as tapering ${\displaystyle f(x)}$  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(6πt) e^−πt2 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. 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 ${\displaystyle {\hat {f}}(3)}$  we must integrate e^−i2π(3t)f(t). The second image shows the plot of the real and imaginary parts of this function. The real part of the integrand is almost always positive, because when f(t) is negative, the real part of e^−i2π(3t) is negative as well. Because they oscillate at the same rate, when f(t) is positive, so is the real part of e^−i2π(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in this case 1/2). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at ${\displaystyle {\hat {f}}(5)}$ , 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 f(t).

•  

  Original function showing oscillation 3 Hz.

•  

  Real and imaginary parts of integrand for Fourier transform at 3 Hz

•  

  Real and imaginary parts of integrand for Fourier transform at 5 Hz

•  

  Magnitude of Fourier transform, with 3 and 5 Hz labeled.

Properties of the Fourier transform

Here we assume f(x), g(x) and h(x) are integrable functions: Lebesgue-measurable on the real line satisfying:

$${\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}$$

 

We denote the Fourier transforms of these functions as f̂(ξ), ĝ(ξ) and ĥ(ξ) respectively.

Basic properties

The Fourier transform has the following basic properties:^[9]

Linearity

For any complex numbers a and b, if h(x) = a f (x) + b g(x), then ĥ(ξ) = a f̂(ξ) + b ĝ(ξ).

Translation / time shifting

 

Animation showing the Fourier Transform of a time shifted signal. [Top] the original signal (yellow), is continuously time shifted (blue). [Bottom] The resultant Fourier Transform of the time shifted signal. Note how the higher frequency components revolve in complex plane faster than the lower frequency components.

For any real number x₀, if h(x) = f(x − x₀), then ĥ(ξ) = e^{−i2πx₀ξ} f̂(ξ).

Modulation / frequency shifting

For any real number ξ₀, if h(x) = e^i2πξ₀x f(x), then ĥ(ξ) = f̂(ξ − ξ₀).

Time scaling

For a non-zero real number a, if h(x) = f(ax), then

$${\displaystyle {\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right).}$$

 

The case a = −1 leads to the time-reversal property, which states: if h(x) = f(−x), then ĥ(ξ) = f̂(−ξ).

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: ${\displaystyle {\begin{aligned}{\mathsf {Time\ domain}}\quad &\ f\quad &=\quad &f_{_{RE}}\quad &+\quad &f_{_{RO}}\quad &+\quad i\ &f_{_{IE}}\quad &+\quad &\underbrace {i\ f_{_{IO}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}\quad &{\hat {f}}\quad &=\quad &{\hat {f}}_{RE}\quad &+\quad &\overbrace {i\ {\hat {f}}_{IO}} \quad &+\quad i\ &{\hat {f}}_{IE}\quad &+\quad &{\hat {f}}_{RO}\end{aligned}}}$ 

From this, various relationships are apparent, for example:

• The transform of a real-valued function (f_RE+ f_RO) is the even symmetric function f̂_RE+ i f̂_IO. Conversely, an even-symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function (i f_IE+ i f_IO) is the odd symmetric function f̂_RO+ i f̂_IE, and the converse is true.
• The transform of an even-symmetric function (f_RE+ i f_IO) is the real-valued function f̂_RE+ f̂_RO, and the converse is true.
• The transform of an odd-symmetric function (f_RO+ i f_IE) is the imaginary-valued function i f̂_IE+ i f̂_IO, and the converse is true.

Conjugation

If h(x) = f(x), then

$${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(-\xi )}}.}$$

 

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

$${\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}},}$$

 

that is, f̂ is a Hermitian function. And if f is purely imaginary, then

$${\displaystyle {\hat {f}}(-\xi )=-{\hat {f}}(\xi ).}$$

 

Real and imaginary part in time

• If ${\displaystyle h(x)=\Re {(f(x))}}$ , then ${\displaystyle {\hat {h}}(\xi )={\frac {1}{2}}\left({\hat {f}}(\xi )+{\overline {{\hat {f}}(-\xi )}}\right)}$ .
• If ${\displaystyle h(x)=\Im {(f(x))}}$ , then ${\displaystyle {\hat {h}}(\xi )={\frac {1}{2i}}\left({\hat {f}}(\xi )-{\overline {{\hat {f}}(-\xi )}}\right)}$ .

The zero frequency component

Substituting ξ = 0 in the definition, we obtain

$${\displaystyle {\hat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}$$

 

That is the same as the integral of f over all its domain and is also known as the average value or DC bias of the function.

Invertibility and periodicity

Further information: Fourier inversion theorem and Fractional Fourier transform

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

More precisely, defining the parity operator ${\displaystyle {\mathcal {P}}}$  such that ${\displaystyle ({\mathcal {P}}f)(x)=f(-x)}$ , we have:

${\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}$ 

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 x-axis and frequency as the y-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 SL₂(R) on the time–frequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. This approach is particularly studied in signal processing, under time–frequency analysis.

Units and duality

See also: Spectral density § Units

The frequency variable must have inverse units to the units of the original function's domain (typically named t or x). For example, if t is measured in seconds, ξ should be in cycles per second or hertz. If the scale of time is in units of 2π seconds, then another greek letter ω typically is used instead to represent angular frequency (where ω = 2πξ) in units of radians per second. If using x for units of length, then ξ 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 t and measured in units of t, and the other which is the range of ξ and measured in inverse units to the units of t. 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, ξ 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 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 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.

Let ${\displaystyle {\hat {f}}_{1}(\xi )}$  be the form of the Fourier transform in terms of ordinary frequency ξ.

Because ${\displaystyle \xi ={\tfrac {\omega }{2\pi }}}$ , the alternative form ${\displaystyle {\hat {f}}_{3}(\omega )}$  (which Fourier transform § Other conventions calls the non-unitary form in angular frequency) has no factor in its definition

${\displaystyle {\hat {f}}_{3}(\omega )\mathrel {\stackrel {\mathrm {def} }{=}} \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\hat {f}}_{1}\left({\frac {\omega }{2\pi }}\right)}$ 

but has a factor of ${\displaystyle {\tfrac {1}{2\pi }}}$  in its corresponding inversion formula

${\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}_{3}(\omega )\,e^{i\omega \,x}\,d\omega .}$ 

An alternative form ${\displaystyle {\hat {f}}_{2}(\omega )}$  (which Fourier transform § Other conventions calls the unitary form in angular frequency) has a factor of ${\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}$  in its definition

${\displaystyle {\hat {f}}_{2}(\omega )\mathrel {\stackrel {\mathrm {def} }{=}} {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\,e^{-i\omega \,x}\,dx}$ 

and also has that same factor of ${\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}$  in its corresponding inversion formula, producing a symmetrical relationship

${\displaystyle f(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {f}}_{2}(\omega )\,e^{i\omega \,x}\,d\omega .}$ 

In other conventions, the Fourier transform has i in the exponent instead of −i, and vice versa for the inversion formula. This convention is common in modern physics^[10] 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 ${\displaystyle {\hat {f}}(\xi )}$  is the amplitude of the wave  ${\displaystyle e^{-i2\pi \xi x}}$   instead of the wave  ${\displaystyle e^{i2\pi \xi x}}$  (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 i have it replaced by −i. In Electrical engineering the letter j is typically used for the imaginary unit instead of i because i 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 Φ of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential, and since the units of x are ignored, there is no 2π either:

${\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}$ 

(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 rectangular function is Lebesgue integrable.

 

The sinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.

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 f̂ of any integrable function f is uniformly continuous and^[11]

${\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}$ 

By the Riemann–Lebesgue lemma,^[12]

${\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}$ 

However, ${\displaystyle {\hat {f}}}$  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 f and ${\displaystyle {\hat {f}}}$  are integrable, the inverse equality

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi }$ 

holds almost everywhere. That is, the Fourier transform is injective on L¹(R). (But if f is continuous, then equality holds for every x.)

Plancherel theorem and Parseval's theorem

Let f(x) and g(x) be integrable, and let f̂(ξ) and ĝ(ξ) be their Fourier transforms. If f(x) and g(x) are also square-integrable, then the Parseval formula follows:^[13]

${\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}$ 

where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that^[14]

${\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}$ 

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L²(R). On L¹(R) ∩ L²(R), this extension agrees with original Fourier transform defined on L¹(R), thus enlarging the domain of the Fourier transform to L¹(R) + L²(R) (and consequently to L^p(R) for 1 ≤ p ≤ 2). 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

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 f,

${\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}$ 

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 f(x) is an absolutely continuous differentiable function, and both f and its derivative f′ are integrable. Then the Fourier transform of the derivative is given by

${\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}$ 

More generally, the Fourier transformation of the nth derivative f⁽ⁿ⁾ is given by

${\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}$ 

Analogically, ${\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}$ 

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 "f(x) is smooth if and only if f̂(ξ) quickly falls to 0 for |ξ| → ∞." By using the analogous rules for the inverse Fourier transform, one can also say "f(x) quickly falls to 0 for |x| → ∞ if and only if f̂(ξ) is smooth."

Convolution theorem

Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) are integrable functions with Fourier transforms f̂(ξ) and ĝ(ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms f̂(ξ) and ĝ(ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if:

${\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}$ 

where ∗ denotes the convolution operation, then:

${\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}$ 

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system with input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case, ĝ(ξ) represents the frequency response of the system.

Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms p̂(ξ) and q̂(ξ).

Cross-correlation theorem

Main article: Cross-correlation

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

${\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}$ 

then the Fourier transform of h(x) is:

${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}$ 

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

${\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}$ 

for which

${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}$ 

Eigenfunctions

See also: Mehler kernel and Hermite polynomials § Hermite functions as eigenfunctions of the Fourier transform

The Fourier transform is a linear transform which has eigenfunctions obeying ${\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,}$  with ${\displaystyle \lambda \in \mathbb {C} .}$ 

A set of eigenfunctions is found by noting that the homogeneous differential equation

${\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0}$ 

leads to eigenfunctions ${\displaystyle \psi (x)}$  of the Fourier transform ${\displaystyle {\mathcal {F}}}$  as long as the form of the equation remains invariant under Fourier transform.^{[note 5]} In other words, every solution ${\displaystyle \psi (x)}$  and its Fourier transform ${\displaystyle {\hat {\psi }}(\xi )}$  obey the same equation. Assuming uniqueness of the solutions, every solution ${\displaystyle \psi (x)}$  must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if ${\displaystyle U(x)}$  can be expanded in a power series in which for all terms the same factor of either one of ${\displaystyle \pm 1,\pm i}$  arises from the factors ${\displaystyle i^{n}}$  introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable ${\displaystyle U(x)=x}$  leads to the standard normal distribution.^[15]

More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation

${\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}$ 

with ${\displaystyle C}$  constant and ${\displaystyle W(x)}$  being a non-constant even function remains invariant in form when applying the Fourier transform ${\displaystyle {\mathcal {F}}}$  to both sides of the equation. The simplest example is provided by ${\displaystyle W(x)=x^{2}}$  which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator.^[16] The corresponding solutions provide an important choice of an orthonormal basis for L²(R) and are given by the "physicist's" Hermite functions. Equivalently one may use

${\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}$ 

where He_n(x) are the "probabilist's" Hermite polynomials, defined as

${\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}$ 

Under this convention for the Fourier transform, we have that

${\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}$ 

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L²(R).^[9][17] However, this choice of eigenfunctions is not unique. Because of ${\displaystyle {\mathcal {F}}^{4}=\mathrm {id} }$  there are only four different eigenvalues of the Fourier transform (the fourth roots of unity ±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.^[18] As a consequence of this, it is possible to decompose L²(R) as a direct sum of four spaces H₀, H₁, H₂, and H₃ where the Fourier transform acts on He_k simply by multiplication by i^k.

Since the complete set of Hermite functions ψ_n 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:

${\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}$ 

This approach to define the Fourier transform was first proposed by Norbert Wiener.^[19] 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.^[20] In physics, this transform was introduced by Edward Condon.^[21] 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 ${\displaystyle N}$  via^[22]

${\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}$ 

The operator ${\displaystyle N}$  is the number operator of the quantum harmonic oscillator written as^[23][24]

${\displaystyle N\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right).}$ 

It can be interpreted as the generator of fractional Fourier transforms for arbitrary values of t, and of the conventional continuous Fourier transform ${\displaystyle {\mathcal {F}}}$  for the particular value ${\displaystyle t=\pi /2,}$  with the Mehler kernel implementing the corresponding active transform. The eigenfunctions of ${\displaystyle N}$  are the Hermite functions ${\displaystyle \psi _{n}(x)}$  which are therefore also eigenfunctions of ${\displaystyle {\mathcal {F}}.}$ 

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 L²(R) of square integrable complex valued functions f on the real line, generated by the translations (T_y f)(x) = f (x + y) and multiplication by e^i2πξx, (M_ξ f)(x) = e^i2πξx f (x). These operators do not commute, as their (group) commutator is

${\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{i2\pi \xi y}f(x)}$ 

which is multiplication by the constant (independent of x) e^i2πξy ∈ U(1) (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional Lie group of triples (x, ξ, z) ∈ R² × U(1), with the group law

${\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{i2\pi \left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}$ 

Denote the Heisenberg group by H₁. The above procedure describes not only the group structure, but also a standard unitary representation of H₁ on a Hilbert space, which we denote by ρ : H₁ → B(L²(R)). Define the linear automorphism of R² by

${\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}$ 

so that J² = −I. This J can be extended to a unique automorphism of H₁:

${\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-i2\pi \xi x}\right).}$ 

According to the Stone–von Neumann theorem, the unitary representations ρ and ρ ∘ j are unitarily equivalent, so there is a unique intertwiner W ∈ U(L²(R)) such that

${\displaystyle \rho \circ j=W\rho W^{*}.}$ 

This operator W is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.^[25] For example, the square of the Fourier transform, W², is an intertwiner associated with J² = −I, and so we have (W²f)(x) = f (−x) is the reflection of the original function f.

Complex domain

The integral for the Fourier transform

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}$ 

can be studied for complex values of its argument ξ. Depending on the properties of f, this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of ξ = σ + iτ, or something in between.^[26]

The Paley–Wiener theorem says that f is smooth (i.e., n-times differentiable for all positive integers n) and compactly supported if and only if f̂ (σ + iτ) is a holomorphic function for which there exists a constant a > 0 such that for any integer n ≥ 0,

${\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi )\right\vert \leq Ce^{a\vert \tau \vert }}$ 

for some constant C. (In this case, f is supported on [−a, a].) This can be expressed by saying that f̂ is an entire function which is rapidly decreasing in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).^[27]

(If f is not smooth, but only L², the statement still holds provided n = 0.^[28]) The space of such functions of a complex variable is called the Paley—Wiener space. This theorem has been generalised to semisimple Lie groups.^[29]

If f is supported on the half-line t ≥ 0, then f 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 f̂ extends to a holomorphic function on the complex lower half-plane τ < 0 which tends to zero as τ goes to infinity.^[30] The converse is false and it is not known how to characterise the Fourier transform of a causal function.^[31]

Laplace transform

See also: Laplace transform § Fourier transform

The Fourier transform f̂(ξ) is related to the Laplace transform F(s), which is also used for the solution of differential equations and the analysis of filters.

It may happen that a function f 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 f(t) is of exponential growth, i.e.,

${\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}$ 

for some constants C, a ≥ 0, then^[32]

${\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}$ 

convergent for all 2πτ < −a, is the two-sided Laplace transform of f.

The more usual version ("one-sided") of the Laplace transform is

${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}$ 

If f is also causal, and analytical, then: ${\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).}$  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 s = i2πξ.

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

If f̂ is complex analytic for a ≤ τ ≤ b, then

${\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{i2\pi \xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{i2\pi \xi t}\,d\sigma }$ 

by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.^[33]

Theorem: If f(t) = 0 for t < 0, and |f(t)| < Ce^a|t| for some constants C, a > 0, then

${\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{i2\pi \xi t}\,d\sigma ,}$ 

for any τ < −a/2π.

This theorem implies the Mellin inversion formula for the Laplace transformation,^[32]

${\displaystyle f(t)={\frac {1}{i2\pi }}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}$ 

for any b > a, where F(s) is the Laplace transform of f(t).

The hypotheses can be weakened, as in the results of Carleson and Hunt, to f(t) e^−at being L¹, provided that f is of bounded variation in a closed neighborhood of t (cf. Dirichlet–Dini theorem), the value of f at t 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.^[34]

L² versions of these inversion formulas are also available.^[35]

Fourier transform on Euclidean space

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

${\displaystyle {\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} }$ 

where x and ξ are n-dimensional vectors, and x · ξ is the dot product of the vectors. Alternatively, ξ can be viewed as belonging to the dual vector space ${\displaystyle \mathbb {R} ^{n\star }}$ , in which case the dot product becomes the contraction of x and ξ, usually written as ⟨x, ξ⟩.

All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.^[12]

Uncertainty principle

Further information: Gabor limit

Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform f̂(ξ) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.

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

${\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}$ 

It follows from the Plancherel theorem that f̂(ξ) is also normalized.

The spread around x = 0 may be measured by the dispersion about zero^[36] defined by

${\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}$ 

In probability terms, this is the second moment of |f(x)|² about zero.

The uncertainty principle states that, if f(x) is absolutely continuous and the functions x·f(x) and f′(x) are square integrable, then^[9]

${\displaystyle D_{0}(f)D_{0}\left({\hat {f}}\right)\geq {\frac {1}{16\pi ^{2}}}}$ .

The equality is attained only in the case

${\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}$ 

where σ > 0 is arbitrary and C₁ = ⁴√2/√σ so that f is L²-normalized.^[9] In other words, where f is a (normalized) Gaussian function with variance σ²/2π, centered at zero, and its Fourier transform is a Gaussian function with variance σ⁻²/2π.

In fact, this inequality implies that:

${\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}}$ 

for any x₀, ξ₀ ∈ R.^[37]

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.^[38]

A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:

${\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}$ 

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

${\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}$ 

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

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 f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically^[39]) λ by

${\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}$ 

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):

${\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}$ 

and

${\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}$ 

Older literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b.

The function f can be recovered from the sine and cosine transform using

${\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}$ 

together with trigonometric identities. This is referred to as Fourier's integral formula.^{[32][40][41][42]}

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree k on Rⁿ be denoted by A_k. The set A_k consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e^−π|x|2P(x) for some P(x) in A_k, then f̂(ξ) = i^−k f(ξ). Let the set H_k be the closure in L²(Rⁿ) of linear combinations of functions of the form f(|x|)P(x) where P(x) is in A_k. The space L²(Rⁿ) is then a direct sum of the spaces H_k and the Fourier transform maps each space H_k to itself and is possible to characterize the action of the Fourier transform on each space H_k.^[12]

Let f(x) = f₀(|x|)P(x) (with P(x) in A_k), then

${\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}$ 

where

${\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}$ 

Here J_{(n + 2k − 2)/2} denotes the Bessel function of the first kind with order n + 2k − 2/2. When k = 0 this gives a useful formula for the Fourier transform of a radial function.^[43] This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n^[44] 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 L²(Rⁿ) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in L^p for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in Rⁿ is of particular interest. In this case the Tomas–Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in Rⁿ is a bounded operator on L^p provided 1 ≤ p ≤ 2n + 2/n + 3.

One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets E_R indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function f, consider the function f_R defined by:

${\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}$ 

Suppose in addition that f ∈ L^p(Rⁿ). For n = 1 and 1 < p < ∞, if one takes E_R = (−R, R), then f_R converges to f in L^p as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that E_R is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball E_R = {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in L^p(Rⁿ). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2.^[19] In fact, when p ≠ 2, this shows that not only may f_R fail to converge to f in L^p, but for some functions f ∈ L^p(Rⁿ), f_R is not even an element of L^p.

Fourier transform on function spaces

On L^p spaces

On L¹

The definition of the Fourier transform by the integral formula

${\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx}$ 

is valid for Lebesgue integrable functions f; that is, f ∈ L¹(Rⁿ).

The Fourier transform F : L¹(Rⁿ) → L^∞(Rⁿ) is a bounded operator. This follows from the observation that

${\displaystyle \left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}$ 

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 L¹ is a subset of the space C₀(Rⁿ) 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 L²

Since compactly supported smooth functions are integrable and dense in L²(Rⁿ), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L²(Rⁿ) by continuity arguments. The Fourier transform in L²(Rⁿ) is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an L² function f,

${\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-i2\pi \xi \cdot x}\,dx}$ 

where the limit is taken in the L² sense. (More generally, you can take a sequence of functions that are in the intersection of L¹ and L² and that converges to f in the L²-norm, and define the Fourier transform of f as the L² -limit of the Fourier transforms of these functions.^[45])

Many of the properties of the Fourier transform in L¹ carry over to L², by a suitable limiting argument.

Furthermore, F : L²(Rⁿ) → L²(Rⁿ) is a unitary operator.^[46] For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any f, g ∈ L²(Rⁿ) we have

${\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}$ 

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

On other L^p

The definition of the Fourier transform can be extended to functions in L^p(Rⁿ) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L² plus a fat body part in L¹. In each of these spaces, the Fourier transform of a function in L^p(Rⁿ) is in L^q(Rⁿ), where q = p/p − 1 is the Hölder conjugate of p (by the Hausdorff–Young inequality). However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in L^p for the range 2 < p < ∞ requires the study of distributions.^[11] In fact, it can be shown that there are functions in L^p with p > 2 so that the Fourier transform is not defined as a function.^[12]

Tempered distributions

Main article: Distribution (mathematics) § Tempered distributions and Fourier transform

One might consider enlarging the domain of the Fourier transform from L¹ + L² by considering generalized functions, or distributions. A distribution on Rⁿ is a continuous linear functional on the space C_c(Rⁿ) of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on C_c(Rⁿ) and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map C_c(Rⁿ) to C_c(Rⁿ). In fact the Fourier transform of an element in C_c(Rⁿ) can not vanish on an open set; see the above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartz functions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.^[12] The tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.

For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let f̂ and ĝ be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,^[12]

${\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}$ 

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

${\displaystyle T_{f}(\varphi )=\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}$ 

for all Schwartz functions φ. So it makes sense to define Fourier transform T̂_f of T_f by

${\displaystyle {\hat {T}}_{f}(\varphi )=T_{f}\left({\hat {\varphi }}\right)}$ 

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

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

Generalizations

Fourier–Stieltjes transform

The Fourier transform of a finite Borel measure μ on Rⁿ is given by:^[47]

${\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu .}$ 

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures.^[11] In the case that dμ = f(x) dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take e^iξx instead of e^−i2πξx.^[9] In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants.

The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.^[11]

Furthermore, the Dirac delta function, although not a function, is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).

Locally compact abelian groups

Main article: Pontryagin duality

The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group that is at the same time a locally compact Hausdorff topological space so that the group operation is continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its characters. With its natural group structure and the topology of pointwise convergence, the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L¹(G), its Fourier transform is defined by^[11]

${\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \quad {\text{for any }}\xi \in {\hat {G}}.}$ 

The Riemann–Lebesgue lemma holds in this case; f̂(ξ) is a function vanishing at infinity on Ĝ.

The Fourier transform on T = R/Z is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) ${\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}}$  where ${\displaystyle e_{k}(x)=e^{i2\pi kx}}$  for ${\displaystyle x\in T}$ .

The character of such representation, that is the trace of ${\displaystyle e_{k}(x)}$  for each ${\displaystyle x\in T}$  and ${\displaystyle k\in Z}$ , is ${\displaystyle e^{i2\pi kx}}$  itself. In the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma. Now the group T is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function ${\displaystyle e_{k}(x)}$  of ${\displaystyle x\in T,}$  and the inner product between two class functions (all functions being class functions since T is abelian) ${\displaystyle f,g\in L^{2}(T,d\mu )}$  is defined as ${\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)}$  with the normalizing factor ${\displaystyle |T|=1}$ . The sequence ${\displaystyle \{e_{k}\mid k\in Z\}}$  is an orthonormal basis of the space of class functions ${\displaystyle L^{2}(T,d\mu )}$ .

For any representation V of a finite group G, ${\displaystyle \chi _{v}}$  can be expressed as the span ${\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}}$  (${\displaystyle V_{i}}$  are the irreps of G), such that ${\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}$ . Similarly for ${\displaystyle G=T}$  and ${\displaystyle f\in L^{2}(T,d\mu )}$ , ${\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}$ . The Pontriagin dual ${\displaystyle {\hat {T}}}$  is ${\displaystyle \{e_k}$

March 21, 2023