Editing Fourier transform (section)

== Properties ==

Let <math>f(x)</math> and <math>h(x)</math> represent ''integrable functions'' [[Lebesgue-measurable]] on the real line satisfying:
<math display="block">\int_{-\infty}^\infty |f(x)| \, dx < \infty.</math>
We denote the Fourier transforms of these functions as <math>\hat f(\xi)</math> and <math>\hat h(\xi)</math> respectively.

=== Basic properties ===
The Fourier transform has the following basic properties:<ref name="Pinsky-2002">{{harvnb|Pinsky|2002}}</ref>

==== Linearity ====

<math display="block">a\ f(x) + b\ h(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ a\ \widehat f(\xi) + b\ \widehat h(\xi);\quad \ a,b \in \mathbb C</math>

==== Time shifting ====

<math display="block">f(x-x_0)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ e^{-i 2\pi x_0 \xi}\ \widehat f(\xi);\quad \ x_0 \in \mathbb R</math>

==== Frequency shifting ====

<math display="block">e^{i 2\pi \xi_0 x} f(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f(\xi - \xi_0);\quad \ \xi_0 \in \mathbb R</math>

==== Time scaling ====

<math display="block">f(ax)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \frac{1}{|a|}\widehat{f}\left(\frac{\xi}{a}\right);\quad \ a \ne 0 </math>
The case <math>a=-1</math> leads to the ''time-reversal property'':
<math display="block">f(-x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f (-\xi)</math>

<div class="skin-invert">{{Annotated image
| caption=The transform of an even-symmetric real-valued function <math>(f(t) = f_{RE})</math> is also an even-symmetric real-valued function <math>(\hat f_{RE}).</math>  The time-shift, <math>(g(t) = g_{RE} + g_{RO}),</math> creates an imaginary component, <math>i\cdot \hat g_{IO}.</math>  (see {{slink||Symmetry}}.
| image=Fourier_unit_pulse.svg
| image-width = 300
| outer-css = color: black;
| annotations =
{{Annotation|20|40|<math>\scriptstyle f(t)</math>}}
{{Annotation|170|40|<math>\scriptstyle \widehat{f}(\omega)</math>}}
{{Annotation|20|140|<math>\scriptstyle g(t)</math>}}
{{Annotation|170|140|<math>\scriptstyle \widehat{g}(\omega)</math>}}
{{Annotation|130|80|<math>\scriptstyle t</math>}}
{{Annotation|280|85|<math>\scriptstyle \omega</math>}}
{{Annotation|130|192|<math>\scriptstyle t</math>}}
{{Annotation|280|180|<math>\scriptstyle \omega</math>}}
}}</div>

==== Symmetry ====
When the real and imaginary parts of a complex function are decomposed into their [[Even and odd functions#Even–odd decomposition|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:<ref name="ProakisManolakis1996">{{cite book|last1=Proakis|first1=John G. |last2=Manolakis|first2=Dimitris G.|author2-link= Dimitris Manolakis |title=Digital Signal Processing: Principles, Algorithms, and Applications|url=https://archive.org/details/digitalsignalpro00proa|url-access=registration|year=1996|publisher=Prentice Hall|isbn=978-0-13-373762-2|edition=3rd|page=[https://archive.org/details/digitalsignalpro00proa/page/291 291]}}</ref>

<math>
\begin{array}{rlcccccccc}
\mathsf{Time\ domain} & f & = & f_{_{\text{RE}}} & + & f_{_{\text{RO}}} & + & i\ f_{_{\text{IE}}} & + & \underbrace{i\ f_{_{\text{IO}}}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & \widehat f & = & \widehat f_{_\text{RE}} & + & \overbrace{i\ \widehat f_{_\text{IO}}\,} & + & i\ \widehat f_{_\text{IE}} & + & \widehat f_{_\text{RO}}
\end{array}
</math>

From this, various relationships are apparent, for example''':'''
* The transform of a real-valued function <math>(f_{_{RE}}+f_{_{RO}})</math> is the ''[[Even and odd functions#Complex-valued functions|conjugate symmetric]]'' function <math>\hat f_{RE}+i\ \hat f_{IO}.</math>  Conversely, a ''conjugate symmetric'' transform implies a real-valued time-domain.
* The transform of an imaginary-valued function <math>(i\ f_{_{IE}}+i\ f_{_{IO}})</math> is the ''[[Even and odd functions#Complex-valued functions|conjugate antisymmetric]]'' function <math>\hat f_{RO}+i\ \hat f_{IE},</math> and the converse is true.
* The transform of a ''[[Even and odd functions#Complex-valued functions|conjugate symmetric]]'' function <math>(f_{_{RE}}+i\ f_{_{IO}})</math> is the real-valued function <math>\hat f_{RE}+\hat f_{RO},</math> and the converse is true.
* The transform of a ''[[Even and odd functions#Complex-valued functions|conjugate antisymmetric]]'' function <math>(f_{_{RO}}+i\ f_{_{IE}})</math> is the imaginary-valued function <math>i\ \hat f_{IE}+i\hat f_{IO},</math> and the converse is true.

==== Conjugation ====
<math display="block">\bigl(f(x)\bigr)^*\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \left(\widehat{f}(-\xi)\right)^*</math>
(Note: the ∗ denotes [[Complex conjugate|complex conjugation]].)

In particular, if <math>f</math> is '''real''', then <math>\widehat f</math> is [[Even and odd functions#Complex-valued functions|even symmetric]] (aka [[Hermitian function]]):
<math display="block">\widehat{f}(-\xi)=\bigl(\widehat f(\xi)\bigr)^*.</math>

And if <math>f</math> is purely imaginary, then <math>\widehat f</math> is [[Even and odd functions#Complex-valued functions|odd symmetric]]:
<math display="block">\widehat f(-\xi) = -(\widehat f(\xi))^*.</math>

==== Real and imaginary parts ====
<math display="block">\operatorname{Re}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ 
\tfrac{1}{2} \left( \widehat f(\xi) + \bigl(\widehat f (-\xi) \bigr)^* \right)</math>
<math display="block">\operatorname{Im}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ 
\tfrac{1}{2i} \left( \widehat f(\xi) - \bigl(\widehat f (-\xi) \bigr)^* \right)</math>

==== Zero frequency component ====
Substituting <math>\xi = 0</math> in the definition, we obtain:
<math display="block">\widehat{f}(0) = \int_{-\infty}^{\infty} f(x)\,dx.</math>

The integral of <math>f</math> over its domain is known as the average value or [[DC bias]] of the function.

=== Uniform continuity and the Riemann–Lebesgue lemma ===
[[File:Rectangular function.svg|class=skin-invert-image|thumb|The [[rectangular function]] is [[Lebesgue integrable]].]]
[[File:Sinc function (normalized).svg|class=skin-invert-image|thumb|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 <math>\hat{f}</math> of any integrable function <math>f</math> is [[uniformly continuous]] and{{sfn|Katznelson|2004|p=134}}{{sfn|Stein|Weiss|1971|p=2}}
<math display="block">\left\|\hat{f}\right\|_\infty \leq \left\|f\right\|_1</math>

By the ''[[Riemann–Lebesgue lemma]]'',<ref name="Stein-Weiss-1971">{{harvnb|Stein|Weiss|1971}}</ref>
<math display="block">\hat{f}(\xi) \to 0\text{ as }|\xi| \to \infty.</math>

However, <math>\hat{f}</math> 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 integral]]s behave analogously to the [[alternating harmonic series]], in converging to a sum without being [[absolutely convergent]].

It is not generally possible to write the ''inverse transform'' as a [[Lebesgue integral]]. However, when both <math>f</math> and <math>\hat{f}</math> are integrable, the inverse equality
<math display="block">f(x) = \int_{-\infty}^\infty \hat f(\xi) e^{i 2\pi x \xi} \, d\xi</math> holds for almost every {{mvar|x}}.  As a result, the Fourier transform is [[injective]] on {{math|[[Lp space|''L''<sup>1</sup>('''R''')]]}}.

=== Plancherel theorem and Parseval's theorem ===
{{main|Plancherel theorem|Parseval's theorem}}
Let {{math|''f''(''x'')}} and {{math|''g''(''x'')}} be integrable, and let {{math|''f̂''(''ξ'')}} and {{math|''ĝ''(''ξ'')}} be their Fourier transforms. If {{math|''f''(''x'')}} and {{math|''g''(''x'')}} are also [[square-integrable]], then the Parseval formula follows:<ref>{{harvnb|Rudin|1987|p=187}}</ref>
<math display="block">\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,</math>
where the bar denotes [[complex conjugation]].

The [[Plancherel theorem]], which follows from the above, states that<ref>{{harvnb|Rudin|1987|p=186}}</ref>
<math display="block">\|f\|^2_{L^{2}} = \int_{-\infty}^\infty \left| f(x) \right|^2\,dx = \int_{-\infty}^\infty \left| \hat{f}(\xi) \right|^2\,d\xi. </math>

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a [[unitary operator]] on {{math|''L''<sup>2</sup>('''R''')}}. On {{math|''L''<sup>1</sup>('''R''') ∩ ''L''<sup>2</sup>('''R''')}}, this extension agrees with original Fourier transform defined on {{math|''L''<sup>1</sup>('''R''')}}, thus enlarging the domain of the Fourier transform to {{math|''L''<sup>1</sup>('''R''') + ''L''<sup>2</sup>('''R''')}} (and consequently to {{math|{{math|''L''{{i sup|''p''}}}}('''R''')}} for {{math|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.

=== Convolution theorem ===
{{Main|Convolution theorem}}

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

This means that if:
<math display="block">h(x) = (f*g)(x) = \int_{-\infty}^\infty f(y)g(x - y)\,dy,</math>
where {{math|∗}} denotes the convolution operation, then:
<math display="block">\hat{h}(\xi) = \hat{f}(\xi)\, \hat{g}(\xi).</math>

In [[LTI system theory|linear time invariant (LTI) system theory]], it is common to interpret {{math|''g''(''x'')}} as the [[impulse response]] of an LTI system with input {{math|''f''(''x'')}} and output {{math|''h''(''x'')}}, since substituting the [[Dirac delta function|unit impulse]] for {{math|''f''(''x'')}} yields {{math|1=''h''(''x'') = ''g''(''x'')}}. In this case, {{math|''ĝ''(''ξ'')}} represents the [[frequency response]] of the system.

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

=== Cross-correlation theorem ===
{{Main|Cross-correlation|Wiener–Khinchin_theorem}}

In an analogous manner, it can be shown that if {{math|''h''(''x'')}} is the [[cross-correlation]] of {{math|''f''(''x'')}} and {{math|''g''(''x'')}}:
<math display="block">h(x) = (f \star g)(x) = \int_{-\infty}^\infty \overline{f(y)}g(x + y)\,dy</math>
then the Fourier transform of {{math|''h''(''x'')}} is:
<math display="block">\hat{h}(\xi) = \overline{\hat{f}(\xi)} \, \hat{g}(\xi).</math>

As a special case, the [[autocorrelation]] of function {{math|''f''(''x'')}} is:
<math display="block">h(x) = (f \star f)(x) = \int_{-\infty}^\infty \overline{f(y)}f(x + y)\,dy</math>
for which
<math display="block">\hat{h}(\xi) = \overline{\hat{f}(\xi)}\hat{f}(\xi) = \left|\hat{f}(\xi)\right|^2.</math>

=== Differentiation ===
Suppose {{math|''f''(''x'')}} is an absolutely continuous differentiable function, and both {{math|''f''}} and its derivative {{math|''f′''}} are integrable. Then the Fourier transform of the derivative is given by
<math display="block">\widehat{f'\,}(\xi) = \mathcal{F}\left\{ \frac{d}{dx} f(x)\right\} = i 2\pi \xi\hat{f}(\xi).</math>
More generally, the Fourier transformation of the {{mvar|n}}th derivative {{math|''f''{{isup|(''n'')}}}} is given by
<math display="block">\widehat{f^{(n)}}(\xi) = \mathcal{F}\left\{ \frac{d^n}{dx^n} f(x) \right\} = (i 2\pi \xi)^n\hat{f}(\xi).</math>

Analogously, <math>\mathcal{F}\left\{ \frac{d^n}{d\xi^n} \hat{f}(\xi)\right\} = (i 2\pi x)^n f(x)</math>, so <math>\mathcal{F}\left\{ x^n f(x)\right\} = \left(\frac{i}{2\pi}\right)^n \frac{d^n}{d\xi^n} \hat{f}(\xi).</math>

By applying the Fourier transform and using these formulas, some [[ordinary differential equation]]s can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "{{math|''f''(''x'')}} is smooth [[if and only if]] {{math|''f̂''(''ξ'')}} quickly falls to 0 for {{math|{{abs|''ξ''}} → ∞}}." By using the analogous rules for the inverse Fourier transform, one can also say "{{math|''f''(''x'')}} quickly falls to 0 for {{math|{{abs|''x''}} → ∞}} if and only if {{math|''f̂''(''ξ'')}} is smooth."

=== Eigenfunctions ===
{{see also|Mehler kernel|Hermite polynomials#Hermite functions as eigenfunctions of the Fourier transform}}
The Fourier transform is a linear transform which has eigenfunctions obeying <math>\mathcal{F}[\psi] = \lambda \psi,</math> with <math> \lambda \in \mathbb{C}.</math>

A set of eigenfunctions is found by noting that the homogeneous differential equation 
<math display="block">\left[ U\left( \frac{1}{2\pi}\frac{d}{dx} \right) + U( x ) \right] \psi(x) = 0</math> 
leads to eigenfunctions <math>\psi(x)</math> of the Fourier transform <math>\mathcal{F}</math> as long as the form of the equation remains invariant under Fourier transform.<ref group=note>The operator <math>U\left( \frac{1}{2\pi}\frac{d}{dx} \right)</math> is defined by replacing <math>x</math> by <math>\frac{1}{2\pi}\frac{d}{dx}</math> in the [[Taylor series|Taylor expansion]] of <math>U(x).</math></ref> In other words, every solution <math>\psi(x)</math> and its Fourier transform <math>\hat\psi(\xi)</math>  obey the same equation. Assuming [[Ordinary differential equation#Existence and uniqueness of solutions|uniqueness]] of the solutions, every solution <math>\psi(x)</math> must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if <math>U(x)</math> can be expanded in a power series in which for all terms the same factor of either one of <math>\pm 1, \pm i</math> arises from the factors <math>i^n</math> introduced by the [[#Differentiation|differentiation]] rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable <math>U(x)=x</math> leads to the [[Normal distribution#Fourier transform and characteristic function|standard normal distribution]].<ref>{{harvnb|Folland|1992|p=216}}</ref>

More generally, a set of eigenfunctions is also found by noting that the [[#Differentiation|differentiation]] rules imply that the [[ordinary differential equation]] 
<math display="block">\left[ W\left( \frac{i}{2\pi}\frac{d}{dx} \right) + W(x) \right] \psi(x) = C \psi(x)</math>
with <math>C</math> constant and <math>W(x)</math> being a non-constant even function remains invariant in form when applying the Fourier transform <math>\mathcal{F}</math> to both sides of the equation. The simplest example is provided by <math>W(x) = x^2</math> which is equivalent to considering the Schrödinger equation for the [[Quantum harmonic oscillator#Natural length and energy scales|quantum harmonic oscillator]].<ref>{{harvnb|Wolf|1979|p=307ff}}</ref> The corresponding solutions provide an important choice of an orthonormal basis for {{math|[[Square-integrable function|''L''<sup>2</sup>('''R''')]]}} and are given by the "physicist's" [[Hermite polynomials#Hermite functions as eigenfunctions of the Fourier transform|Hermite functions]]. Equivalently one may use
<math display="block">\psi_n(x) = \frac{\sqrt[4]{2}}{\sqrt{n!}} e^{-\pi x^2}\mathrm{He}_n\left(2x\sqrt{\pi}\right),</math>
where {{math|He<sub>''n''</sub>(''x'')}} are the "probabilist's" [[Hermite polynomial]]s, defined as
<math display="block">\mathrm{He}_n(x) = (-1)^n e^{\frac{1}{2}x^2}\left(\frac{d}{dx}\right)^n e^{-\frac{1}{2}x^2}.</math>

Under this convention for the Fourier transform, we have that
<math display="block">\hat\psi_n(\xi) = (-i)^n \psi_n(\xi).</math>

In other words, the Hermite functions form a complete [[orthonormal]] system of [[eigenfunctions]] for the Fourier transform on {{math|''L''<sup>2</sup>('''R''')}}.<ref name="Pinsky-2002" /><ref>{{harvnb|Folland|1989|p=53}}</ref> However, this choice of eigenfunctions is not unique. Because of <math>\mathcal{F}^4 = \mathrm{id}</math> there are only four different [[eigenvalue]]s of the Fourier transform (the fourth roots of unity ±1 and ±{{mvar|i}}) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.<ref>{{harvnb|Celeghini|Gadella|del Olmo|2021}}</ref> As a consequence of this, it is possible to decompose {{math|''L''<sup>2</sup>('''R''')}} as a direct sum of four spaces {{math|''H''<sub>0</sub>}}, {{math|''H''<sub>1</sub>}}, {{math|''H''<sub>2</sub>}}, and {{math|''H''<sub>3</sub>}} where the Fourier transform acts on {{math|He<sub>''k''</sub>}} simply by multiplication by {{math|''i''<sup>''k''</sup>}}.

Since the complete set of Hermite functions {{math|''ψ<sub>n</sub>''}} 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:
<math display="block">\mathcal{F}[f](\xi) = \int dx f(x) \sum_{n \geq 0} (-i)^n \psi_n(x) \psi_n(\xi) ~.</math>

This approach to define the Fourier transform was first proposed by [[Norbert Wiener]].<ref name="Duoandikoetxea-2001">{{harvnb|Duoandikoetxea|2001}}</ref> 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.<ref name="Boashash-2003">{{harvnb|Boashash|2003}}</ref> In [[physics]], this transform was introduced by [[Edward Condon]].<ref>{{harvnb|Condon|1937}}</ref> This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right [[#Other conventions|conventions]]. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator <math>N</math> via<ref>{{harvnb|Wolf|1979|p=320}}</ref>
<math display="block">\mathcal{F}[\psi] = e^{-i t N} \psi.</math>

The operator <math>N</math> is the [[Quantum harmonic oscillator#Ladder operator method|number operator]] of the quantum harmonic oscillator written as<ref name="auto">{{harvnb|Wolf|1979|p=312}}</ref><ref>{{harvnb|Folland|1989|p=52}}</ref>
<math display="block">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).</math>

It can be interpreted as the [[symmetry in quantum mechanics|generator]] of [[Mehler kernel#Fractional Fourier transform|fractional Fourier transforms]] for arbitrary values of {{mvar|t}}, and of the conventional continuous Fourier transform <math>\mathcal{F}</math> for the particular value <math>t = \pi/2,</math> with the [[Mehler kernel#Physics version|Mehler kernel]] implementing the corresponding [[active and passive transformation#In abstract vector spaces|active transform]]. The eigenfunctions of <math> N</math> are the [[Hermite polynomials#Hermite functions|Hermite functions]] <math>\psi_n(x)</math> which are therefore also eigenfunctions of <math>\mathcal{F}.</math>

Upon extending the Fourier transform to [[distribution (mathematics)|distributions]] the [[Dirac comb#Fourier transform|Dirac comb]] is also an eigenfunction of the Fourier transform.

=== Inversion and periodicity ===
{{Further|Fourier inversion theorem|Fractional Fourier transform}}

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

More precisely, defining the ''parity operator'' <math>\mathcal{P}</math> such that <math>(\mathcal{P} f)(x) = f(-x)</math>, we have:
<math display="block">\begin{align}
  \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{align}</math>
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 {{mvar|x}}-axis and frequency as the {{mvar|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 transformation]]s, which can be visualized as the action of the [[special linear group]] {{math|[[SL2(R)|SL<sub>2</sub>('''R''')]]}} on the time–frequency plane, with the preserved symplectic form corresponding to the [[#Uncertainty principle|uncertainty principle]], below. This approach is particularly studied in [[signal processing]], under [[time–frequency analysis]].

=== Connection with the Heisenberg group ===
The [[Heisenberg group]] is a certain [[group (mathematics)|group]] of [[unitary operator]]s on the [[Hilbert space]] {{math|''L''<sup>2</sup>('''R''')}} of square integrable complex valued functions {{mvar|f}} on the real line, generated by the translations {{math|1=(''T<sub>y</sub> f'')(''x'') = ''f'' (''x'' + ''y'')}} and multiplication by {{math|''e''<sup>''i''2π''ξx''</sup>}}, {{math|1=(''M<sub>ξ</sub> f'')(''x'') = ''e''<sup>''i''2π''ξx''</sup> ''f'' (''x'')}}. These operators do not commute, as their (group) commutator is
<math display="block">\left(M^{-1}_\xi T^{-1}_y M_\xi T_yf\right)(x) = e^{i 2\pi\xi y}f(x)</math>
which is multiplication by the constant (independent of {{mvar|x}}) {{math|''e''<sup>''i''2π''ξy''</sup> ∈ ''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 {{math|(''x'', ''ξ'', ''z'') ∈ '''R'''<sup>2</sup> × ''U''(1)}}, with the group law
<math display="block">\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^{i 2\pi \left(x_1 \xi_1 + x_2 \xi_2 + x_1 \xi_2\right)}\right).</math>

Denote the Heisenberg group by {{math|''H''<sub>1</sub>}}. The above procedure describes not only the group structure, but also a standard [[unitary representation]] of {{math|''H''<sub>1</sub>}} on a Hilbert space, which we denote by {{math|''ρ'' : ''H''<sub>1</sub> → ''B''(''L''<sup>2</sup>('''R'''))}}. Define the linear automorphism of {{math|'''R'''<sup>2</sup>}} by
<math display="block">J \begin{pmatrix}
   x \\
  \xi
\end{pmatrix} = \begin{pmatrix}
  -\xi \\
    x
\end{pmatrix}</math>
so that {{math|1=''J''{{isup|2}} = −''I''}}. This {{mvar|J}} can be extended to a unique automorphism of {{math|''H''<sub>1</sub>}}:
<math display="block">j\left(x, \xi, t\right) = \left(-\xi, x, te^{-i 2\pi\xi x}\right).</math>

According to the [[Stone–von Neumann theorem]], the unitary representations {{mvar|ρ}} and {{math|''ρ'' ∘ ''j''}} are unitarily equivalent, so there is a unique intertwiner {{math|''W'' ∈ ''U''(''L''<sup>2</sup>('''R'''))}} such that
<math display="block">\rho \circ j = W \rho W^*.</math>
This operator {{mvar|W}} is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.<ref>{{harvnb|Howe|1980}}</ref> For example, the square of the Fourier transform, {{math|''W''{{isup|2}}}}, is an intertwiner associated with {{math|1=''J''{{isup|2}} = −''I''}}, and so we have {{math|1=(''W''{{i sup|2}}''f'')(''x'') = ''f'' (−''x'')}} is the reflection of the original function {{mvar|f}}.