Editing Fourier transform (section)

=== 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.