Editing Fourier transform (section)

=== 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}}.