Editing Plancherel theorem (section)

== Formal definition ==
The [[Fourier transform]] of an [[Lp space|''L''<sup>''1''</sup>]] function <math>f</math> on the [[real line]] <math>\mathbb R</math> is defined as the [[Lebesgue integral]]
<math display="block">\hat f(\xi) = \int_{\mathbb R} f(x)e^{-2\pi i x\xi}dx.</math>
If <math>f</math> belongs to both <math>L^1</math> and <math>L^2</math>, then the Plancherel theorem states that <math>\hat f</math> also belongs to <math>L^2</math>, and the Fourier transform is an [[isometry]] with respect to the ''L''<sup>2</sup> norm, which is to say that
<math display="block">\int_{-\infty}^\infty |f(x)|^2 \, dx = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2  \, d\xi</math>

This implies that the Fourier transform restricted to <math>L^1(\mathbb{R}) \cap L^2(\mathbb{R})</math>  has a unique extension to  a [[Linear isometry|linear isometric map]] <math>L^2(\mathbb{R}) \mapsto L^2(\mathbb{R})</math>, sometimes called the Plancherel transform.  This isometry is actually a [[unitary operator|unitary]] map.  In effect, this makes it possible to speak of Fourier transforms of [[quadratically integrable function]]s.

A proof of the theorem is available from ''Rudin (1987, Chapter 9)''.  The basic idea is to prove it for [[Gaussian distribution]]s, and then use density. But a standard Gaussian is transformed to itself under the Fourier transformation, and the theorem is trivial in that case.  Finally, the standard transformation properties of the Fourier transform then imply Plancherel for all Gaussians.

Plancherel's theorem remains valid as stated on ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>. The theorem also holds more generally in [[locally compact abelian group]]s. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions.  This is the subject of [[non-commutative harmonic analysis]].

Due to the [[polarization identity]], one can also apply Plancherel's theorem to the [[Lp space|<math>L^2(\mathbb{R})</math>]] [[inner product]] of two functions. That is, if <math>f(x)</math> and <math>g(x)</math> are two <math>L^2(\mathbb{R})</math> functions, and <math> \mathcal P</math> denotes the Plancherel transform, then
<math display="block">\int_{-\infty}^\infty f(x)\overline{g(x)} \, dx = \int_{-\infty}^\infty (\mathcal P f)(\xi) \overline{(\mathcal P g)(\xi)} \, d\xi,</math>
and if <math>f(x)</math> and <math>g(x)</math> are furthermore <math>L^1(\mathbb{R})</math> functions, then 
<math display="block"> (\mathcal P f)(\xi) = \widehat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i \xi x} \, dx ,</math>
and 
<math display="block"> (\mathcal P g)(\xi) = \widehat{g}(\xi) = \int_{-\infty}^\infty g(x) e^{-2\pi i \xi x} \, dx ,</math>
so
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