Plancherel theorem: Difference between revisions

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== Formal definition ==

A more precise formulation is that if a function is in both [[Lp space|''L''^''p'' spaces]] <math>L^1(\mathbb{R})</math> and <math>L^2(\mathbb{R})</math>, then its Fourier transform is in <math>L^2(\mathbb{R})</math> and the Fourier transform is an [[isometry]] with respect to the ''L''² norm. 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.

The [[Fourier transform]] of an [[Lp space|''L''^''1'']] 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''² 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>

A more precise formulation is that if a function is in both [[Lp space|''L''^''p'' spaces]] <math>L^1(\mathbb{R})</math> and <math>L^2(\mathbb{R})</math>, then its Fourier transform is in <math>L^2(\mathbb{R})</math> and the Fourier transform is an [[isometry]] with respect to the ''L''² norm. 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.

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==Locally compact groups==

There is also a Plancherel theorem for the Fourier transform in [[locally compact group]]s. In the case of an [[abelian group]] <math>G</math>, there is a [[PontrjaginPontryagin dual]] group <math>\widehat G</math> of characters on <math>G</math>. Given a [[Haar measure]] on <math>G</math>, the Fourier transform of a function in <math>L^1(G)</math> is

<math display="block">\hat f(\chi) = \int_G \overline{\chi(g)}f(g)\,dg</math>

for <math>\chi</math> a character on <math>G</math>.

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In full generality, a Plancherel theorem is

<math display="block">\|f\|^2_G = \int_{\hat G} \|\hat f(\rho)\|_{HS}^2d\mu(g\rho)</math>

where the norm is the [[Hilbert-Schmidt norm]] of the operator

<math display="block">\hat f(\rho) = \int_G f(g)\rho(g)^*\,dg</math>

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==See also==

* [[Carleson's theorem]]

*[[Plancherel theorem for spherical functions]]