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Tito Omburo (talk | contribs) →Precise formulation: abelian groups Tag: Reverted |
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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.
==Abelian topological groups
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]].
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