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Tito Omburo (talk | contribs) gave a sketch of the proof |
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A more precise formulation is that if a function is in both [[Lp space|''L''<sup>''p''</sup> 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''<sup>2</sup> 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
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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