Plancherel theorem: Difference between revisions

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Revision as of 20:10, 2 September 2023

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A more precise formulation is that if a function is in both [[Lp space|Lp''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 map is an isometry with respect to the ''L''² norm. This implies that the Fourier transform map restricted to <math>L^1(\mathbb{R}) \cap L^2(\mathbb{R})</math> has a unique extension to a 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.

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]].