Plancherel theorem 454315 194320115 2008-02-27T01:52:39Z Cwkmail 4154028 /* References */ Added citation of Plancherel article (1910) In [[mathematics]], the '''Plancherel theorem''' is a result in [[harmonic analysis]], first proved by [[Michel Plancherel]] [1]. In its simplest form it states that if a function ''f'' is in both [[Lp space|''L''<sup>1</sup>('''R''')]] and [[Lp space|''L''<sup>2</sup>('''R''')]], then its [[Fourier transform]] is in ''L''<sup>2</sup>('''R'''); moreover the Fourier transform map is isometric. This implies that the Fourier transform map restricted to ''L''<sup>1</sup>('''R''') &cap; ''L''<sup>2</sup>('''R''') has a unique extension to a linear isometric map ''L''<sup>2</sup>('''R''') &rarr;''L''<sup>2</sup>('''R'''). This isometry is actually a [[unitary operator|unitary]] map. Here Plancherel's version concerns spaces of functions on the [[real line]]. The theorem is valid in abstract versions, on [[locally compact abelian group]]s in general. Even more generally, there is 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]]. The unitarity of the Fourier transform is often called [[Parseval's theorem]] in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the [[Fourier series]]. == References == * J. Dixmier, ''Les C*-algèbres et leurs Représentations'', Gauthier Villars, 1969 * K. Yosida, ''Functional Analysis'', Springer Verlag, 1968 [1] Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," ''Rendiconti del Circolo Matematico di Palermo'', vol. 30, pages 298-335. [[Category:Functional analysis]] [[Category:Mathematical theorems]] [[Category:Fourier analysis]] [[fr:théorème de Plancherel]] [[it:Teorema di Plancherel]] [[pt:Teorema de Plancherel]]