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''') ∩ ''L''<sup>2</sup>('''R''') has a unique extension to a linear isometric map ''L''<sup>2</sup>('''R''') →''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]]