Plancherel theorem

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In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel in 1910. In its simplest form it states that if a function is in both L¹(R) and L²(R), then its Fourier transform is in L²(R), and the Fourier transform map is an isometry with respect to the L² norm. This implies that the Fourier transform map restricted to L¹(R) ∩ L²(R) has a unique extension to a linear isometric map L²(R) →L²(R). This isometry is actually a unitary map.

Plancherel's theorem remains valid as stated on n-dimensional Euclidean space Rⁿ. The theorem also holds more generally in locally compact abelian groups. 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.

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.

[edit] See also

• Plancherel theorem for spherical functions

[edit] References

• Plancherel, Michel (1910), "Contribution à l'étude de la représentation d'une fonction arbitraire par les intégrales définies", Rendiconti del Circolo Matematico di Palermo 30: 298–335 .
• Dixmier, J. (1969), Les C*-algèbres et leurs Représentations, Gauthier Villars .
• Yosida, K. (1968), Functional Analysis, Springer Verlag .

[edit] External links

• Plancherel's Theorem on Mathworld

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