- Plancherel theorem
In
mathematics , the Plancherel theorem is a result inharmonic analysis , first proved byMichel Plancherel [1] . In its simplest form it states that if a function "f" is in both "L"1(R) and "L"2(R), then itsFourier transform is in "L"2(R); moreover the Fourier transform map is isometric. This implies that the Fourier transform map restricted to "L"1(R) ∩ "L"2(R) has a unique extension to a linear isometric map "L"2(R) →"L"2(R). This isometry is actually a unitary map.Here Plancherel's version concerns spaces of functions on the
real line . The theorem is valid in abstract versions, onlocally 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 ofnon-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 theFourier 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.
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