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Plancherel theorem

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In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science and engineering, proving the unitarity of the Fourier transform.

The theorem states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if is a function on the real line, and is its frequency spectrum, then

A more precise formulation is that if a function is in both Lp spaces and , then its Fourier transform is in and the Fourier transform is an isometry with respect to the L2 norm. This implies that the Fourier transform restricted to has a unique extension to a linear isometric map , sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.

A proof of the theorem is available from Rudin (1987, Chapter 9).

Plancherel's theorem remains valid as stated on n-dimensional Euclidean space . 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.

Due to the polarization identity, one can also apply Plancherel's theorem to the inner product of two functions. That is, if and are two functions, and denotes the Plancherel transform, then and if and are furthermore functions, then and so

See also

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References

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  • Plancherel, Michel (1910), "Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies", Rendiconti del Circolo Matematico di Palermo, 30 (1): 289–335, doi:10.1007/BF03014877, S2CID 122509369.
  • Dixmier, J. (1969), Les C*-algèbres et leurs Représentations, Gauthier Villars.
  • Yosida, K. (1968), Functional Analysis, Springer Verlag.
  • Rudin, Walter (1987), "9 Fourier Transforms", Real and Complex Analysis (3 ed.), McGraw-Hill Book Company.
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