Plancherel theorem: Difference between revisions

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{{Short description|Theorem in harmonic analysis}}

In [[mathematics]], the '''Plancherel theorem''' (sometimes called the [[Marc-Antoine Parseval|Parseval]]–Plancherel'''Parseval–Plancherel identity<ref>{{cite book |author1=Cohen-Tannoudji, Claude |author2=Dupont-Roc, Jacques |author3=Grynberg, Gilbert |title=Photons and Atoms : Introduction to Quantum Electrodynamics |year=1997 |url=https://archive.org/details/photonsatomsintr00cohe_398 |url-access=limited |publisher=Wiley |isbn=0-471-18433-0 |page=[https://archive.org/details/photonsatomsintr00cohe_398/page/n39 11]}}</ref>''') is a result in [[harmonic analysis]], proven by [[Michel Plancherel]] in 1910. It statesis thata the integralgeneralization of a function[[Parseval's [[squared modulustheorem]]; isoften equalused toin the integralfields of thescience squaredand modulusengineering, ofproving itsthe [[frequencyunitary spectrumtransformation|unitarity]]. That is, if <math>f(x) </math> is a function onof the real[[Fourier line, and <math>\widehat{f}(\xi)</math> is its frequency spectrum, thentransform]].

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 <math>f(x) </math> is a function on the real line, and <math>\widehat{f}(\xi)</math> is its frequency spectrum, then

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A more precise formulation is that if a function is in both [[Lp space|''L''^''p'' spaces]] <math>L^1(\mathbb{R})</math> and <math>L^2(\mathbb{R})</math>, then its [[Fourier transform]] is in <math>L^2(\mathbb{R})</math>, and the Fourier transform map is an [[isometry]] with respect to the ''L''² norm. This implies that the Fourier transform map restricted to <math>L^1(\mathbb{R}) \cap L^2(\mathbb{R})</math> has a unique extension to a [[Linear isometry|linear isometric map]] <math>L^2(\mathbb{R}) \mapsto L^2(\mathbb{R})</math>, sometimes called the Plancherel transform. This isometry is actually a [[unitary operator|unitary]] map. In effect, this makes it possible to speak of Fourier transforms of [[quadratically integrable function]]s.

Plancherel's theorem remains valid as stated on ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>. The theorem also holds more generally in [[locally compact abelian group]]s. 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 [[unitary transformation|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]].

Due to the [[polarization identity]], one can also apply Plancherel's theorem to the [[Lp space|<math>L^2(\mathbb{R})</math>]] [[inner product]] of two functions. That is, if <math>f(x)</math> and <math>g(x)</math> are two <math>L^2(\mathbb{R})</math> functions, and <math> \mathcal P</math> denotes the Plancherel transform, then