Plancherel theorem

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 ${\displaystyle f(x)}$ is a function on the real line, and ${\displaystyle {\widehat {f}}(\xi )}$ is its frequency spectrum, then

A more precise formulation is that if a function is in both L^p spaces ${\displaystyle L^{1}(\mathbb {R} )}$ and ${\displaystyle L^{2}(\mathbb {R} )}$, then its Fourier transform is in ${\displaystyle L^{2}(\mathbb {R} )}$ and the Fourier transform is an isometry with respect to the L² norm. This implies that the Fourier transform restricted to ${\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} )}$ has a unique extension to a linear isometric map ${\displaystyle L^{2}(\mathbb {R} )\mapsto L^{2}(\mathbb {R} )}$, 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). The basic idea is to prove it for Gaussian distributions, and then use density. But a standard Gaussian is transformed to itself under the Fourier transformation, and the theorem is trivial in that case. Finally, the standard transformation properties of the Fourier transform then imply Plancherel for all Gaussians.

Plancherel's theorem remains valid as stated on n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. 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 ${\displaystyle L^{2}(\mathbb {R} )}$ inner product of two functions. That is, if ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are two ${\displaystyle L^{2}(\mathbb {R} )}$ functions, and ${\displaystyle {\mathcal {P}}}$ denotes the Plancherel transform, then $${\displaystyle \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }({\mathcal {P}}f)(\xi ){\overline {({\mathcal {P}}g)(\xi )}}\,d\xi ,}$$ and if ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are furthermore ${\displaystyle L^{1}(\mathbb {R} )}$ functions, then $${\displaystyle ({\mathcal {P}}f)(\xi )={\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi i\xi x}\,dx,}$$ and $${\displaystyle ({\mathcal {P}}g)(\xi )={\widehat {g}}(\xi )=\int _{-\infty }^{\infty }g(x)e^{-2\pi i\xi x}\,dx,}$$ so