Plancherel theorem explained

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 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). 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.

. 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$$\backslash int\_^\backslash infty\; f(x)\backslash overline\; \backslash ,\; dx\; =\; \backslash int\_^\backslash infty\; (\backslash mathcal\; P\; f)(\backslash xi)\; \backslash overline\; \backslash ,\; d\backslash xi,$$and if and are furthermore functions, then $$(\backslash mathcal\; P\; f)(\backslash xi)\; =\; \backslash widehat(\backslash xi)\; =\; \backslash int\_^\backslash infty\; f(x)\; e^\; \backslash ,\; dx,$$and $$(\backslash mathcal\; P\; g)(\backslash xi)\; =\; \backslash widehat(\backslash xi)\; =\; \backslash int\_^\backslash infty\; g(x)\; e^\; \backslash ,\; dx,$$so

Locally compact groups

, there is a Pontrjagin dual group of characters on . Given a Haar measure on , the Fourier transform of a function in is $$\backslash hat\; f(\backslash chi)\; =\; \backslash int\_G\; \backslash overlinef(g)\backslash ,dg$$for a character on .

The Plancherel theorem states that there is a Haar measure on

, the dual measure such that$$\backslash |f\backslash |\_G^2\; =\; \backslash |\backslash hat\; f\backslash |\_^2$$for all (and the Fourier transform is also in ).

The theorem also holds in many non-abelian locally compact groups, except that the set of irreducible unitary representations

may not be a group. For example, when is a finite group, is the set of irreducible characters. From basic character theory, if is a class function, we have the Parseval formula$$\backslash |f\backslash |\_G^2\; =\; \backslash |\backslash hat\; f\backslash |\_^2$$$$\backslash |f\backslash |\_G^2\; =\; \backslash frac$$\sum_ |f(g)|^2, \quad \|\hat f\|_^2 = \sum_ (\dim\rho)^2|\hat f(\rho)|^2.More generally, when is not a class function, the norm is$$\backslash |\backslash hat\; f\backslash |\_^2\; =\; \backslash sum\_\; \backslash dim\backslash rho\backslash ,\backslash operatorname(\backslash hat\; f(\backslash rho)^*\backslash hat\; f(\backslash rho))$$so the Plancherel measure weights each representation by its dimension.

In full generality, a Plancherel theorem is$$\backslash |f\backslash |^2\_G\; =\; \backslash int\_\; \backslash |\backslash hat\; f(\backslash rho)\backslash |\_^2d\backslash mu(\backslash rho)$$where the norm is the Hilbert-Schmidt norm of the operator$$\backslash hat\; f(\backslash rho)\; =\; \backslash int\_G\; f(g)\backslash rho(g)^*\backslash ,dg$$and the measure

, if one exists, is called the Plancherel measure.

See also

• Plancherel theorem for spherical functions

References

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External links

• • Plancherel's Theorem on Mathworld