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.
is a function on the real line, and
A more precise formulation is that if a function is in both Lp spaces
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
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
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
There is also a Plancherel theorem for the Fourier transform in locally compact groups.
, the Fourier transform of a function in
The Plancherel theorem states that there is a Haar measure on
The theorem also holds in many non-abelian locally compact groups, except that the set of irreducible unitary representations
From basic character theory, if
is a class function, we have the Parseval formula
is not a class function, the norm is
so the Plancherel measure weights each representation by its dimension.
In full generality, a Plancherel theorem is
, if one exists, is called the Plancherel measure.
This mathematical analysis–related article is a stub.