In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Plancherel theorem (sometimes called the
Parseval–Plancherel identity
) is a result in
harmonic analysis, proven by
Michel Plancherel in 1910. It states that the integral of a function's
squared modulus is equal to the integral of the squared modulus of its
frequency spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
. 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
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...
and
, then its
Fourier transform is in
, and the Fourier transform map is an isometry with respect to the ''L''
2 norm. This implies that the Fourier transform map restricted to
has a unique extension to a linear isometric map
, sometimes called the Plancherel transform. This isometry is actually a
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
map. In effect, this makes it possible to speak of Fourier transforms of
quadratically integrable functions.
Plancherel's theorem remains valid as stated on ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. The theorem also holds more generally in
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
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
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin dualit ...
.
The
unitarity
In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quant ...
of the
Fourier transform is often called
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
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
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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
*
Plancherel theorem for spherical functions In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analy ...
References
* .
* .
* .
External links
*
Plancherel's Theoremon Mathworld
Theorems in functional analysis
Theorems in harmonic analysis
Theorems in Fourier analysis
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