In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, proven by
Michel Plancherel
Michel Plancherel (; 16 January 1885 – 4 March 1967) was a Swiss people, Swiss mathematician.
Biography
He was born in Bussy, Fribourg, Bussy (Canton of Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribou ...
in 1910. It is a generalization of
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 ...
; often used in the fields of science and engineering, proving the
unitarity of the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
.
The theorem states that the integral of a function's
squared modulus
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square o ...
is equal to the integral of the squared modulus of its
frequency spectrum
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed int ...
. That is, if
is a function on the real line, and
is its frequency spectrum, then
Formal definition
The
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of an
''L''''1'' function
on the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
is defined as the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
If
belongs to both
and
, then the Plancherel theorem states that
also belongs to
, and the Fourier transform is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
with respect to the ''L''
2 norm, which is to say that
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
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 semigr ...
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 distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
f(x ...
s, 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. 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
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 ...
.
Due to the
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
, 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, ofte ...
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
Locally compact groups
There is also a Plancherel theorem for the Fourier transform in
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
s. In the case of an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
, there is a
Pontryagin dual
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
group
of characters on
. Given a
Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfr� ...
on
, the Fourier transform of a function in
is
for
a character on
.
The Plancherel theorem states that there is a Haar measure on
, the ''dual measure'' such that
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
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...
, if
is a
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugati ...
, we have the Parseval formula
More generally, when
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
where the norm is the
Hilbert-Schmidt norm of the operator
and the measure
, if one exists, is called the Plancherel measure.
See also
*
Carleson's theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the ( Lebesgue) pointwise almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the ...
*
Plancherel theorem for spherical functions
References
* .
* .
* .
* .
External links
*
Plancherel's Theoremon Mathworld
Theorems in functional analysis
Theorems in harmonic analysis
Theorems in Fourier analysis
Lp spaces
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