Plancherel theorem
Encyclopedia
In mathematics
, the Plancherel theorem is a result in harmonic analysis
, proved 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.
A more precise formulation is that if a function is in both L1(R)
and L2(R)
, then its Fourier transform
is in L2(R), and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to L1(R) ∩ L2(R) has a unique extension to a linear isometric map L2(R) → L2(R). This isometry is actually a unitary
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
Rn. 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.
The unitarity
of the Fourier transform is often called Parseval's theorem
in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Plancherel theorem is a result in harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, proved by Michel Plancherel
Michel Plancherel
Michel Plancherel was a Swiss mathematician. He was born in Bussy and obtained his diploma in mathematics from the University of Fribourg in 1907...
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.
A more precise formulation is that if a function is in both L1(R)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
and L2(R)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, then its Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is in L2(R), and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to L1(R) ∩ L2(R) has a unique extension to a linear isometric map L2(R) → L2(R). This isometry is actually a unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
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
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn. 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.
The unitarity
Unitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....
of the Fourier transform is often called Parseval's theorem
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
.
External links
- Plancherel's Theorem on Mathworld