Parseval's identity
Encyclopedia
In mathematical analysis
, Parseval's identity is a fundamental result on the summability of the Fourier series
of a function. Geometrically, it is the
Pythagorean theorem
for inner-product spaces.
Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function,
where the Fourier coefficients cn of ƒ are given by
More formally, the result holds as stated provided ƒ is square-integrable
or, more generally, in L2[−π,π
]. A similar result is the Plancherel theorem
, which asserts that the integral of the square of the Fourier transform
of a function is equal to the integral of the square of the function itself. In one-dimension, for ,
in the more general setting of a separable Hilbert space
as follows. Suppose that H is a Hilbert space with inner product 〈•,•〉. Let (en) be an orthonormal basis
of H; i.e., the linear span
of the en is dense
in H, and the en are mutually orthonormal:
Then Parseval's identity asserts that for every x ∈ H,
This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting H be the Hilbert space L2[−π,π], and setting en = e−inx for
More generally, Parseval's identity holds in any inner-product space, not just separable Hilbert spaces. Thus suppose that H is an inner-product space. Let B be an orthonormal basis
of H; i.e., an orthonormal set which is total in the sense that the linear span of B is dense in H. Then
The assumption that B is total is necessary for the validity of the identity. If B is not total, then the equality in Parseval's identity must be replaced yielding Bessel's inequality
. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem
.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, Parseval's identity is a fundamental result on the summability 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...
of a function. Geometrically, it is the
Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
for inner-product spaces.
Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function,
where the Fourier coefficients cn of ƒ are given by
More formally, the result holds as stated provided ƒ is square-integrable
Square-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
or, more generally, in L2[−π,π
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
]. A similar result is the Plancherel theorem
Plancherel theorem
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....
, which asserts that the integral of the square of the 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...
of a function is equal to the integral of the square of the function itself. In one-dimension, for ,
Generalization of the Pythagorean theorem
The identity is related to the Pythagorean theoremPythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
in the more general setting of a separable Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
as follows. Suppose that H is a Hilbert space with inner product 〈•,•〉. Let (en) be an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of H; i.e., the linear span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
of the en is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in H, and the en are mutually orthonormal:
Then Parseval's identity asserts that for every x ∈ H,
This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting H be the Hilbert space L2[−π,π], and setting en = e−inx for
More generally, Parseval's identity holds in any inner-product space, not just separable Hilbert spaces. Thus suppose that H is an inner-product space. Let B be an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of H; i.e., an orthonormal set which is total in the sense that the linear span of B is dense in H. Then
The assumption that B is total is necessary for the validity of the identity. If B is not total, then the equality in Parseval's identity must be replaced yielding Bessel's inequality
Bessel's inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence....
. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem
Riesz–Fischer theorem
In mathematics, the Riesz–Fischer theorem in real analysis refers to a number of closely related results concerning the properties of the space L2 of square integrable functions...
.