Parseval's identity


In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalised
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 Lp space|L2. 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,

Generalization of the Pythagorean theorem

The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. Suppose that H is a Hilbert space with inner product 〈•,•〉. Let 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, 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.