Parseval's identity

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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 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,

\sum_{n=-\infty}^\infty |c_n|^2 = \frac{1}{2\pi}\int_{-\pi}^\pi |f(x)|^2 \, dx,

where the Fourier coefficients cn of ƒ are given by

c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) \mathrm{e}^{-inx} \, dx.

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 ƒL2(R),

\int_{-\infty}^\infty |\hat{f}(\xi)|^2\,d\xi = \int_{-\infty}^\infty |f(x)|^2\, dx.

Generalization of the Pythagorean theorem[edit]

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 (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:

\langle e_m, e_n\rangle = \begin{cases}1&\mbox{if}\ m=n\\
0&\mbox{if}\ m \not= n.\end{cases}

Then Parseval's identity asserts that for every x ∈ H,

\sum_n |\langle x, e_n\rangle|^2 = \|x\|^2.

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 nZ.

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

\|x\|^2=\langle x,x\rangle=\sum_{v\in B}\left|\langle x,v\rangle\right|^2.

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 by ≥, yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem.

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