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 (which can have an uncountable infinity of basis vectors).

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,

\Vert f\Vert _{L^{2}(-\pi ,\pi )}^{2}=\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=2\pi \sum _{n=-\infty }^{\infty }|c_{n}|^{2}

where the Fourier coefficients c_n of ƒ are given by

c_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)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 ƒ ∈ L²(R),

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

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 (e_n) be an orthonormal basis of H; i.e., the linear span of the e_n is dense in H, and the e_n 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 e_n = e^−inx for n ∈ Z.

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.

References

"Parseval equality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Johnson, Lee W.; Riess, R. Dean (1982), Numerical Analysis (2nd ed.), Reading, Mass.: Addison-Wesley, ISBN 0-201-10392-3.
Titchmarsh, E (1939), The Theory of Functions (2nd ed.), Oxford University Press.
Zygmund, Antoni (1968), Trigonometric Series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0-521-35885-9.
Siktar, Joshua (2019), Recasting the Proof of Parseval's Identity, Turkish Journal of Inequalities. [1]

Parseval's identity

Generalization of the Pythagorean theorem

See also

References