# 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 generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors).

Informally, the identity asserts that the sum of 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 $f$ 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 $f$ is a square-integrable function or, more generally, in Lp space $L^{2}[-\pi ,\pi ].$ 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 $f\in L^{2}(\mathbb {R} ),$ $\int _{-\infty }^{\infty }|{\hat {f}}(\xi )|^{2}\,d\xi =\int _{-\infty }^{\infty }|f(x)|^{2}\,dx.$ ## 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 $\langle \,\cdot \,,\,\cdot \,\rangle .$ Let $\left(e_{n}\right)$ 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:

$\langle e_{m},e_{n}\rangle ={\begin{cases}1&{\mbox{if}}~m=n\\0&{\mbox{if}}~m\neq n.\end{cases}}$ Then Parseval's identity asserts that for every $x\in H,$ $\sum _{n}\left|\left\langle x,e_{n}\right\rangle \right|^{2}=\|x\|^{2}.$ 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 $L^{2}[-\pi ,\pi ],$ and setting $e_{n}=e^{-inx}$ for $n\in \mathbb {Z} .$ 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 $H$ ; that is, 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 $\,\geq ,$ yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem.