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. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes). Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors).

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, $${\displaystyle \Vert f\Vert _{L^{2}(-\pi ,\pi )}^{2}=\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=2\pi \sum _{n=-\infty }^{\infty }|c_{n}|^{2}}$$ where the Fourier coefficients ${\displaystyle c_{n}}$ of ${\displaystyle f}$ are given by $${\displaystyle c_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.}$$

The result holds as stated provided ${\displaystyle f}$ is a square-integrable function or, more generally, in Lp space ${\displaystyle L^{2}[-\pi ,\pi ].}$ 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 ${\displaystyle f\in L^{2}(\mathbb {R} ),}$ $${\displaystyle \int _{-\infty }^{\infty }|{\hat {f}}(\xi )|^{2}\,d\xi =\int _{-\infty }^{\infty }|f(x)|^{2}\,dx.}$$