Editing Parseval's identity (section)

{{Short description|The energy of a periodic function is the same in the time and frequency domain.}}
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 space]]s (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,
<math display="block">
 \Vert f \Vert^2_{L^2(-\pi,\pi)} = \frac1{2\pi}\int_{-\pi}^\pi |f(x)|^2 \, dx = \sum_{n=-\infty}^\infty |\hat f(n)|^2,
</math>
where the Fourier coefficients <math>\hat f(n)</math> of <math>f</math> are given by
<math display="block">
 \hat f(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx.
</math>

The result holds as stated, provided <math>f</math> is a [[square-integrable function]] or, more generally, in [[Lp space|''L''<sup>''p''</sup> space]] <math>L^2[-\pi, \pi].</math> 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 <math>f \in L^2(\R),</math>
<math display="block">
 \int_{-\infty}^\infty |\hat{f}(\xi)|^2 \,d\xi = \int_{-\infty}^\infty |f(x)|^2 \,dx.
</math>