# Parseval equality

## 0.1 Parseval’s Equality

Theorem. – If $\{e_{j}\!:\;j\in J\}$ is an orthonormal basis of an Hilbert space $H$, then for every $x\in H$ the following equality holds:

 $\|x\|^{2}=\sum_{j\in J}|\langle x,e_{j}\rangle|^{2}.$

The above theorem is a more sophisticated form of Bessel’s inequality (where the inequality is in fact an equality). The difference is that for Bessel’s inequality it is only required that the set $\{e_{j}:j\in J\}$ is an orthonormal set, not necessarily an orthonormal basis.

## 0.2 Parseval’s Theorem

Applying Parseval’s equality on the Hilbert space $L^{2}([-\pi,\pi])$ (http://planetmath.org/LpSpace), of square integrable functions on the interval $[-\pi,\pi]$, with the orthonormal basis consisting of trigonometric functions, we obtain

Theorem (Parseval’s theorem). – Let $f$ be a Riemann square integrable function from $[-\pi,\pi]$ to $\mathbb{R}$.  The following equality holds

 $\frac{1}{\pi}\int_{-\pi}^{\pi}f^{2}(x)dx=\frac{(a_{0}^{f})^{2}}{2}+\sum_{k=1}^% {\infty}[(a_{k}^{f})^{2}+(b_{k}^{f})^{2}],$

where $a_{0}^{f}$, $a_{k}^{f}$, $b_{k}^{f}$ are the Fourier coefficients of the function $f$.

The function $f$ can be a Lebesgue-integrable function, if we use the Lebesgue integral in place of the Riemann integral.

 Title Parseval equality Canonical name ParsevalEquality Date of creation 2013-03-22 13:57:10 Last modified on 2013-03-22 13:57:10 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 11 Author asteroid (17536) Entry type Theorem Classification msc 42B05 Synonym Parseval equation Synonym Parseval identity Synonym Lyapunov equation Related topic BesselInequality Related topic ValueOfTheRiemannZetaFunctionAtS2 Defines Parseval theorem