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:

$${\parallel x\parallel }^2=\sum _{j\in J}{|⟨x,e_j⟩|}^2.$$

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 $ℝ$.  The following equality holds

$$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{f}^2(x)𝑑x=\frac{{(a_0^f)}^2}{2}+\sum _{k=1}^{\mathrm{\infty }}[{(a_k^f)}^2+{(b_k^f)}^2],$$

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