View source for Parseval's identity

{{Short description|The energy of a periodic function is the same in the time and frequency domain.}}
In [[Mathematical analysis|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|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)}= \int_{-\pi}^\pi |f(x)|^2 \, dx=2\pi\sum_{n=-\infty}^\infty |c_n|^2</math>
where the Fourier coefficients <math>c_n</math> of <math>f</math> are given by
<math display="block">c_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|square-integrable function]] or, more generally, in [[Lp 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>

== Generalization of the Pythagorean theorem ==

The [[Identity (mathematics)|identity]] is related to the [[Pythagorean theorem]] in the more general setting of a separable [[Hilbert space]] as follows.  Suppose that <math>H</math> is a Hilbert space with inner product <math>\langle \,\cdot\,, \,\cdot\, \rangle.</math>  Let <math>\left(e_n\right)</math> be an [[Orthonormal basis|orthonormal basis]] of <math>H</math>; i.e., the [[Linear span|linear span]] of the <math>e_n</math> is [[Dense set|dense]] in <math>H,</math> and the <math>e_n</math> are mutually orthonormal:
:<math>\langle e_m, e_n\rangle = \begin{cases}
1 & \mbox{if}~ m = n \\
0 & \mbox{if}~ m \neq n.
\end{cases}</math>

Then Parseval's identity asserts that for every <math>x \in H,</math>
<math display="block">\sum_n \left|\left\langle x, e_n \right\rangle\right|^2 = \|x\|^2.</math>

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 <math>H</math> be the Hilbert space <math>L^2[-\pi, \pi],</math> and setting <math>e_n = \frac{e^{-i n x}}{\sqrt{2 \pi}}</math> for <math>n \in \Z.</math>

More generally, Parseval's identity holds in any [[Inner product space|inner product space]], not just separable Hilbert spaces.  Thus suppose that <math>H</math> is an inner-product space. Let <math>B</math> be an [[Orthonormal basis|orthonormal basis]] of <math>H</math>; that is, an orthonormal set which is {{em|total}} in the sense that the linear span of <math>B</math> is dense in <math>H.</math>  Then
<math display="block">\|x\|^2 = \langle x,x\rangle = \sum_{v\in B}\left|\langle x, v\rangle\right|^2.</math>

The assumption that <math>B</math> is total is necessary for the validity of the identity. If <math>B</math> is not total, then the equality in Parseval's identity must be replaced by <math>\, \geq,</math> yielding [[Bessel's inequality]].  This general form of Parseval's identity can be proved using the [[Riesz–Fischer theorem]].

== See also ==

* {{annotated link|Parseval's theorem}}

==References==
{{reflist}}

* {{springer|title=Parseval equality|id=p/p071590}}
* {{citation|last1=Johnson|first1=Lee W.|first2=R. Dean|last2=Riess|title=Numerical Analysis|year=1982|edition=2nd|publisher=Addison-Wesley|location=Reading, Mass.|isbn=0-201-10392-3}}.
* {{citation|last=Titchmarsh|first=E|title=The Theory of Functions|year=1939|edition=2nd|publisher=Oxford University Press}}.
* {{citation|title=Trigonometric Series|title-link=Trigonometric Series|first=Antoni|last=Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.

{{Functional analysis}}

[[Category:Fourier series]]
[[Category:Theorems in functional analysis]]

{{Sourceattribution|Parseval's identity}}