Editing Parseval's identity

{{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>

== Generalization of the Pythagorean theorem ==

The [[Identity (mathematics)|identity]] is related to the [[Pythagorean theorem]] in the more general setting of a [[Separable (topology)|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]] of <math>H</math>; i.e., the [[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 = e^{i n x}</math> for <math>n \in \Z.</math>

More generally, Parseval's identity holds for arbitrary [[Hilbert space|Hilbert spaces]], not necessarily separable. When the Hilbert space is not separable any orthonormal basis is uncountable and we need to generalize the concept of a series to an unconditional sum as follows: let <math>\{e_r\}_{r\in \Gamma}</math> an orthonormal basis of a Hilbert space (where <math>\Gamma</math> have arbitrary cardinality), then we say that <math display="inline">\sum_{r\in \Gamma} a_r e_r</math> converges unconditionally if for every <math>\epsilon>0</math> there exists a finite subset <math>A\subset \Gamma</math> such that
<math display="block">
\left\| \sum_{r\in B}a_re_r-\sum_{r\in C}a_r e_r\right\|<\epsilon
</math>
for any pair of finite subsets <math>B,C\subset\Gamma</math> that contains <math>A</math> (that is, such that <math>A\subset B\cap C</math>). Note that in this case we are using a [[Net (mathematics)|net]] to define the unconditional sum.

== 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|authorlink=Edward Charles Titchmarsh|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|authorlink=Antoni Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.

{{Lp spaces}}
{{Functional analysis}}
{{Hilbert space}}

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