Parseval's theorem expresses the squared L2-norm of a function in terms of its Fourier
series coefficients. Let 
be a square integrable real function on ![[-pi,pi]](/images/equations/ParsevalsTheorem/Inline2.svg)
, with 
for 
and 
for 
. Define its finite partial
sum by
 |
(1)
|
Squaring this finite sum gives
![[S_N(x)]^2=1/4a_0^2+a_0sum_(n=1)^N[a_ncos(nx)+b_nsin(nx)]+sum_(n=1)^Nsum_(m=1)^N[a_na_mcos(nx)cos(mx)+a_nb_mcos(nx)sin(mx)+a_mb_nsin(nx)cos(mx)+b_nb_msin(nx)sin(mx)].](/images/equations/ParsevalsTheorem/NumberedEquation2.svg) |
(2)
|
Integrating and using that the trigonometric functions are orthogonal, with 
the Kronecker delta,
![int_(-pi)^pi[S_N(x)]^2dx=1/4a_0^2int_(-pi)^pidx+a_0int_(-pi)^pisum_(n=1)^N[a_ncos(nx)+b_nsin(nx)]dx+int_(-pi)^pisum_(n=1)^Nsum_(m=1)^N[a_na_mcos(nx)cos(mx)+a_nb_mcos(nx)sin(mx)+a_mb_nsin(nx)cos(mx)+b_nb_msin(nx)sin(mx)]dx
=1/4a_0^2(2pi)+0+sum_(n=1)^Nsum_(m=1)^N[a_na_mpidelta_(nm)+0+0+b_nb_mpidelta_(nm)],](/images/equations/ParsevalsTheorem/NumberedEquation3.svg) |
(3)
|
so
![1/piint_(-pi)^pi[S_N(x)]^2dx=1/2a_0^2+sum_(n=1)^N(a_n^2+b_n^2).](/images/equations/ParsevalsTheorem/NumberedEquation4.svg) |
(4)
|
The trigonometric complete orthogonal system gives 
in L2 (Axler 2026, p. 356). Hence
the norms converge, and taking 
yields
![1/piint_(-pi)^pi[f(x)]^2dx=1/2a_0^2+sum_(n=1)^infty(a_n^2+b_n^2).](/images/equations/ParsevalsTheorem/NumberedEquation5.svg) |
(5)
|
This establishes equality in Bessel's inequality without assuming pointwise convergence of the Fourier series or interchanging products of infinite series and integrals.
For a generalized Fourier series of a normalized complete orthogonal system,
the analogous identity is 
, using the inner
product linear in its first argument (Axler 2026, pp. 244-245).
For a square integrable complex function, use the complex exponential Fourier series coefficients
 |
(6)
|
The corresponding normalization gives
 |
(7)
|
Both normalizations agree with the conventions for Fourier series (NIST, §1.8).
