If a function has a Fourier series given by
 |
(1)
|
then Bessel's inequality becomes an equality
known as Parseval's theorem. From (1),
![[f(x)]^2=1/4a_0^2+a_0sum_(n=1)^infty[a_ncos(nx)+b_nsin(nx)]+sum_(n=1)^inftysum_(m=1)^infty[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
![int_(-pi)^pi[f(x)]^2dx=1/4a_0^2int_(-pi)^pidx+a_0int_(-pi)^pisum_(n=1)^infty[a_ncos(nx)+b_nsin(nx)]dx+int_(-pi)^pisum_(n=1)^inftysum_(m=1)^infty[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)^inftysum_(m=1)^infty[a_na_mpidelta_(nm)+0+0+b_nb_mpidelta_(nm)],](/images/equations/ParsevalsTheorem/NumberedEquation3.svg) |
(3)
|
so
![1/piint_(-pi)^pi[f(x)]^2dx=1/2a_0^2+sum_(n=1)^infty(a_n^2+b_n^2).](/images/equations/ParsevalsTheorem/NumberedEquation4.svg) |
(4)
|
For a generalized Fourier series of a complete orthogonal system
, an analogous relationship holds.
For a complex Fourier
series,
 |
(5)
|
See also
Bessel's Inequality,
Complete Orthogonal System,
Fourier Series,
Generalized
Fourier Series,
Plancherel's Theorem,
Power Spectrum
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
p. 1101, 2000.Kaplan, W. Advanced
Calculus, 4th ed. Reading, MA: Addison-Wesley, p. 501, 1992.Referenced
on Wolfram|Alpha
Parseval's Theorem
Cite this as:
Weisstein, Eric W. "Parseval's Theorem."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParsevalsTheorem.html
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