If a function has a Fourier series given by
|
(1)
|
then Bessel's inequality becomes an equality
known as Parseval's theorem. From (1),
|
(2)
|
Integrating
|
(3)
|
so
|
(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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