## Parseval's Theorem

Let be a continuous function and and be Fourier Transform pairs so that

 (1) (2)

Then
 (3)
where is the Delta Function.

For finite Fourier Transform pairs and ,

 (4)

If a function has a Fourier Series given by

 (5)

then Bessel's Inequality becomes an equality known as Parseval's theorem. From (5),
 (6)
Integrating
 (7)
so
 (8)

For a generalized Fourier Series with a Complete Basis , an analogous relationship holds. For a Complex Fourier Series,
 (9)

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1101, 1979.