If is piecewise Continuous and has a general Fourier Series

(1) 
with Weighting Function , it must be true that

(2) 

(3) 
But the Coefficient of the generalized Fourier Series is given by

(4) 
so

(5) 

(6) 
Equation (6) is an inequality if the functions are not Complete. If they are
Complete, then the inequality (2) becomes an equality, so (6) becomes an
equality and is known as Parseval's Theorem. If has a simple Fourier Series expansion with
Coefficients , , , and , ..., , then

(7) 
The inequality can also be derived from Schwarz's Inequality

(8) 
by expanding in a superposition of Eigenfunctions of ,
. Then

(9) 

(10) 
If is normalized, then
and

(11) 
See also Schwarz's Inequality, Triangle Inequality
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 526527, 1985.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA:
Academic Press, p. 1102, 1980.
© 19969 Eric W. Weisstein
19990526