Bessel's Inequality

If is piecewise continuous and has a generalized 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 do not form a complete orthogonal system. If they are a complete orthogonal system, 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)

and

			(10)
			(11)

where is the complex conjugate. If is normalized, then and

(12)