Bessel’s Inequality and Parseval’s Theorem

Bessel's Inequality

Let f (x) be a piecewise continuous function defined on the interval [−π, π], so that its Fourier series is given by

Bessel's inequality states that

From here we can conclude that the series $$ is convergent.

Parseval's Theorem

If $$ is a square-integrable function on the interval $$ such that

then the Bessel's inequality becomes equality. In this case we have Parseval's formula:

Parseval's Formula in Complex Form

Let again $$ be a square-integrable function on the interval $$ and let $$ be complex coefficients such that

where

Then the Parseval's formula can be written in the form

Note that the energy of a $$-periodic wave $$ is

Solved Problems

Solution.

Fourier series expansion of the function $$ on the interval $$ is given by

(See Example $$ on the page Definition of Fourier Series and Typical Examples.)

Here the Fourier coefficients are $$ $$ (since the function $$ is odd) and $$

Using Parseval's formula, we have

Note that $$ is called Riemann zeta function $$ Thus, we have proved that

Solution.

We have found in Example $$ in the section Definition of Fourier Series and Typical Examples that the Fourier series of the function $$ on the interval $$ is given by

where

Applying Parseval's formula to the function, we obtain

The series $$ is known as Riemann zeta function $$ Consequently,