Engineering Analysis/Bessel Equation and Parseval Theorem

Bessel's equation relates the original function to the fourier coefficients a_n:

[Bessel's Equation]

$\sum _{n=1}^{\infty }a_{n}^{2}\leq \|f(x)\|^{2}$

If the basis set is infinitely orthogonal, and if an infinite sum of the basis functions perfectly reproduces the function f(x), then the above equation will be an equality, known as Parseval's Theorem:

[Parseval's Theorem]

$\sum _{n=1}^{\infty }a_{n}^{2}=\|f(x)\|^{2}$

Engineers may recognize this as a relationship between the energy of the signal, as represented in the time and frequency domains. However, parseval's rule applies not only to the classical Fourier series coefficients, but also to the generalized series coefficients as well.