Parseval Equality for Functions and Fourier Series

The Parseval Equality for functions is used primarily in Fourier Series and Wavelets and derives its key concept from the Parseval Equality for Vectors.

If $f$ is an integrable function on $\text{[}0,2\pi ]$, then the norm of $f$ is:

$$||f|{|}^2=2\pi a\_0^2+\pi \sum \_{k=1}^{\infty }a\_k^2+\pi \sum \_{k=1}^{\infty }b\_k^2$$

Proof

The formula is derived from the idea that $||p\_n(x)||=⟨p\_n,p\_n⟩$ and that $||f(x)-p\_n(x)||\to 0$ as $n\to \infty$, such that one can state, $p\_n(x)\to f(x)$ as $n\to \infty$ and therefore, when summing over an interval of $n=\infty$, $||f|{|}^2=⟨f,f⟩$ will eventually equal $||f|{|}^2=⟨p\_n,p\_n⟩=||p\_n|{|}^2$.t