Complete Orthonormal Set

Definition

Suppose that $s_N=\sum _{n=1}^N{c}_n^2{\varphi }_n(x)$ and $Y\subset C_p(a,b)$.
If $\forall f\in Y$, $\underset{N\to \infty }{\lim }||f-s_N||=0$ is satisfied, we say $\{{\varphi }_n|n\in N\}$ is complete

Facts

Parseval’s Equation

Definition

$||f|{|}^2=\underset{N\to \infty }{\lim }{\sum }_{n=1}^N{c}_n^2=\underset{N\to \infty }{\lim }||{\sum }_{n=1}^N{c}_n{\varphi }_n|{|}^2$

Facts

Parseval’s equation is a sufficient and necessary condition for orthonormal set to be complete

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