Generalized Fourier series

Definition

Assume that $\{{\varphi }_n|n\in N\}\subset C_p(a,b)$ is an Orthonormal Set A function $f\in C_p(a,b)$ can be represented by a linear combination of $\{{\varphi }_n|n\in N\}$
$f(x)=\sum _{n=1}^{\infty }{c}_n{\varphi }_n(x)$, where $c_n={\int }_a^{b}f(x){\varphi }_n(x)dx$: generalized Fourier series

Facts

Best approximation in the mean

Let $f\in C_p(a,b)$ and $\{{\varphi }_n|n\in N\}$ an orthonormal set and ${\Phi }_N(x)=\sum _{n=1}^N{\gamma }_n{\varphi }_n(x)$
$E=||f-{\Phi }_N|{|}^2=||f|{|}^2+\sum _{n=1}^N({\gamma }_n-c_n{)}^2-\sum _{n=1}^N{c}_n^2$ where $c_n={\int }_a^{b}f(x){\varphi }_n(x)dx$
Therefore, the best approximation in the mean to $f(x)$ is $\sum _{n=1}^N{c}_n{\varphi }_n(x)$