Given a periodic signal $x$ in CT, we can estimate the signal using the formula: 

$$ \hat{x}(t) = \sum \limits_{k=-N}^{N}\alpha_k e^{ik\omega_0 t}   $$

How can we determine the coefficients that make the estimates closest in terms of error energy? Let $W\in\R^{2N+1}$ be a subspace spanned by the set of orthogonal basis vectors $\Psi_0, \Psi_1, \Psi_{-1} \dots, \Psi_{N}, \Psi_{-N}$. If $x$ is a vector that is not in the column space of $W$, then we can project $x$ onto the column space of $W$, producing an approximation vector $\hat{x}$, which has a distance of $\abs{\left| \mathcal{E}_N\right|}$ from the vector $x$. The vectors $\mathcal{E}_N$ and $\hat{x}$ are orthogonal.

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We want to minimize $\abs{\left| \mathcal{E}_N\right|}$ to make the best approximation. $\mathcal{E}_N = x - \hat{x}_N$, hence $\abs{\left| \mathcal{E}_N\right|} = \abs{\left| x-\hat{x}_N\right|}$. Since $\mathcal{E} \perp W$, then we can use the inner product and the properties of orthogonality to solve for the coefficients. Since $W \in \R^{2N+1}$, we have $2N+1$ equations and $2N+1$ coefficients.

\begin{align*}
\langle \hat{x}_N, \Psi_\ell\rangle  &= \langle \sum \limits_{k=-N}^{N}\alpha_k e^{ik\omega_0 t} , \Psi_\ell\rangle  \\
\langle \hat{x}_N, \Psi_\ell\rangle  &= \sum \limits_{k=-N}^{N}\alpha_k \langle e^{ik\omega_0 t} , \Psi_\ell\rangle  \\
\langle \hat{x}_N, \Psi_\ell\rangle  &= \alpha_\ell \langle e^{i\ell\omega_0 t} , \Psi_\ell\rangle  \\
\langle \hat{x}_N, \Psi_\ell\rangle  &= \alpha_\ell \langle \Psi_\ell , \Psi_\ell\rangle  \\
\alpha_\ell &= \frac{\langle \hat{x}_N, \Psi_\ell\rangle }{\langle \Psi_\ell , \Psi_\ell\rangle } \\
\end{align*}

if $\hat{x}(t) = \sum \limits_{k=0}^{M-1}\alpha_k \Psi_k$, where $x$ is a $p$-periodic signal, which $\Psi_k$'s would you choose? Since we are taking a subset, the larger exponentials are better since they dominate. Otherwise, using $\{0,1,\dots,M\}\in\Z$ is not using very much. Pick the largest in magnitude FS coefficient, and then go down from there, but it doesn't have to be continguous. 


Approximations