Let us apply the above formal results to the case \( N=2 \). This means that we can approximate a function \( f(x) \) with a polynomial \( P_3(x) \) of order \( 2N-1=3 \).
The mesh points are the zeros of \( L_2(x)=1/2(3x^2-1) \). These points are \( x_0=-1/\sqrt{3} \) and \( x_1=1/\sqrt{3} \).
Specializing Eq. (16) $$ \begin{equation*} Q_{N-1}(x_k)=\sum_{i=0}^{N-1}\alpha_iL_i(x_k) \hspace{1cm} k=0,1,\dots, N-1. \end{equation*} $$ to \( N=2 \) yields $$ \begin{equation*} Q_1(x_0)=\alpha_0-\alpha_1\frac{1}{\sqrt{3}}, \end{equation*} $$ and $$ \begin{equation*} Q_1(x_1)=\alpha_0+\alpha_1\frac{1}{\sqrt{3}}, \end{equation*} $$ since \( L_0(x=\pm 1/\sqrt{3})=1 \) and \( L_1(x=\pm 1/\sqrt{3})=\pm 1/\sqrt{3} \).