Application to the case \( N=2 \)
Obviously, there is no problem in changing the numbering of the matrix elements \( i,k=0,1,2,\dots,N-1 \) to
\( i,k=1,2,\dots,N \). We have chosen to start from zero, since we deal with polynomials of degree \( N-1 \).
Summarizing, for Legendre polynomials with \( N=2 \) we have
weights
$$
\begin{equation*}
\omega : \left\{1,1\right\},
\end{equation*}
$$
and mesh points
$$
\begin{equation*}
x : \left\{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right\}.
\end{equation*}
$$