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 $$\begin{equation*} x : \left\{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right\}. \end{equation*}$$

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