Integration points and weights with orthogonal polynomials
Instead of an integration problem we need now to define the coefficient \( \alpha_0 \).
Since we know the values of \( Q_{N-1} \) at the zeros of \( L_N \), we may rewrite
Eq.
(15) as
$$
\begin{equation}
Q_{N-1}(x_k)=\sum_{i=0}^{N-1}\alpha_iL_i(x_k)=\sum_{i=0}^{N-1}\alpha_iL_{ik} \hspace{1cm} k=0,1,\dots, N-1.
\tag{16}
\end{equation}
$$
Since the Legendre polynomials are linearly independent of each other, none
of the columns in the matrix \( L_{ik} \) are linear combinations of the others.