Orthogonal polynomials, Legendre
The orthogonality relation above is important in our discussion
on how to obtain the weights and mesh points. Suppose we have an arbitrary
polynomial \( Q_{N-1} \) of order \( N-1 \) and a Legendre polynomial \( L_N(x) \) of
order \( N \). We could represent \( Q_{N-1} \)
by the Legendre polynomials through
$$
\begin{equation}
Q_{N-1}(x)=\sum_{k=0}^{N-1}\alpha_kL_{k}(x),
\tag{13}
\end{equation}
$$
where \( \alpha_k \)'s are constants.
Using the orthogonality relation of Eq. (10) we see that
$$
\begin{equation}
\int_{-1}^1L_N(x)Q_{N-1}(x)dx=\sum_{k=0}^{N-1} \int_{-1}^1L_N(x) \alpha_kL_{k}(x)dx=0.
\tag{14}
\end{equation}
$$
We will use this result in our construction of mesh points and weights
in the next subsection.