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.