Integration points and weights with orthogonal polynomials
If we identify the weights with \( 2(L^{-1})_{0i} \), where the points \( x_i \) are
the zeros of \( L_N \), we have an integration formula of the type
$$
\begin{equation*}
\int_{-1}^1P_{2N-1}(x)dx=\sum_{i=0}^{N-1}\omega_iP_{2N-1}(x_i)
\end{equation*}
$$
and if our function \( f(x) \) can be approximated by a polynomial \( P \) of degree
\( 2N-1 \), we have finally that
$$
\begin{equation*}
\int_{-1}^1f(x)dx\approx \int_{-1}^1P_{2N-1}(x)dx=\sum_{i=0}^{N-1}\omega_iP_{2N-1}(x_i) .
\end{equation*}
$$
In summary, the mesh points \( x_i \) are defined by the zeros of an orthogonal polynomial of degree \( N \), that is
\( L_N \), while the weights are
given by \( 2(L^{-1})_{0i} \).