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} .