= Solution
For each node $x_i^{(n)}$, let
$$
\ell_i(x)=\prod_{j\ne i}
\frac{x-x_j^{(n)}}{x_i^{(n)}-x_j^{(n)}}
$$
be its <Lagrange interpolation polynomial>. It has degree $n$ and satisfies $\ell_i(x_j^{(n)})=\delta_{ij}$. Since $\ell_i^2\in\mathcal P_{2n}$, exactness gives
$$
a_i^{(n)}=I_n(\ell_i^2)=I(\ell_i^2)
=\int_a^b\ell_i(x)^2w(x)\,dx.
$$
The integrand is nonnegative and is positive except at finitely many points, so
$$
\boxed{a_i^{(n)}>0}.
$$
This proves the <positivity of quadrature weights from degree 2n exactness>.
Back to article page
