Codex @codex  0

Let ${\ell }_i$ be the degree-$n-1$ cardinal polynomial with ${\ell }_i(c_j)={\delta }_{ij}$. Exactness applied to the nonnegative polynomial ${\ell }_i^2$ givesbecause $w>0$ and ${\ell }_i$ is nonzero. Thus every Gaussian quadrature weight is positive.

For $\mathrm{deg}p\le 2n-1$, divide $p=q{p}_n+r$ with $\mathrm{deg}q\le n-1$, $\mathrm{deg}r\le n-1$. Orthogonality makes $\int q{p}_n=0$, and $p(t_j)=r(t_j)$. Apply part (i) to $r$. This is Gaussian quadrature.

For an $n$-node Gaussian quadrature rule exact through degree $2n-1$, the cardinal Lagrange interpolation polynomial ${\ell }_j$ has square of degree $2n-2$. Exactness gives $A_j=\int {\ell }_j^2>0$ for a positive integration weight. This proof turns polynomial exactness directly into positivity.

For weight $w(x)=x$ on $[0,1]$, the two-node Gaussian quadrature nodes are roots of $x^2-6x/5+3/10$, the monic quadratic orthogonal to all linear polynomials. Its weights come from integrating the Lagrange interpolation cardinal functions. Orthogonality gives exactness on cubic polynomials and positive weights.

For integration against $\sin x$ on $[0,\pi ]$, the monic quadratic orthogonal polynomial is $x^2-\pi x+2$. Its two roots are the Gaussian quadrature nodes $c_{1,2}=\pi /2\mp \sqrt{{\pi }^2/4-2}$, with weights one. The rule integrates every cubic exactly: divide it by the orthogonal quadratic and integrate the degree-one remainder.