Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 2I (Topics in Analysis)

Let ${\ell }_j$ be the Lagrange basis polynomials for the nodes. Necessity and existence give uniquely $A_j={\int }_{-1}^1{\ell }_j(t)dt$, because every polynomial of degree at most $n-1$ is ${\sum }_{j}p(t_j){\ell }_j(t)$.

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.

Apply exactness to ${\ell }_j^2$: $A_j={\sum }_k{A}_k{\ell }_j(t_k{)}^2=\int {\ell }_j^2>0$. Applying it to 1 gives $\sum A_j=2$. For continuous $f$ and any admissible $p$,