Past exam of the numerical analysis course of the University of Oxford / 2015 / 2 / b / Solution

Assume $f\in C^{n+1}[a,b]$ and set $\omega (t)={\prod }_{i=0}^n(t-x_i)$, a monic polynomial of polynomial degree $n+1$. Fix $x\in [a,b]$ distinct from all the interpolation nodes, and chooseThe auxiliary functionvanishes at the $n+2$ distinct points $x_0,\dots ,x_n,x$. Order those points and repeatedly apply Rolle's theorem. After $n+1$ applications, there is some $\xi$ strictly between their smallest and largest values, hence in $(a,b)$, such that $F^{(n+1)}(\xi )=0$.

Because $p_n$ has polynomial degree at most $n$ and $\omega$ is monic of polynomial degree $n+1$,Substituting the resulting value of $C$ proves the polynomial interpolation error formulaAt an interpolation node, both sides are zero, so any $\xi \in (a,b)$ works. Such an interior point exists because $n\ge 1$ and the interpolation nodes are distinct. The selected $\xi$ generally depends on $x$; the theorem makes an existence assertion, not a regularity assertion about that dependence.