Past exam of the options II course of the University of Oxford / 2009 / P / P3

Interpolation means $p_n(x_i)=f(x_i)$ for every node. Define the Lagrange basis polynomialsDistinctness of the nodes makes every denominator nonzero, and ${\ell }_i(x_j)={\delta }_{ij}$. Thusis a polynomial of degree at most $n$ with the required values. If two such polynomials existed, their difference would have $n+1$ distinct roots while having degree at most $n$; the polynomial root bound makes it identically zero. Hence existence and uniqueness hold. The actual degree can be smaller than $n$, despite the conventional name.

Multiplication by the displayed Vandermonde matrix maps the coefficient vector of a polynomial of degree at most $n$ to its vector of values at the nodes. Part (a) proves that every vector of node values is attained uniquely, so this square linear map is bijective. Therefore $A_n$ is invertible. Equivalently, a kernel vector would define a polynomial of degree at most $n$ with $n+1$ distinct roots and would have to vanish.

If $x$ is a node the error and the product both vanish, and any interior $\xi$ works. Otherwise put $\omega (t)={\prod }_{i=0}^n(t-x_i)$ and $C=(f(x)-p_n(x))/\omega (x)$. The functionhas $n+2$ distinct zeros, at the nodes and at $x$. Repeated Rolle's theorem gives a point $\xi$ between the extreme zeros, and therefore in $(a,b)$, where $R^{(n+1)}(\xi )=0$. Since $p_n^{(n+1)}=0$ and ${\omega }^{(n+1)}=(n+1)!$, one gets $C=f^{(n+1)}(\xi )/(n+1)!$. Substitution gives

Every derivative of order $n+1$ of $\sin (2x)$ has absolute value at most $2^{n+1}$. The equally spaced nodes are symmetric about $1/2$. Pair each node $a\le 1/2$ with $1-a$; for $x\in [0,1]$,If an unpaired middle node remains, its factor satisfies $|x-1/2|\le 1/2$. Consequently the complete nodal product is at most $2^{-(n+1)}$. The error formula actually gives the stronger estimateThe first bound tends to zero, proving uniform convergence on the whole interval. There is no appeal to a general claim that equispaced interpolation always converges; the derivative and symmetric-product bounds are particular to this analytic function and these nodes.