Past exam of the options II course of the University of Oxford / 2012 / P / P4

Subtract the two finite sums. In the shifted sum replace the summation index $k$ by $k-1$. For every interior index the new coefficient isby Pascal's identity; the two endpoint coefficients agree as well. ThereforeFor a degree-$d$ polynomial with leading coefficient $c\ne 0$, the backward difference has leading termSince $h>0$, this coefficient is nonzero for $d\ge 1$. Starting with $Q_{(0)}=P$, iteration gives degree $n-r$ and leading coefficient $c{h}^{r}n!/(n-r)!$ for $0\le r\le n$. The next difference of the resulting constant is zero:

A natural cubic spline with the given knots is a $C^2$ function that is polynomial of degree at most three on each interval between consecutive knots and has $S^{\prime \prime }(x_0)=S^{\prime \prime }(x_4)=0$. With the usual whole-line convention, it extends affinely outside the extreme knots. The word natural refers to these endpoint second-derivative conditions; the interpolation values must be specified separately.

For $H_k(x)=(x-x_k{)}_{+}^3$, direct differentiation on either side of the knot givesBoth one-sided limits of the function and its first two derivatives at $x_k$ are zero, so these formulas are continuous there andIts third derivative jumps from zero to six, which is permitted for a cubic spline.

The finite sum of truncated cubes is $C^2$ by part (b) and is cubic on each knot interval. For $x\le 0$, every term vanishes. For $x\ge 4h$, every truncated power equals its ordinary polynomial, so part (a) applied to $P(x)=x^3$ givesIt is therefore zero on both exterior intervals. Its $C^2$ regularity makes the endpoint second derivatives zero too, proving naturality.

Evaluation at the five knots givesFor example the three interior values in units of $h^3$ are $1$, $8-4=4$, and $27-4\cdot 8+6=1$. At $4h$ the cancellation is $64-4\cdot 27+6\cdot 8-4=0$.

For $h=1$ an explicit graph description isIt is symmetric about $x=2$, increases from zero to its maximum four at the centre, and decreases symmetrically back to zero. It joins the zero tails with zero first and second derivatives.