Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 61 / 3 / b / Solution

We derive the Cox-de Boor recurrence from the Leibniz rule for divided differences. Fix $t$, let $u_j=t_{i+j}$, and setAway from the spline knots, $h(s)=(s-t{)}_{+}^{k-1}$. IntroduceThe defining divided-difference recurrence says $(u_k-u_0)D=R-L$. Since $s-t=(s-u_0)+(u_0-t)$ and the first factor is linear, the Leibniz rule for divided differences givesMultiplication by $u_k-u_0$ consequently yieldsReplace $L$ and $R$ by the corresponding lower-order B-splines divided by their support widths. This provesThe denominators are positive for the distinct spline knots of this question. The recursion starts with $N_{i,1}$, the interval indicator on $[t_i,t_{i+1})$. For $k\ge 2$, values at the spline knots follow by the continuous extension of the left side; the usual half-open convention for the order-one factors gives the same result. This proves the recurrence rather than assuming it as a definition.