Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 6 / b / Solution

Write $N_{i,k}$ for the order-$k$ B-spline. At order one, $N_{i,1}=1_{[t_i,t_{i+1})}$, and the finite sum is one on its basic knot interval. For larger order, the Cox-de Boor recurrence iswith a zero-denominator term defined as zero. Sum over $i=1,\dots ,n$. For each interior lower-order index $j=2,\dots ,n$, the combined coefficient isThe two unmatched lower-order terms have supports $[t_1,t_k]$ and $[t_{n+1},t_{n+k}]$, respectively, and vanish on the open basic interval $(t_k,t_{n+1})$. Induction therefore givesFor simple knots and $k\ge 2$, continuity extends the identity to both endpoints. In the order-one case and at admissible repeated endpoint knots, use the one-sided value from the basic interval. Extending the knot sequence in both directions also proves the subpartition of unity for B-splines, $0\le {\sum }_i{N}_i\le 1$ everywhere: the finite sum is a subset of a nonnegative full partition. That weaker global inequality will be needed in Question 7(a).