Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 61 / 4 / b / Solution

Uniform knots give translation invariance $N_{i,k}(x)=N_{0,k}(x-i)$. Put $v_{k,j}=N_{0,k}(j)$ and set missing values to zero. The recurrence in part (a) gives the exact numerical recursionThe order-two starting row is $(0,1,0)$. For example, the order-four middle value is $(2\cdot \frac{1}{2}+2\cdot \frac{1}{2})/3=2/3$, and the order-six central value is $(3\cdot \frac{11}{24}+3\cdot \frac{11}{24})/5=66/120$. Iteration gives the requested array, listing $N_{0,k}(0),\dots ,N_{0,k}(k)$:The symmetry follows from reflection of the uniform Cardinal B-spline about the midpoint of its support. The row sums equal one, consistent with sampling the uniform partition of unity. These are partition-normalized values; normalizing each spline to have peak one would produce a different array.