Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 71 / 4 / 2 / a / Solution

Use the Cox-de Boor recurrence. For unit knots the order-two hat is $L(u)=u$ on $[0,1]$, $L(u)=2-u$ on $[1,2]$, and zero elsewhere. The order-three quadratic cardinal B-spline isThus $N_j(t)=q(t-j)$. At $x_i=i+3/2$, the only possible arguments in its support are $3/2$ and $1/2,5/2$ at neighboring indices. Their values giveIndices outside $1,\dots ,n$ are absent, so the first and last rows have just one neighboring nonzero value.