Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 69 / 3 / 2 / a / Solution

For unit-spaced knots, the quadratic cardinal B-spline is a translate ofFor example, the Cox-de Boor recurrence gives $N_{j,3}(t)=(t-j)N_{j,2}(t)/2+(j+3-t)N_{j+1,2}(t)/2$ from the two linear hats. Substituting their linear pieces gives the three expressions above. At $x_i=i+3/2$, only the B-splines with indices $j=i-1,i,i+1$ have nonzero values, and their values are respectivelyOnly indices $1\le j\le n$ belong to the given basis, so entries outside that range are omitted at the boundary.