Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 62 / 5 / 2

For unit-spaced knots, the quadratic cardinal B-spline with support $[i,i+3]$ is $N_i(t)=B(t-i)$, where the Cox-de Boor recurrence givesIn particular $B(1)=B(2)=1/2$ and $B(0)=B(3)=0$. At $x_i=i+2$, the only nonzero entries in row $i$ are $A_{ii}=1/2$ and, if $i<n$, $A_{i,i+1}=1/2$. Write $U$ for the upper shift matrix, with ones immediately above its diagonal. Thensince $U^n=0$. Thus $(A_x^{-1}{)}_{ij}=2(-1{)}^{j-i}$ for $j\ge i$, and zero otherwise. The largest absolute row sum is the first, giving $\Vert A_x^{-1}{\Vert }_{{\ell }_{\infty }}=2n$.

Using (1) and the permitted $d_3=3$ proves the stronger linear growth of shifted quadratic spline interpolation estimateTherefore $\Vert P_x\Vert =\Theta (n)$ and is not uniformly bounded. An $O(n)$ upper bound alone would not imply unboundedness; the lower bound is essential.