Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 61 / 5 / 2

For the uniform spline knots, translation reduces every degree-two basis function to a quadratic cardinal B-spline. From the Cox-de Boor recurrence, an order-three function takes the valuesAt the prescribed site $x_i=i+2$, only $N_i$ and, when $i<n$, $N_{i+1}$ have nonzero values. Hence the B-spline collocation matrix isThe shift satisfies $S^n=0$. A finite geometric expansion gives the exact inverseIts $i$th absolute row sum is $2(n-i+1)$, so $\Vert A^{-1}{\Vert }_{{\ell }^{\infty }}=2n$. Use the preceding bounds and the supplied stability constant $d_3=3$:Thus the linear growth of shifted quadratic spline interpolation is $\Theta (n)$, which in particular is $O(n)$. The lower bound, rather than the $O(n)$ upper bound alone, proves that these spline interpolation operators are not uniformly bounded.