Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 70 / 3 / iv / Solution

The rule is linear in the ordinates and translation invariant on the uniform grid, so write $s(x)={\sum }_i{y}_i\beta ((x-x_i)/h)$. Find its Chaikin basis function by setting $y_0=1$ and every other ordinate to zero. The central local piece just derived has $D_0=-2$ and gives $\beta (t)=3/4-t^2$ for $|t|\le 1/2$. On the next interval, the interpolating quadratic through values $1,0,0$ at abscissae $0,1,2$ is $\frac{1}{2}(t-1)(t-2)$. Adding its limiting offset $1/8$ gives $\frac{1}{2}(3/2-t{)}^2$. Reflection gives the left interval, and locality gives zero outside. ThusThis is a centered quadratic cardinal B-spline. It is nonnegative, and its integer translates form a partition of unity because refinement preserves constant data. Its support has width three, and its first derivative agrees at the half-integer knots while its second generally jumps, consistent with the earlier calculations.