= 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 $|t|\leq1/2$. On the next interval, the interpolating quadratic through values $1,0,0$ at abscissae $0,1,2$ is $\tfrac12(t-1)(t-2)$. Adding its limiting offset $1/8$ gives $\tfrac12(3/2-t)^2$. Reflection gives the left interval, and locality gives zero outside. Thus
$$
\boxed{\beta(t)=\begin{cases}
\frac34-t^2,&|t|\leq\frac12,\\
\frac12(\frac32-|t|)^2,&\frac12\leq|t|\leq\frac32,\\
0,&|t|\geq\frac32.
\end{cases}}
$$
This 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-70-spline-bases.png]
{title=Four cubic B-spline Bézier pieces and the convergence of a Chaikin impulse to its quadratic basis function}
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