= Peano kernel of the third forward difference
{c}
{title2=$K_h(t)=h^2B_2(t/h)$}
For $h>0$, the third <forward difference> $L_hu=-u(0)+3u(h)-3u(2h)+u(3h)$ annihilates <polynomials> of degree at most two. Its <Peano kernel> is $K_h(t)=\tfrac12[3(h-t)_+^2-3(2h-t)_+^2+(3h-t)_+^2]$ for $0\leq t\leq3h$, extended by zero outside that interval. It is $h^2$ times the <quadratic cardinal B-spline> evaluated at $t/h$, is positive on $(0,3h)$, and has <integral> $h^3$. The <Peano kernel theorem> therefore represents $L_hu$ as a positive weighted average of the third <derivative>: $L_hu=h^3u^{(3)}(\xi)$ for some interior $\xi$ when that <derivative> is continuous. The kernel is the smooth error weight hidden inside a discrete difference.
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