= Sharp error constants for a Peano kernel
{title2=$r=\frac{\|K\|_\infty}{k!},\quad s=\frac{\|K\|_1}{k!}$}
If $\lambda(f)=k!^{-1}\int K f^{(k+1)}$, the best unrestricted derivative-norm bounds are $|\lambda(f)|\le \|K\|_\infty\|f^{(k+1)}\|_1/k!$ and $|\lambda(f)|\le\|K\|_1\|f^{(k+1)}\|_\infty/k!$. Continuous concentrated bumps approach the first constant, and continuous approximations to $\operatorname{sgn}K$ approach the second. For the <three-point asymmetric second-derivative formula>, these constants are $5/6$ and $7/9$.
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