Sharp error constants for a Peano kernel ($r=\frac{\Vert K{\Vert }_{\infty }}{k!},s=\frac{\Vert K{\Vert }_1}{k!}$)

If $\lambda (f)=k{!}^{-1}\int K{f}^{(k+1)}$, the best unrestricted derivative-norm bounds are $|\lambda (f)|\le \Vert K{\Vert }_{\infty }\Vert f^{(k+1)}{\Vert }_1/k!$ and $|\lambda (f)|\le \Vert K{\Vert }_1\Vert f^{(k+1)}{\Vert }_{\infty }/k!$. Continuous concentrated bumps approach the first constant, and continuous approximations to $\mathrm{sgn}K$ approach the second. For the three-point asymmetric second-derivative formula, these constants are $5/6$ and $7/9$.