Peano kernel theorem

The Peano kernel theorem represents a linear approximation error $L$ that annihilates polynomials below degree $r$ as

The Peano kernel of a linear error functional $L$ that annihilates all polynomials of degree at most $k$ isIt represents the error by integrating $K$ against the $(k+1)$st derivative.

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 $7/9$.

At unit spacing, exactness through cubic polynomials gives

If the fourth-order Peano kernel of the four-point endpoint formula is nonnegative, its sharp sup-norm error constant is its integral. Evaluating the error on $(x+1{)}^4/24$ gives