= Error bound for one-sided differentiation
{title2=$\|D_h f^\delta-f'\|_2\leq\sqrt6\delta/h+\|f''\|_\infty h/2$}
For $f\in C^2[0,1]$ and $\|f^\delta-f\|_2\leq\delta$, two translated half-interval integrals have overlap multiplicity at most two. The inequality $|a-b|^2\leq2|a|^2+2|b|^2$ gives $\|D_h\|\leq\sqrt6/h$. Averaging $f'$ over the relevant interval and applying $|f'(x+t)-f'(x)|\leq\|f''\|_\infty|t|$ gives the second term. For this to estimate a <Moore–Penrose inverse of an operator>, also require $f(0)=0$; otherwise the derivative is still approximated but $K^\dagger f$ is not defined.
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