= Differentiation by one-sided difference quotients
{title2=$D_h f(x)=\frac{f(x+h)-f(x)}h\ \text{or}\ \frac{f(x)-f(x-h)}h$}
On $(0,1/2)$ use the forward difference and on $(1/2,1)$ the backward difference, with $0<h<1/2$. This defines a <bounded linear operator> on $L^2(0,1)$ and $\|D_h\|\leq\sqrt6/h$. For $f$ in the <range of the Volterra integration operator>, these differences are local averages of $f'$ and converge to $f'$ in $L^2$ by continuity of translations. Hence $D_h$ gives a <linear regularization> of differentiation. No pointwise evaluation of an arbitrary $L^2$ equivalence class is needed: translated functions are defined almost everywhere.
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