= Central-difference noise-bias bound
{title2=$\|R_\alpha f^\delta-f'\|_2\leq\sqrt6\delta/\alpha+\sqrt{17}\alpha c/4$}
For the mixed forward, <central finite difference>, and backward derivative regularizer on $[0,1]$, with dividing points $(1-\alpha)/2$ and $(1+\alpha)/2$, the <overlap multiplicity bound for piecewise difference operators> has $m=3$. On the middle interval, the error kernel is $\operatorname{sgn}(s)(\alpha/2-|s|)/\alpha$ for $|s|\leq\alpha/2$, with <L1 norm> $\alpha/4$. <Young's convolution inequality> bounds that part of the bias by $\alpha\|f''\|_2/4$. If the two outer intervals together have squared error at most $\alpha^2\|f''\|_2^2$, combining the three pieces gives the displayed estimate for $\|f''\|_2\leq c$ and $\|f^\delta-f\|_2\leq\delta$. Interpreting $f'$ as the inverse of the <Volterra integration operator> requires the <range of the Volterra integration operator> boundary condition $f(0)=0$.
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