= Uniform noise bound for relaxed Landweber iteration
{title2=$\|x_n^\delta-x_n\|\leq\delta\sqrt{2n\tau}$}
For $0<\tau<2/\|A\|^2$, the <Landweber full inverse filter> obeys
$$
|g_n(\sigma)|\leq\min(n\tau\sigma,2/\sigma)\leq\sqrt{2n\tau}.
$$
Indeed, with $t=\tau\sigma^2\in[0,2]$, the telescoping <geometric series> and $|1-t|\leq1$ give $|1-(1-t)^n|\leq\min(nt,2)$. Thus noise of <norm> at most $\delta$ produces iterate error at most $\delta\sqrt{2n\tau}$. If $\tau\|A\|^2\leq1$, replace 2 by 1 to obtain $\delta\sqrt{n\tau}$. Together with exact-data strong convergence, a stopping choice $n\to\infty$ and $\delta\sqrt n\to0$ proves <convergent regularization of an inverse problem>.
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