= Strong convergence of relaxed Landweber iteration
{title2=$x_n\to x^\dagger$}
For nonzero <compact operator> $A$, step $0<\tau<2/\|A\|^2$, and data admitting a <minimum-norm least-squares solution> $x^\dagger$, zero-initialized <Landweber iteration> satisfies
$$
\|x_n-x^\dagger\|^2=\sum_i|1-\tau\sigma_i^2|^{2n}\frac{|\langle y,u_i\rangle|^2}{\sigma_i^2}\longrightarrow0.
$$
The <Picard criterion> makes the majorant summable, so the <dominated convergence theorem> proves this limit. No mesh-independent or singular-value-independent strict contraction factor is required. At $\tau=2/\|A\|^2$, the top singular mode has multiplier $-1$ and may not converge. Without solvable projected data, the iterates need not have a limit in the solution <Hilbert space>, even when their residuals approach the least-squares infimum.
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