= Solution
Starting from zero, the <Landweber iteration> is
$$
\boxed{u_\delta^{(0)}=0,\qquad
u_\delta^{(k+1)}=u_\delta^{(k)}+\tau K^*(f^\delta-Ku_\delta^{(k)}).}
$$
For $K\neq0$, take the <step size> $0<\tau<2/\|K\|^2$. Iterating the linear update gives
$$
R_{1/k}=\tau\sum_{m=0}^{k-1}(I-\tau K^*K)^mK^*,
$$
and its <Landweber spectral filter> in a <singular system of a compact operator> is
$$
\boxed{R_{1/k}f^\delta=
\sum_j\frac{1-(1-\tau\sigma_j^2)^k}{\sigma_j}
\langle f^\delta,w_j\rangle v_j.}
$$
The strict upper <step size> bound makes $|1-\tau\sigma_j^2|<1$ for every positive <singular value>. A common more restrictive choice is $0<\tau\leq\|K\|^{-2}$, giving nonnegative damping factors. Each finite iterate is bounded and linear; <early stopping of Landweber iteration> controls the amplification of noise as smaller <singular values> are progressively inverted. If $K=0$, every iterate is zero for any positive <step size>.
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