Repeated substitution in <Landweber iteration> from $u_0=0$ gives
$$
u_k
=\tau\sum_{j=0}^{k-1}(I-\tau A^*A)^jA^*f.
$$
On the $i$th singular vector, $A^*A$ has <eigenvalue> $\sigma_i^2$ and $A^*y_i=\sigma_ix_i$. The finite <geometric series> therefore gives
$$
\boxed{
u_k=\sum_i
\frac{1-(1-\tau\sigma_i^2)^k}{\sigma_i}
\langle f,y_i\rangle x_i}.
$$
This is a <spectral regularization method> with filter
$$
g_k(\lambda)=\frac{1-(1-\tau\lambda)^k}{\lambda},
\qquad
u_k=g_k(A^*A)A^*f.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 1 / 1 / 3 / a / Solution

A spectral regularization method applies a bounded scalar filter to the spectrum of $A^*A$. In a <singular system of a compact operator>, it has the form
$$
R_\alpha f=\sum_jg_\alpha(\sigma_j^2)\sigma_j\langle f,u_j\rangle v_j,
$$
where the filters approximate $1/\lambda$ while controlling amplification near $\lambda=0$.

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 Incoming links: Spectral regularization method