= Solution
Let $(\sigma_j,u_j,v_j)$ be a <singular system of a compact operator>, with
$$
Av_j=\sigma_ju_j,
\qquad
A^*u_j=\sigma_jv_j.
$$
The <Moore–Penrose inverse of an operator> is the generally unbounded map
$$
\boxed{
A^\dagger g
=\sum_j\frac{\langle g,u_j\rangle}{\sigma_j}v_j},
$$
defined when the <Picard criterion> holds, with the component in $\ker A^*$ sent to zero. It is the <minimum-norm least-squares solution> of $Af=g$.
A <regularization of an inverse problem> consists of bounded maps $R_\alpha:Y\to X$ and a parameter rule $α=α(δ,g^{(δ)})$ such that
$$
\alpha\to0,
\qquad
R_\alpha g^{(\delta)}\to A^\dagger g
$$
whenever $\|g^{(\delta)}-g\|\leq\delta$ and $g$ is in the domain of $A^\dagger$.
For <Tikhonov regularization>, minimizing
$$
\|Af-g^{(\delta)}\|^2+\alpha\|f\|^2
$$
gives
$$
R_\alpha g^{(\delta)}
=(A^*A+\alpha I)^{-1}A^*g^{(\delta)}
=\sum_j\frac{\sigma_j}{\sigma_j^2+\alpha}
\langle g^{(\delta)},u_j\rangle v_j.
$$
The scalar <spectral filter> satisfies
$$
\sup_{\sigma\geq0}\frac{\sigma}{\sigma^2+\alpha}
=\frac{1}{2\sqrt\alpha},
$$
and consequently
$$
\|R_\alpha(g^{(\delta)}-g)\|
\leq\frac{\delta}{2\sqrt\alpha}.
$$
For exact data, each filter factor $\sigma_j^2/(\sigma_j^2+\alpha)$ tends to one, so $R_\alpha g\to A^\dagger g$. Choosing
$$
\boxed{\alpha(\delta)\to0,
\qquad \frac{\delta}{\sqrt{\alpha(\delta)}}\to0}
$$
therefore makes both the approximation error and propagated data error vanish. For example, $α(δ)=δ$ is an admissible <a priori regularization parameter choice>.
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