= Solution
A <regularization of an inverse problem> replaces the unbounded generalized inverse by bounded operators $R_\alpha:Y\to X$ satisfying $R_\alpha y\to A^\dagger y$ as $\alpha\downarrow0$ on admissible exact data. A <spectral regularization method> can be written
$$
\boxed{x_\alpha=R_\alpha y=\sum_j\frac{f_\alpha(\sigma_j)}{\sigma_j}\langle y,u_j\rangle v_j,}
$$
where $f_\alpha(\sigma)\to1$ for fixed $\sigma>0$ while $f_\alpha(\sigma)/\sigma$ remains bounded for each fixed $\alpha$. A uniform bound on $f_\alpha$ supplies the usual dominated-convergence argument on exact data.
This defines $f_\alpha$ as the dimensionless damping factor. In the alternative convention calling the entire inverse coefficient the <spectral filter>, that filter is $f_\alpha(\sigma)/\sigma$. For data error bounded by $\delta$, a sufficient <regularization parameter choice> also requires $\delta\|R_{\alpha(\delta)}\|\to0$; consistency alone does not control noise.
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