= Singular-system Tikhonov filter
{title2=$g_\alpha(\sigma)=\frac{\sigma}{\sigma^2+\alpha}$}
For a <singular system of a compact operator> with $Av_i=\sigma_i u_i$, <Tikhonov regularization> gives $x_\alpha=\sum_i g_\alpha(\sigma_i)(y,u_i)v_i$, where $g_\alpha(\sigma)=\sigma/(\sigma^2+\alpha)$ and the <inner product> is linear in the first argument. The bounded gain yields a norm-convergent reconstruction; components in $\ker A^*$ are discarded and no component in $\ker A$ is introduced.
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