= Closed form of a Tikhonov-initialized Landweber iterate
{title2=$g_{\alpha,n}(\sigma)=\sigma^{-1}\left[1-\frac{\alpha}{\alpha+\sigma^2}(1-\sigma^2/\alpha)^n\right]$}
In a <singular system of a compact operator>, <Landweber iteration with a Tikhonov initial value> has coefficients $g_{\alpha,n}(\sigma_i)(y,u_i)$. Summing the finite geometric series gives the displayed filter. For fixed $n$, $g_{\alpha,n}(\sigma)=(n+1)\sigma/\alpha+O(\sigma^3)$ at zero, so each finite iterate is bounded. Under $\alpha>\|A\|^2/2$ the filter approaches $1/\sigma$ on every positive singular mode; noisy data require <early stopping of Landweber iteration>.
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