Closed form of a Tikhonov-initialized Landweber iterate ($g_{\alpha ,n}(\sigma )={\sigma }^{-1}\left[1-\frac{\alpha }{\alpha +{\sigma }^2}(1-{\sigma }^2/\alpha {)}^n\right]$)

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 $\alpha >\Vert A{\Vert }^2/2$ the filter approaches $1/\sigma$ on every positive singular mode; noisy data require early stopping of Landweber iteration.