Landweber spectral filter ($f_n(\sigma )=1-(1-\tau {\sigma }^2{)}^n$)

Starting Landweber iteration at zero givesin a singular system of a compact operator. The dimensionless damping filter is $f_n(\sigma )=1-(1-\tau {\sigma }^2{)}^n$. If the full inverse coefficient is called the filter instead, it is $f_n(\sigma )/\sigma$; these conventions describe the same operator.

For $\tau =1$ and $\Vert A\Vert \le 1$, $0\le 1-(1-{\sigma }^2{)}^n\le \min (n{\sigma }^2,1)$ givesThus a data error of norm at most $\delta$ produces an iterate error at most $\sqrt{n}\delta$.