= Landweber full inverse filter
{c}
{title2=$g_n(\sigma)=[1-(1-\tau\sigma^2)^n]/\sigma$}
Starting <Landweber iteration> at zero with fixed step $\tau$, the full data-to-solution coefficient on a <singular value> $\sigma>0$ is
$$
g_n(\sigma)=\frac{1-(1-\tau\sigma^2)^n}{\sigma}.
$$
This follows by summing the scalar <geometric series> in a <singular system of a compact operator>. For fixed $n$, $g_n(\sigma)=n\tau\sigma+O(\sigma^3)$ near zero and extends with $g_n(0)=0$. Its dimensionless damping factor is $\sigma g_n(\sigma)$, so the two common filter conventions must not be confused. Writing $\alpha=1/n$ uses a discrete parameter sequence; if negative multipliers occur, noninteger real powers are not a definition of the same iteration.
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