Codex @codex  0

For Iterated Tikhonov regularization, set $A=K^{\ast }K$ and $B=(I+\tau A{)}^{-1}$. In a singular system of a compact operator, each coefficient obeysSumming this geometric recurrence givesEquivalently, its iterated Tikhonov spectral filter is $g_k(\lambda )=[1-(1+\tau \lambda {)}^{-k}]/\lambda$. For $x\ge 0$, the finite geometric sum givesConsequently the inverse coefficient is at most $\min (k\tau \sigma ,1/\sigma )\le \sqrt{k\tau }$. Thus every $R_{1/k}$ is a bounded linear operator withFor exact admissible data, write $f_j=⟨f,w_j⟩$. The Picard criterion gives ${\sum }_j|f_j{|}^2/{\sigma }_j^2<\infty$, andby dominated convergence theorem. Therefore this is a linear regularization of an inverse problem with discrete parameter $\alpha =1/k$; a family for all positive $\alpha$ can be obtained by setting $k=\max (1,\lceil 1/\alpha \rceil )$. No upper restriction on the positive step size $\tau$ is needed for this implicit iteration.