= Tikhonov regularization with a coercive penalty operator
{c}
{title2=$R_\alpha=(K^*K+\alpha B^*B)^{-1}K^*$}
= Tikhonov–Phillips regularization
{c}
{synonym}
= Tikhonov–Philipps regularization
{c}
{synonym}
For <bounded linear operators> between <Hilbert spaces> with $\|Bu\|\geq\beta\|u\|$ and $\beta>0$, the operator $K^*K+\alpha B^*B$ is coercive and invertible for $\alpha>0$. The reconstruction uniquely minimizes $\|Ku-f\|^2/2+\alpha\|Bu\|^2/2$. Testing its <normal equation for coercive Tikhonov regularization> by $u$ gives $\|R_\alpha\|\leq1/(2\beta\sqrt\alpha)$. If $K$ is injective, exact-data consistency and $\alpha\to0$, $\delta/\sqrt\alpha\to0$ give <norm convergence> to $K^\dagger f$.
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