= Solution
A <regularization of an inverse problem> replaces the possibly unbounded inverse by a family of continuous reconstruction operators $R_\alpha:Y\to X$. For compatible exact data it must recover the minimum-norm solution $x^\dagger\in(\ker A)^\perp$ as $\alpha\downarrow0$. For noisy data $\|y^\delta-y\|\leq\delta$, a parameter rule $\alpha=\alpha(\delta)$ must make both bias and amplified noise tend to zero. The target is the minimum-norm solution because data cannot determine a component in the <kernel of a linear map>.
<Tikhonov regularization> minimizes
$$
J_\alpha(x)=\|Ax-y^\delta\|_Y^2+\alpha\|x\|_X^2,\qquad \alpha>0.
$$
The <quadratic norm penalty> makes this <functional> <strictly convex> and a <coercive function>. Its first variation, applied to arbitrary real and imaginary perturbations, gives
$$
(A^*A+\alpha I)x_\alpha^\delta=A^*y^\delta.
$$
The <positive operator> $A^*A+\alpha I$ is bounded below by $\alpha I$ and has a bounded inverse. \b[The unique <Tikhonov regularization> solution is]
$$
\boxed{x_\alpha^\delta=R_\alpha y^\delta,\qquad
R_\alpha=(A^*A+\alpha I)^{-1}A^*.}
$$
The <Tikhonov stability bound> follows from the singular-value gain:
$$
\|R_\alpha\|\leq\sup_{\sigma\geq0}\frac{\sigma}{\sigma^2+\alpha}
=\frac1{2\sqrt\alpha},\qquad
\boxed{\|x_\alpha^\delta-x_\alpha\|\leq\frac{\delta}{2\sqrt\alpha}.}
$$
Thus reconstruction is stable for each fixed positive $\alpha$. Stability is not uniform as $\alpha\downarrow0$: that limit restores the unbounded inverse. For exact compatible data,
$$
x_\alpha-x^\dagger=-\alpha(A^*A+\alpha I)^{-1}x^\dagger\longrightarrow0.
$$
The <singular system of a compact operator> makes this strong convergence transparent: each positive singular-value coefficient tends to zero and is bounded by the corresponding coefficient of $x^\dagger$. Consequently
$$
\|x_\alpha^\delta-x^\dagger\|
\leq\|x_\alpha-x^\dagger\|+\frac{\delta}{2\sqrt\alpha}\longrightarrow0
$$
whenever $\alpha(\delta)\downarrow0$ and $\delta/\sqrt{\alpha(\delta)}\to0$, for example $\alpha(\delta)=\delta$. This is a convergent <regularization of an inverse problem>, not merely a bounded formula.
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