= Solution
Substitution into the <Tikhonov regularization> series gives an explicit reconstruction for every $y\in L^2(0,1)$:
$$
\boxed{x_\alpha(x)=\sum_{n=1}^{\infty}\frac{a_n}{1+\alpha a_n^2}\left[\int_0^1 y(t)\sqrt2\sin(a_nt)\,dt\right]\sqrt2\cos(a_nx),\qquad a_n=(n-\tfrac12)\pi.}
$$
Equivalently the summand is $\sigma_n\langle y,v_n\rangle u_n/(\alpha+\sigma_n^2)$. The <Tikhonov stability bound> makes the series converge in $L^2$ for every positive $\alpha$. It suppresses high-frequency components of unstable differentiation, rather than differentiating arbitrary noisy data pointwise.
The <range of the Volterra integration operator> is $\{y\in H^1(0,1):y(0)=0\}$. On exactly these data, $A^\dagger y=y'$ and the series converges to $y'$ as $\alpha\downarrow0$. For arbitrary $L^2$ data outside that range, a finite regularized solution still exists, but a finite-norm unregularized solution need not exist. This is <Spectral Tikhonov differentiation for the Volterra operator>.
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