 Solution

Substitution into the Tikhonov regularization series gives an explicit reconstruction for every $y\in L^2(0,1)$:Equivalently the summand is ${\sigma }_n⟨y,v_n⟩u_n/(\alpha +{\sigma }_n^2)$. 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, $A^{†}y=y^{\prime }$ and the series converges to $y^{\prime }$ as $\alpha \downarrow 0$. 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.