Spectral Tikhonov differentiation for the Volterra operator ($x_{\alpha }={\sum }_n\frac{a_n}{1+\alpha a_n^2}⟨y,v_n⟩u_n$)

The Tikhonov regularization of the Volterra integration operator replaces the unbounded differentiation gain $a_n$ by $a_n/(1+\alpha a_n^2)$. The series converges in $L^2$ for every datum and $\alpha >0$. On the range of the Volterra integration operator, it converges to the weak derivative of the datum as $\alpha \downarrow 0$. No pointwise differentiation of arbitrary noisy L2 space data is required.