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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.

The weak gradient is the vector of first-order weak derivatives of a function. Its $i$th component is specified by $\int u{\partial }_i\phi =-\int ({\partial }_{i}u)\phi$ for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.