On the total-variation convergence of regularizing algorithms for ill-posed problems

It is well known that ill-posed problems in the space $V[a,b]$ of functions of bounded variation cannot generally be regularized and the approximate solutions do not converge to the exact one with respect to the variation. However, this convergence can be achieved on separable subspaces of $V[a,b]$. It is shown that the Sobolev spaces $W_1^m[a,b]$, $m\in\mathbb N$ can be used as such subspaces. The classes of regularizing functionals are indicated that guarantee that the approximate solutions produced by the Tikhonov variational scheme for ill-posed problems converge with respect to the norm of $W_1^m[a,b]$. In turn, this ensures the convergence of the approximate solutions with respect to the variation and the higher order total variations.