Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems. Proceedings of the Steklov Institute of Mathematics Volume 280, Issue SUPPL.1, 2013, Pages 119-133

In the space of functions of two variables with Hardy-Krause property, new notions of higher-order total variations and Banach spaces of functions of two variables with bounded higher variations are introduced. The connection of these spaces with the Sobolev spaces W1 m, m ∈ ℕ, is studied. In the Sobolev spaces, a wide class of integral functionals with the weak regularization properties and the H-property is defined. It is proved that the application of these functionals in the Tikhonov variational scheme generates for m ≥ 3 the convergence of approximate solutions with respect to the total variation of order m - 3. The results are naturally extended to the case of functions of N variables. © 2013 Pleiades Publishing, Ltd.
Author keywords
higher-order total variations for functions of several variables; regularization of ill-posed problems