Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems

A. S. Leonov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 198-212

Résumé

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 Sobolev spaces $W^m_1$, $m\in\mathbb N$, is studied. In Sobolev spaces, a wide class of integral functionals with the weak regularization properties and the $H$-property is isolated. It is proved that the application of these functionals in the Tikhonov variational scheme generates for $m\ge3$ 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.

Détail

Keywords: higher-order total variations for functions of several variables, regularization of ill-posed problems.

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