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On the total variation for functions of several variables and a multidimensional analog of Helly’s selection principle. (English. Russian original) Zbl 0924.26007
Math. Notes 63, No. 1, 61-71 (1998); translation from Mat. Zametki 63, No. 1, 69-80 (1998).
The author starts with the remark that, with respect to the large number of approaches to the idea of variation for functions of several variables, still remain important properties of functions with bounded variation, of one variable, that don’t have their analog in the multidimensional case; an example is Helly’s selection principle. In order to bridge this gap, one considers the Hardy class \(H\) of functions of bounded variation in several variables, but its definition is modified in order to make possible an associated notion (absent to Hardy) of “variation of a function in several variables”. The new notion of total variation for such functions leads to a description of \(H\) as the set of functions of bounded variation, in such a way that some additional properties can be established, for instance the multidimensional analog of the inequality \(| z(x)- z(y)|\leq \bigvee^y_x(z)\), \(\forall x,y\in[a, b]\), well-known for functions of bounded variation in one variable. The same framework allows to obtain a perfect multidimensional analog of Helly’s selection principle. Important applications, for instance the solution of the piecewise-uniform regularization problem for multidimensional ill-posed problems by means of multidimensional functions of bounded variation.

26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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