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Article

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Journal of Mathematical Analysis and Applications. 2010. Vol. 370. No. 2. P. 672-686.
Chistyakov V., Tretyachenko Y.

Given two points a=(a1,…,an) and b=(b1,…,bn) from Rn with a<b componentwise and a map f from the rectangle View the MathML source into a metric semigroup M=(M,d,+), we study properties of the total variationView the MathML source of f on View the MathML source introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61–71] (nN) for real-valued functions of n variables.