Maps of several variables of finite total variation. I. Mixed differences and the total variation |
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Authors: | Vyacheslav V Chistyakov Yuliya V Tretyachenko |
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Institution: | Department of Applied Mathematics and Informatics, State University Higher School of Economics, Bol'shaya Pechërskaya Street 25/12, Nizhny Novgorod 603155, Russian Federation |
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Abstract: | Given two points a=(a1,…,an) and b=(b1,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on 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] (n∈N) for real-valued functions of n variables. |
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Keywords: | Maps of several variables Total variation Selection principle Metric semigroup Pointwise convergence |
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