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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:370

Article

Chistyakov, Vyacheslav V.¹  Tretyachenko, Yuliya V.¹ 

[1] State Univ Higher Sch Econ, Dept Appl Math & Informat, Nizhnii Novgorod 603155, Russia

关键词: Maps of several variables;    Total variation;    Selection principle;    Metric semigroup;    Pointwise convergence;   

DOI  :  10.1016/j.jmaa.2010.04.055

来源: Elsevier

【 摘 要 】

Given two points a (a(1), ..., a(n)) and b = (b(1), ..., b(n)) from R-n with a < b componentwise and a map f from the rectangle I-a(b) = [a(1), b(1)] x [a(n), b(n)] X Lan, NI into a metric semigroup M = (M, d, +), we study properties of the total variation TV(f,, l) of f on lit; 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, 19631 (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) 61-711 (n is an element of N) for real-valued functions of n variables. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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