A pointwise selection principle for maps of several variables via the total joint variation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:402

Article

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

[1] Natl Res Univ, Higher Sch Econ, Dept Appl Math & Comp Sci, Nizhnii Novgorod 603155, Russia

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

DOI  :  10.1016/j.jmaa.2013.02.005

来源: Elsevier

【 摘 要 】

Given a rectangle in the real Euclidean n-dimensional space and two maps f and g defined on it and taking values in a metric semigroup, We introduce the notion of the total joint variation TV(f , g) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) [5,8] and the authors (2010). We prove the following irregular pointwise selection principle in terms of the total joint variation: if a sequence of maps {f(j)}(j=1)(infinity) from the rectangle into a metric semigroup is pointwise precompact and lim sup(j,k ->infinity) TV(f(j), f(k)) is finite, then it admits a pointwise convergent subsequence (whose limit may be a highly irregular, e.g.; everywhere discontinuous, map). This result generalizes some recent pointwise selection principles for real functions and maps of several real variables. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free