Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Chistyakov, Vyacheslav V.; Tretyachenko, Yuliya V.

doi:10.1016/j.jmaa.2010.02.042

Cited by 7 publications

(2 citation statements)

References 27 publications

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“…(2) In this article we examine Definition 3.3 mainly in the case when X is a median pretree. Note that for functions on multidimensional objects (subsets of R n ) there are several definitions for BV functions (see, for example, Vitali-Hardy-Krause type variation in [5,17,7] and references therein). Such definitions and ideas probably would be useful also for abstract median algebras or for metric median spaces with finite rank in the sense of [4].…”

Section: 2mentioning

confidence: 99%

Median pretrees and functions of bounded variation

Megrelishvili¹

2019

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We introduce functions of bounded variation on median algebras and study some properties for median pretrees. We show that if X is a compact median pretree (e.g., a dendron) in its shadow topology then every function f : X → R of bounded variation has the point of continuity property (Baire 1, if X, in addition, is metrizable). We prove a generalized version of Helly's selection theorem for a sequence of functions with total bounded variation defined on a compact metrizable median pretree X.

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Section: 2mentioning

confidence: 99%

Median pretrees and functions of bounded variation

Megrelishvili¹

2019

Preprint

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“…We hope that our notion will find further applications in the research area related to Fourier series, calculus of variations, Koksma-Hlawka type inequalities on the Sierpiński gasket. The reader is encouraged to see [1,3,5,6,7] and references therein for recent works on bounded variation functions defined on non-fractal domains.…”

Section: Introductionmentioning

confidence: 99%

Bounded Variation on the Sierpinski Gasket

Verma¹,

Sahu²

2020

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Under certain continuity conditions, we estimate upper and lower box dimension of graph of a function defined on the Sierpiński gasket. We also give an upper bound for Hausdorff dimension and box dimension of graph of function having finite energy. Further, we introduce two sets of definitions of bounded variation for a function defined on the Sierpiński gasket. We show that fractal dimension of graph of a continuous function of bounded variation is log 3 log 2 . We also prove that the class of all bounded variation functions is closed under arithmetic operations. Furthermore, we show that every function of bounded variation is continuous almost everywhere in the sense of log 3 log 2 −dimensional Hausdorff measure.

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