On the generalized Wiener bounded variation spaces with p-variable

TL;DR

This paper explores the properties of generalized Wiener spaces with variable exponents, establishing key functional characteristics and behaviors of functions within these spaces, including discontinuity points and limit existence.

Contribution

It introduces and analyzes the generalized Wiener bounded variation spaces with p-variable, revealing properties like convexity, reflexivity, and discontinuity characterization.

Findings

Proved uniform convexity and reflexivity of the space

Characterized the set of discontinuity points of functions

Showed existence of functions without right-hand limits in unbounded cases

Abstract

In this paper, we have investigated the generalized Wiener space of bounded variation with $p$-variable. Various results are obtained such as uniform convexity and reflexivity, there was characterized the set of points of discontinuity of functions from this space. For bounded exponents, it is shown the existence of right and left-hand limits in each point. Also, there is an unbounded exponent such that in corresponding generalized Wiener bounded variation space exists a function that does not have the right-hand limit at a point. Also, for some wide classes of exponents in this space additivity of the variation is not fulfilled.