Compactness in the spaces of functions of bounded variation

TL;DR

This paper extends recent results on compactness characterization in BV spaces to more general function spaces like Waterman, Young, and integral variation, showing uniform approximation of norms on compact sets.

Contribution

It generalizes the compactness characterization from BV spaces to Waterman, Young, and integral variation spaces, providing new insights into their structure.

Findings

Norms are uniformly approximated by seminorms on compact sets.

Generalizations of compactness characterization to broader function spaces.

Provides a unified approach to compactness in various variation spaces.

Abstract

Recently the characterization of the compactness in the space $BV([0,1])$ of functions of bounded Jordan variation was given. Here, certain generalizations of this result are given for the spaces of functions of bounded Waterman $\Lambda$-variation, Young $\Phi$-variation and integral variation. It appears that on the compact sets the norm is uniformly approximated by certain seminorms induced by the selection of finitely many intervals in $[0,1]$.