Compactness in the spaces of functions of bounded variation

Abstract

Recently, the characterization of the compactness in the space $\mathrm{BV}([0,1])$ 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 $[0,1]$.