On classes of functions generating absolutely continuous variational measures

Abstract

It is proved that for a wide class of bases in ${\mathbb R}^m$ a function generates a $\sigma$-finite absolutely continuous variational measure if and only if this function belongs to $ACG_\delta$-class. It is also shown that under some additional assumptions on a basis, $\sigma$-finiteness follows from the absolute continuity of the variational measure.