Representations of functions by derivatives

Abstract

Let Δ ′ \Delta ’ be the class of all derivatives. The main goal of this paper is the investigation of the vector space generated by Δ ′ \Delta ’ and O’Malley’s class B 1 ∗ B_1^ \ast ; this space is identical with our system [ Δ ′ ] [\Delta ’] . We show, in particular, that each approximately continuous function and each approximate derivative belongs to [ Δ ′ ] [\Delta ’] and that [ Δ ′ ] [\Delta ’] is the system of all functions of the form g ′ + h k ′ g’ + hk’ , where g g , h h and k k are differentiable.