Approximation and Transfer of Properties Between Translation Invariant Convex Differentiation Bases

Abstract

We prove a transfer of differential properties within the framework of non-negative functions from a translation invariant convex density basis $\textbf{B}$ to a translation invariant convex basis $\textbf{H}$ provided each set $H$ forming $\textbf{H}$ can be approximated by some set $B$ forming $\textbf{B}$ in the sense that the estimate $|H\triangle B|\leqslant c\mkern1mu |H\,{\cap}\, B|$ is valid, where $c$ is a positive constant not depending on $H$. An application of this transference result shows that the study of differential properties in the class of non-negative functions for translation invariant convex density bases can be reduced to that for translation invariant Busemann-Feller density bases formed of multi-dimensional rectangles.