On the differentiation of integrals with respect to translation invariant convex density bases

TL;DR

This paper investigates how the differentiation properties of translation invariant convex density bases are preserved when extended to their Busemann-Feller counterparts, enabling transfer of known results between these classes.

Contribution

It demonstrates that the Busemann-Feller extension maintains key properties of the original basis, allowing results for Busemann-Feller bases to apply to broader classes.

Findings

Busemann-Feller extension closely preserves the differentiation properties of the original basis.

Results for Busemann-Feller bases are transferable to non-Busemann-Feller bases.

The extension ensures the same class of functions is differentiated by both bases.

Abstract

For a translation invariant convex density basis $B$ it is shown that its Busemann-Feller extension $B_{\mathrm{BF}}$ has close to $B$ properties, in particular, $B_{\mathrm{BF}}$ differentiates the same class of non-negative functions as $B$. Using the similarity between properties of the bases $B$ and $B_{\mathrm{BF}}$ some results known for Busemann-Feller bases are transferred to bases without restriction of being Busemann-Feller.