# Convolutions and the geometry of multifractal measures

## Kenneth J Falconer and Toby C O'Neil

Appeared in Mathematische Nachrichten, 1999.

## Abstract

This paper relates multifractal features of a measure in $$\mathbb{R}^n$$ to those of the projection of the measure onto typical $$m$$-dimensional linear subspaces. We achieve this through the introduction of suitably defined convolution kernels. This provides a unified approach to projections of measures and leads to new results on multifractal properties as well as alternative derivations of some existing formulae. These include formulae and estimates for the local dimensions and generalised $$q$$-dimensions of projected measures as well as more precise information about the limiting behaviour of multifractal expressions. We consider briefly how similar ideas may be applied to sections of a measure by $$(n-m)$$-dimensional planes.