New York Journal of Mathematics
Volume 13 (2007) 175-198


Terry A. Loring

Rényi dimension and Gaussian filtering

Published: July 17, 2007
Keywords: Rényi dimension, fractal, regular variation, least squares, Laplacian pyramid, convolution, Gaussian, Matuszewska indices
Subject: 28A80, 28A78


Consider the partition function Sμq(ε) associated in the theory of Rényi dimension to a finite Borel measure μ on Euclidean d-space. This partition function Sμq(ε) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width ε. We further Guérin's investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of μ against an approximate identity of Gaussians. We prove a Lipschitz-type estimate on the partition function. This bound on the partition function leads to results regarding the computation of Rényi dimension. It also shows that the partition function is of O-regular variation.

We find situations where one can or cannot replace the partition function by a discrete version. We discover that the slopes of the least-square best fit linear approximations to the partition function cannot always be used to calculate upper and lower Rényi dimension.


This work was supported in part by DARPA Contract N00014-03-1-0900.

Author information

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA