 

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 


Abstract
Consider the partition function S_{μ}^{q}(ε) associated
in the theory of Rényi dimension to a finite Borel measure μ
on Euclidean dspace. This partition function S_{μ}^{q}(ε)
is the sum of the qth powers of the measure applied to a partition
of dspace into dcubes of width ε. We further Guérin's
investigation of the relation between this partition function and
the Lebesgue L^{p} norm (L^{q} norm) of the convolution of μ
against an approximate identity of Gaussians. We prove a Lipschitztype
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 Oregular variation.
We find situations where one can or cannot replace the partition
function by a discrete version. We discover that the slopes of the
leastsquare best fit linear approximations to the partition function
cannot always be used to calculate upper and lower Rényi dimension.


Acknowledgements
This work was supported in part by DARPA Contract N000140310900.


Author information
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
loring@math.unm.edu
http://www.math.unm.edu/~loring/

