Renyi Dimension and Gaussian Filtering

Abstract

Consider the partition function S(\epsilon) associated in theory of Renyi dimension to a finite Borel measure \mu on Euclidean d-space. This partion function S(\epsilon) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width \epsilon. We further Guerin's investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of \mu against an approximate identity of Gaussians. We prove a Lipschitz-type esimate on the partition function. This bound on the partition function leads to results regarding the computation of Renyi dimension. It also shows that the partion function is of O-regular variation. We find situtations 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 partion function cannot always be used to calculate upper and lower Renyi dimension.