Renyi Dimension and Gaussian Filtering II

Abstract

We consider convolving a Gaussian of a varying scale epsilon against a Borel measure mu on Euclidean delta-dimensional space. The Lq norm of the result is differentiable in epsilon. We calculate this derivative and show how the upper order of its growth relates to its lower Renyi dimension. We assume q is strictly between 1 and infty and that mu is finite with compact support. Consider choosing a sequence epsilon_n of scales for the Gaussians. The differences between the usual Lq norms at adjacent scales can be made to grow more slowly than any positive power of n by setting the epsilon_n by a power rule. The correct exponent in the power rule is determined by the lower Renyi dimension. We calculate and find bounds on the derivative of the Gaussian kernel versions of the correlation integral. We show that a Gaussian Kernel version of the Renyi entropy sum in continuous.