  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.

Functional Analysis