A note on the generalized fractal dimensions of a probability measure

We prove the following result on the generalized fractal dimensions D-q(+/- ) of a probability measure mu on R-n. Let g be a complex-valued measurable function on R-n satisfying the following conditions: (1) g is rapidly decre asing at infinity, (2) g is continuous and nonvanishing at (at least) one p oint, (3) integralg not equal0. Define the partition function Lambda (a)(mu ,q)=a(n(q-1))parallel tog(a)*mu parallel to (q)(q), where g(a)(x)=a(-n)g(a (-1)x) and * is the convolution in R-n. Then for all q>1 we have D-q(+/-)=1 /(q-1)lim(r-->0) (sup)(inf)x[log Lambda (a)mu (r,q)/log r]. (C) 2001 Americ an Institute of Physics.