We prove the following result on the generalized fractal dimensions Dq±
of a probability measure μ on n.
be a complex-valued measurable function on n
satisfying the following conditions: (1) g
is rapidly decreasing at infinity, (2) g
is continuous and nonvanishing at (at least) one point, (3) ∫ g ≠ 0.
Define the partition function Λa(μ,q) = an(q−1)‖ga ∗ μ,
where ga(x) = a−ng(a−1x)
and ∗ is the convolution in n.
Then for all q>1
we have Dq± = 1/(q−1)limr→0 infsup[log Λaμ(r,q)/log r].
© 2001 American Institute of Physics.