Moments of the weighted Cantor measures

Abstract

Based on the seminal work of Hutchinson, we investigate properties of {\em $\alpha$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $\alpha$ is a vector of nonnegative weights summing to $1$, and the corresponding weighted Cantor measure $\mu^\alpha$ is the unique Borel probability measure on $[0,1]$ satisfying $ \mu^\alpha(E) = \sum_{ n=0 }^{N-1} \alpha_n\mu^\alpha( \varphi_n^{-1}(E) )$ where $\varphi_n: x\mapsto (x+n)/N$. In Sections 1 and 2 we examine several general properties of the measure $\mu^\alpha$ and the associated Legendre polynomials in $L_{\mu^\alpha}^2[0,1]$. In Section 3, we (1) compute the Laplacian and moment generating function of $\mu^\alpha$, (2) characterize precisely when the moments $I_m = \int_{[0,1]}x^m\,d\mu^\alpha$ exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first $m$ moments within uniform error $\varepsilon$ in $O( (\log\log(1/\varepsilon))\cdot m\log m )$. We also state analogous results in the natural case where $\alpha$ is {\em palindromic} for the measure $\nu^{\alpha}$ attained by shifting $\mu^{\alpha}$ to $[-1/2,1/2]$.