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- W2968388315 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_nmu^alpha( varphi_n^{-1}(E) )$ where $varphi_n: xmapsto (x+n)/N$. 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 $I_m = int_{[0,1]}x^m,dmu^alpha$ exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first $m$ moments within uniform error $varepsilon$ in $O( (loglog(1/varepsilon))cdot mlog m )$. 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 $[-1/2,1/2]$." @default.
- W2968388315 created "2019-08-22" @default.
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- W2968388315 date "2019-08-14" @default.
- W2968388315 modified "2023-09-27" @default.
- W2968388315 title "Moments of the weighted Cantor measures." @default.
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