Moments of the weighted Cantor measures

[摘要] Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μ α is the unique Borel probability measure on [0, 1] satisfying μα(E)=∑n=0N-1αnμα(ϕn-1(E)){\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕ n : x ↦ ( x + n )/ N . In Sections 1 and 2 we examine several general properties of the measure μ α and the associated Legendre polynomials in Lμα2L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μ α , (2) characterize precisely when the moments I m = ∫ [0,1] x m dμ α exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O ((log log(1/ ε )) · m log m ). We also state analogous results in the natural case where α is palindromic for the measure ν α attained by shifting μ α to [−1/2, 1/2].

[学科分类] 外科医学

[关键词] Cantor;moments;orthogonal polynomials;generating function;iterated function system