Moments of the weighted Cantor measures
Moments of the weighted Cantor measures
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加权康托测度的矩
DOI：
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发表时间：
2019
影响因子：
2
通讯作者：
Alexander W. N. Riasanovsky
中科院分区：
文献类型：
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作者：
Steven N. Harding;Alexander W. N. Riasanovsky
摘要在哈钦森开创性工作的基础上，研究了支撑为单位区间分形的α-加权Cantor测度的性质。这里，α是总和为1的非负权重的向量，并且对应的加权康托测度μα是[0，1]上的唯一Borel概率测度，满足μα（E）=∑n= 0 N-1αnμα（μ n-1（E））{mu ^alpha }（E）= sum
olimits_{n = 0}^{N - 1} {{alpha _n}{mu ^alpha }（varphi _n^{ - 1}（E））}其中n = x <$（x + n）/N。在第1节和第2节中，我们研究了Lμα2L_{mu ^alpha }}^2 [0，1]中测度μα和相应的勒让德多项式的几个一般性质。在第三节中，我们（1）计算了μα的Laplacian和矩母函数，（2）精确地刻画了矩Im =<$[0，1] xmd μα何时呈现多项式衰减或指数衰减，（3）描述了一个在O（（log log（1/ε））·mlog m）中估计一致误差ε内前m阶矩的算法。我们还陈述了在自然情况下的类似结果，其中α对于通过将μα移到[-1/2，1/2]而获得的测度να是回文的。
Abstract 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 μα is the unique Borel probability measure on [0, 1] satisfying μα(E)=∑n=0N-1αnμα(ϕn-1(E)){mu ^alpha }(E) = sum
olimits_{n = 0}^{N - 1} {{alpha _n}{mu ^alpha }(varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα 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 μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα 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 να attained by shifting μα to [−1/2, 1/2].