某类Hausdorff测度的柯西变换的泰勒系数估计

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英文题名：The Estimation of Taylor Coefficients of a Class of Cauchy Transform of Hausdorff Measure 作者：李红光 论文级别：硕士 学科专业名称：基础数学 中文关键词：Cauchy变换 ; 泰勒系数 ; Hausdorff测度 ; 吸引子 ; 自相似集 ; 自相似测度 英文关键词：Cauchy transform ; Taylor coefficient ; Hausdorff measure ; Attractor ; Self-similar set ; Self-similar measure 学位年度：2006 导师：董新汉 学科代码：070101 学位授予单位：湖南师范大学 论文提交日期：2006-03-01

摘要

在这篇文章中,我们主要考虑某类特殊的柯西变换F(z),研究它们的泰勒系数的渐近表示。假设{S_j}_(j=0)~(q-1)是由压缩映射组成的迭代函数系(IFS),其中01内的罗朗系数。本文首先决定了F(z)在z=0的邻域内的解析半径R_q∶当q=2m时,R_(2m)=1-2p,当q=2m+1时,然后研究了F(z)在|z|In this thesis, we consider mainly a class of special Cauchy transform F(z), and study the asymptotic behavior of its Taylor coefficients . Let the iterated function system(IFS) {S_j}_(J=0)~(q-1) be of the formwhere 0 < ρ ≤ ρ_q(q ≥ 4,ρ_q is defined in [1]). Let K be the attractor of {S_j}_(J=0)~(q-1), and μ be Hausdorff measure of surpport on K. The function F(z) = ∫_K(z - w)~(-1)dμ(w) is called Cauchy transform of μ. Recently, the paper [2] has studied the Laurent coefficients of F(z) in |z| > 1. In this paper, we first give analytic radius R_q of F(z) in the neighborhood of z = 0: if q = 2m, R_(2m) = 1-2ρ and if q = 2m + 1,then we study the Taylor expansion of F(z) in |z| < R_q and give asymptotic expression of Taylor coefficients,which is always connected with a multiplicative periodic function. In the other part of the thesis ,we study the properties of these multiplicative periodic functions, getting their analytic scopes and eliminating the measure in integrals and also expressing respectively them as infinite product of a elementary function.

引文

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