一类分形插值函数的最大值问题

发布时间：2018-03-23 19:11

本文选题：迭代函数系　切入点：分形插值函数　出处：《江苏大学》2017年硕士论文

【摘要】：“如何找到分形插值曲线的最高点”是一个在实际应用中经常遇到的基本问题。本文基于迭代函数系的相关理论,从迭代的过程入手,研讨一类分形插值函数的最大值问题。首先构造一个特定的迭代函数系,其次讨论n次迭代之后得到的分形插值函数上的离散点横纵坐标的表示方法。最后按照纵向尺度因子变化,依次给出了不同情形下的该分形插值函数的最大值以及最大值点的分布情况。为实际应用中寻找分形插值曲线的最高点供了一条新的思路。第一章介绍了本文的研究背景、国内国外目前的研讨进展、以及本文探求的主要内容和创新点。第二章回忆了分形学科的基本常识,包含分形集的定义、Cantor三分集、以及迭代函数系(IFS)、分形插值函数(FIF),并介绍了分形插值函数的维数公式。第三章首先对n次迭代之后得到的FIF上的离散点进行讨论,给出了其横纵坐标的表示方法,以此为基础又得到了一类二次函数的分形插值表示,并给出证明。第四章基于纵向尺度因子对分形插值函数的影响,将该类分形插值函数分成几种不同情形,从迭代的过程入手,先讨论n次迭代之后FIF上数值最大的离散点的迭代规律,进一步求出了该类分形插值函数的最大值,最后得到其最大值点的分布情况。第五章对文章的研究内容做了总结,并对该内容的深入研究出了一些后续思考。
[Abstract]:"how to find the highest point of fractal interpolation curve" is a basic problem often encountered in practical application. In this paper, we discuss the maximum value problem of a class of fractal interpolation functions. Secondly, the representation method of the vertical coordinates of discrete points on the fractal interpolation function obtained after n-th iteration is discussed. Finally, according to the variation of the longitudinal scale factor, The maximum value of the fractal interpolation function and the distribution of the maximum point in different cases are given in turn. A new idea for finding the highest point of the fractal interpolation curve in practical application is provided. The first chapter introduces the research background of this paper. The research progress at home and abroad, as well as the main contents and innovations of this paper. Chapter two recalls the basic knowledge of fractal science, including the definition of fractal set and Cantor triple diversity. This paper also introduces the dimension formula of fractal interpolation function. In chapter 3, the discrete points on FIF obtained after n iterations are discussed, and the expression method of its ordinal coordinates is given. On this basis, the fractal interpolation representation of a class of quadratic functions is obtained, and the proof is given. In chapter 4, the fractal interpolation functions are divided into several different cases based on the influence of the longitudinal scaling factors on the fractal interpolation functions. Starting with the iterative process, the iterative law of the largest discrete point on FIF after n iterations is discussed, and the maximum value of this kind of fractal interpolation function is obtained. Finally, the distribution of the maximum point is obtained. Chapter five summarizes the research content of the paper, and gives some follow-up thoughts on the further study of the content.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.42;O189

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