分形插值函数的可微性

发布时间：2017-12-27 21:38

本文关键词：分形插值函数的可微性　出处：《江苏大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文首先从特殊的插值点入手讨论了一种分形插值函数的可微性的问题。这种分形插值函数在一定条件下具有较好的光滑性,文中详细的讨论了纵向尺度因子在给定区间范围内该分形插值函数的可微性问题。其次介绍了一类具有两个参数的分形插值函数,并证明了该分形插值函数有Lipschitz连续性所具有的条件。通过对比发现具有双参数的分形插值函数比M.F.Barnsley定义的仿射FIF在实际生活中应用更具有灵动性,并且可以更好的拟合实验数据。最后介绍了一类具有两个参数多项式分形插值构造,通过调节两个参数的值来控制分形插值曲线的形状,探讨了此类分形插值函数是三次多项式的条件。并推广了n次多项式分形插值的构造,这类分形插值函数是处处可微的。本文共有五章内容如下:第一章阐述了分形几何研究的背景,介绍了目前分形的研究现状,并且给出了本文研究的内容和创新点。第二章介绍了分形插值函数的微积分和可微性相关预备知识。第三章从特殊的插值点入手证明了一种分形插值函数的可微性,构造了一类双参数分形插值函数。由于双参数的灵活性,证明了分形插值函数在给定区间上是可以生成一类三次多项式。第四章讲述了一种多项式的分形插值构造,利用迭代构造原理采用两个参数来构造多项式,并进行了实例对比,在相同插值节点条件下双参数分形构造更具灵活性。第五章是总结与展望。
[Abstract]:In this paper, the problem of the differentiability of a fractal interpolation function is discussed from a special interpolation point. This fractal interpolation function has good smoothness under certain conditions. In this paper, we discuss in detail the differentiability of the fractal dimension interpolation function in a given interval. Secondly, a class of fractal interpolation functions with two parameters is introduced, and the conditions for the fractal interpolation function to have Lipschitz continuity are proved. By comparison, it is found that the fractal interpolation function with two parameters is more intelligent than the affine FIF defined by M.F.Barnsley in real life, and it can better match the experimental data. Finally, a class of fractal interpolation construction with two parameter polynomials is introduced. By adjusting the values of two parameters, we can control the shape of fractal interpolation curves, and discuss the condition that such fractal interpolation functions are three polynomial. The construction of the fractal interpolation of N sub polynomials is also generalized. This kind of fractal interpolation function is differentiable everywhere. There are five chapters in this paper. The contents are as follows: the first chapter describes the background of fractal geometry research, introduces the current research status of fractal, and gives the content and innovation of this paper. In the second chapter, the calculus of fractal interpolation functions and the related preparatory knowledge of differentiability are introduced. In the third chapter, the differentiability of a fractal interpolation function is proved by the special interpolation points, and a kind of double parameter fractal interpolation function is constructed. Because of the flexibility of the two parameters, it is proved that the fractal interpolating function can generate a class of three - order polynomials on a given interval. The fourth chapter describes the construction of a polynomial fractal interpolation, constructs two polynomials based on iterative construction principle, and compares it with examples. Under the same interpolation nodes, the two parameter fractal construction is more flexible. The fifth chapter is the summary and prospect.
【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

【参考文献】