分形插值问题的递归全息方法与递归局息方法
发布时间：2018-11-27 11:55
【摘要】：本文主要研究分形插值问题(FIP),将递归迭代函数系(RIFS)与全息和局息分形插值方法相结合,给出递归全息和递归局息分形插值方法。引入了具有递归形式的离散数据组,定义了全息分形插值RIFS和局息分形插值RIFS,证明了全息与局息吸引子集组的存在唯一性;引入了递归分形插值函数空间、全息与局息分形插值变换,证明了全息与局息递归分形插值函数的存在唯一性。分析了递归局息分形插值方法与递归全息分形插值方法的关系,获得了全息与局息递归迭代函数系(RIFS)的共轭关系,以及全息和局息分形插值变换的共轭关系。给出递归局息递归分形插值函数与其伴随的递归全息递归分形插值函数的维数关系,证明了二者的豪斯道夫维数相等及二者的盒维数相等。提出分形插值维数的定义和计算方法,又给出了递归分形插值函数的绘图及计算维数的实例。递归全息与递归局息分形插值方法,给出了一种规范的形式体系,是对分形插值理论的完善和扩充;其可以灵活地处理局部与局部、局部与整体的相似关系,并构造出全新的具有多段互嵌套相似的递归分形插值函数,为非线性数学的研究和发展带来了新的结构模型和理论依据。
[Abstract]:In this paper, the problem of fractal interpolation is studied. (FIP), combines the recursive iterative function system (RIFS) with holographic and partial fractal interpolation methods, and gives the recursive holographic and recursive partial fractal interpolation methods. In this paper, a discrete data set with recursive form is introduced. The holographic fractal interpolation RIFS and the partial interest fractal interpolation RIFS, are defined to prove the existence and uniqueness of the set of holographic and partial interest attractive subsets. The space of recursive fractal interpolation function, holographic and partial fractal interpolation transformation are introduced, and the existence and uniqueness of holography and partial information recursive fractal interpolation function are proved. The relationship between the recursive partial fractal interpolation method and the recursive holographic fractal interpolation method is analyzed. The conjugate relationship between the holography and the partial information recursive iterative function system (RIFS), and the conjugate relation between the holography and the partial information fractal interpolation transformation is obtained. The dimension relationship between recursive local interest recursive fractal interpolation function and its associated recursive holographic recursive fractal interpolation function is given. It is proved that the Hausdorf dimension is equal and the box dimension is equal. The definition and calculation method of fractal interpolation dimension are put forward. The drawing of recursive fractal interpolation function and an example of calculating dimension are also given. The method of recursive holography and recursive partial fractal interpolation is presented, and a normal formal system is given, which is the perfection and extension of fractal interpolation theory. It can flexibly deal with the similarity between local and local, local and global, and construct a new recursive fractal interpolation function with multi-segment mutual nesting similarity. It provides a new structural model and theoretical basis for the research and development of nonlinear mathematics.
【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.42
本文编号：2360664
[Abstract]:In this paper, the problem of fractal interpolation is studied. (FIP), combines the recursive iterative function system (RIFS) with holographic and partial fractal interpolation methods, and gives the recursive holographic and recursive partial fractal interpolation methods. In this paper, a discrete data set with recursive form is introduced. The holographic fractal interpolation RIFS and the partial interest fractal interpolation RIFS, are defined to prove the existence and uniqueness of the set of holographic and partial interest attractive subsets. The space of recursive fractal interpolation function, holographic and partial fractal interpolation transformation are introduced, and the existence and uniqueness of holography and partial information recursive fractal interpolation function are proved. The relationship between the recursive partial fractal interpolation method and the recursive holographic fractal interpolation method is analyzed. The conjugate relationship between the holography and the partial information recursive iterative function system (RIFS), and the conjugate relation between the holography and the partial information fractal interpolation transformation is obtained. The dimension relationship between recursive local interest recursive fractal interpolation function and its associated recursive holographic recursive fractal interpolation function is given. It is proved that the Hausdorf dimension is equal and the box dimension is equal. The definition and calculation method of fractal interpolation dimension are put forward. The drawing of recursive fractal interpolation function and an example of calculating dimension are also given. The method of recursive holography and recursive partial fractal interpolation is presented, and a normal formal system is given, which is the perfection and extension of fractal interpolation theory. It can flexibly deal with the similarity between local and local, local and global, and construct a new recursive fractal interpolation function with multi-segment mutual nesting similarity. It provides a new structural model and theoretical basis for the research and development of nonlinear mathematics.
【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.42
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