【摘要】：本文主要研究分形插值問題(FIP),將遞歸迭代函數(shù)系(RIFS)與全息和局息分形插值方法相結(jié)合,給出遞歸全息和遞歸局息分形插值方法。引入了具有遞歸形式的離散數(shù)據(jù)組,定義了全息分形插值RIFS和局息分形插值RIFS,證明了全息與局息吸引子集組的存在唯一性;引入了遞歸分形插值函數(shù)空間、全息與局息分形插值變換,證明了全息與局息遞歸分形插值函數(shù)的存在唯一性。分析了遞歸局息分形插值方法與遞歸全息分形插值方法的關(guān)系,獲得了全息與局息遞歸迭代函數(shù)系(RIFS)的共軛關(guān)系,以及全息和局息分形插值變換的共軛關(guān)系。給出遞歸局息遞歸分形插值函數(shù)與其伴隨的遞歸全息遞歸分形插值函數(shù)的維數(shù)關(guān)系,證明了二者的豪斯道夫維數(shù)相等及二者的盒維數(shù)相等。提出分形插值維數(shù)的定義和計(jì)算方法,又給出了遞歸分形插值函數(shù)的繪圖及計(jì)算維數(shù)的實(shí)例。遞歸全息與遞歸局息分形插值方法,給出了一種規(guī)范的形式體系,是對(duì)分形插值理論的完善和擴(kuò)充;其可以靈活地處理局部與局部、局部與整體的相似關(guān)系,并構(gòu)造出全新的具有多段互嵌套相似的遞歸分形插值函數(shù),為非線性數(shù)學(xué)的研究和發(fā)展帶來了新的結(jié)構(gòu)模型和理論依據(jù)。
[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.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O174.42

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本文編號(hào)：2360664