A Study Of Some Analysis Properties For Fractal Interpolation Functions

The iterated function system (IFS) has become a powerful tool for the constructions as well as the analysis of the fractal sets. The ideas of IFS were first put forward by Hutchinson in 1982. After Barnsley's works, the theory of IFS has been perfecting and developing. So far, theory and applications for IFS have become an important research field in fractals, and the achievements of study are greatly enriched. The fractal interpolation functions (FIFs) are the attractors of a special class of IFSs. The free parameters, vertical scaling factors, in IFSs have important influences on FIFs.So, it is worthy of considering and exploring for studying the influence of FIFs on the changes of vertical scaling factors. In addition, the research of analysis properties for FIFs has important significance to the real applications of FIFs.This dissertation consists of four chapters. We simply recall some of the basic concepts, theorems and methods of the fractal geometry, including the definitions of the dimensions of fractal sets, the theory and methods etc. of IFSs and FIFs in Chapter one. In Chapter two, we study the error analysis for bivariate FIFs based on the perturbations of vertical scaling factors. Under certain conditions, we analyze quantitatively the error problem between the FIF generated by the perturbed IFS and the FIF generated by the original IFS. The explicit error expressions are presented, and the upper bound estimate for the error of moments of the two FIFs is obtained. Numerical experiments illustrate the change relations between the vertical scaling factors and the graphs of FIFs. In Chapter three, FIFs with variable parameter vertical scaling factors are constructed and their analysis properties are researched. Based on the traditional IFSs, we introduce a class of IFSs with variable parameter vertical scaling factors. Under some conditions, this kind of IFSs determine a class of FIFs with variable parameter vertical scaling factors. From a theoretical point of view, we research the problem of errors caused by vertical perturbations of interpolation points for this kind of FIFs, and the upper bound of errors is obtained.In addition, the sensitivity of FIFs with variable parameter vertical scaling factors is investigated, meanwhile, the upper bound estimate for the error of moments of the two FIFs is obtained. Numerical experiments show that this kind of FIFs are stable about small vertical perturbations. We believe that the theoretical results on the properties of FIFs will be useful in the applications of the class of FIFs. In Chapter four, We make a conclusion to our studies and prospect the future development of fractals.