| The theory of iterated function systems(IFS)is first proposed by Hutchinson,it has become an effective method for constructing fractal sets.In 1986,Barnsely introduced the concept of fractal interpolation based on the theory of iterated function system.Compared with the traditional interpolation methods,fractal interpolation can use the continuous function which is smooth or non-smooth to fit a given data set.In the application of the IFS theory and fractal interpolation technique in practical problem,we often require the graph of fractal interpolation function(FIF)is located in a designated area.Hence the study of fractal interpolation problems in a restricted domain has some significance to fractal theory and applications.In this thesis,we study the interpolation problem in restricted domain of FIF.The thesis is organized as follows:In chapter 1,we mainly introduce the background and significance of the research and give some reviews of related literature at home and abroad,and meanwhile,we point out the innovation of this thesis.In chapter 2,some basic knowledge and important theories we will use in the thesis about fractal theory is briefly recalled.In chapter 3,we study a restrictive interpolation problem of univariate FIF in a general domain.We give conditions of vertical scaling factors so as to the FIF is restricted in a prescribed area.In addition,several numerical examples are given to illustrate the interpolation effect of change of the vertical scale factors.In chapter 4,the restrictive interpolation problem that studied in the univariate FIF is extended to the bivariate FIF.We investigate the changes of vertical scaling factors that how to influence the fractal interpolation functions.We give the conditions such that the fractal interpolation surface generated by iterated function system is located in a given region.Chapter 5 summarizes the result obtained in this thesis and points out some shortcomings,and further looks forward to the future work. |
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