The Research On Fractal Interpolation Functions

Fractal interpolation function(FIF)is a special type of continuous function which is produced by iterated function system(IFS).We’re able to take advantage of fractal interpolation surface(FIS)which is produced by IFS or recursive iterated function system(RIFS)to fitting all kinds of curves and surfaces in nature.For the FIF,there are plenty of papers give the construction of multivariate FIF and study its existence,uniqueness and so on and obtain many results,as a result of which,we could study FIS.In this article,we discuss the relationship between(?)Ni*(ε)and N*(ε)which is a necessary step in the whole process of achieving the box dimension of affine FIF.For the FIS,which is the attractor of the IFS([0,1]× R,ωi),the box dimension of the FIS is given by the formula dimB(G)= 2+log |d|/log N in paper[2].We prove a accurate relationship between(?)Ni*(ε)and N*(ε)by using the cover of ε columns with scaling appropriately and fill the proof of the whole estimation formula.On the one hand,the properties such as existence,uniqueness of multivariate FIF is studied.On the other hand,we also study the Hausdorff dimension of a class of generalized Cantor Set and some results are obtained.This article is divided into five chapters and the structure is as follows:Chapter 1 Introduction.We introduce the background and current situation of FIF.Chapter 2 Preparatory knowledge where gives the relevant preparatory knowledge and concepts of related FIF and the cover of ε cohrmns according to the problems that we studied.Chapter 3 proves an essential theory when it comes to the complete proof of the box-counting dimension theory of affine fractal interpolation functions.Rather than(?)Ni*(ε)≈N*(ε),a more accurate relationship between(?)Ni*(ε)and N*(ε)is given,which is based on the study on the cover of ε cohrmns.The main result is obtained:Theorem 3.5 If(?)|dk|>1,then there exists constants η,ζ>0,for any 0<ε<(?){xi-xi-1},we haveChapter 4 continues to study multivariate FIF.In this chapter,we study its existence,uniqueness and so on and obtain many results.Theorem 4.1 There exists a continuous function:f:I1×I2×…×IM→R such that f(x1,x2,…,xM)=yilij…iM and G=(?)Wilij…iM(G),where G=Graph(f)={(x1,x2,…,xM,f(x1,x2,…,xM))|(x1,x2,…,xM)∈I1×I2×…×IM} is the graph of f.G is the invariant set with respect to the IFS{K;Wilij…iM,ij=1,2,…,Nj}.Theorem 4.2 Let f be the fractal interpolation function with reapect to the IFS{K;Wilij…iM,ij=1,2,…,Nj} and G=Graph(f)={(x1,…,xM,f(x1,…,xM))|(x1,…,xM)∈I1×I2×…×Im}.hen for any A ∈ H(K),we have(?)h(WR(A),G)=0.f is a unique function.Chapter 5 is the summary and prospect.In this article,we prove a more accurate relationship between(?)Ni*(ε)and N*(ε).We also obtain two result about multivariate FIF and study the Hausdorff dimension of a class of generalized Cantor Set.We summarize up the work of this article and put forward some questions that needed further exploration.