A Class Of Fractal Interpolation Function Of Exponential Type

Barnsley, the mathematician of American, first proposed the concept of fractal interpolation function based on iterated function system theory, provide a new method of non-smooth curve and surface fitting, and achieved great success. The iterated function system in the form of Fij(x,y,z)=ψ(z)·φ(x,y) is mainly discussed in this paper, and constructed a class of multi-parameter exponential function system, as follows:Firstly, we review the generation and development of fractal theory, and summarize the study of this topic status and the main contents and innovation;Secondly, this paper introduce several common dimensions, iterated function systems, fractal interpolation theory, and the box-dimension of fractal interpolation curves;Thirdly, we constructed a class of iterated function system in the form of Fij(x,y,z)=ψ(z)·(p(x,y), proved its attractor is the image of a continuous function under the condition of collinear of interpolation points. Through the discussion of the nature of variation, we estimated the upper bound of dimension;Finally, a class of exponential iterated function systems with multi-parameter is discussed in this paper. And prove that the existence of its attractor under certain conditions, its attractor is the image of a fractal function, and then discuss the continuous of the function dependence on the parameters, give the formula of dimension. The case that practical data are used to fractal interpolation surface, providing a theoretical basis for the study of complex geometry.