Box Dimension And Applications Of Fractal Interpolation Functions

This thesis mainly includes two topics.1.Box dimension of bilinear fractal interpolation surfaces.Barnsley and Massopust con-structed one dimensional bilinear fractal interpolation functions in 2015,and obtained their box dimensions under certain constraint conditions.Ruan and Xu introduced a framework to construct fractal functions on a rectangular grid in 2015,and constructed two dimensional bilinear fractal interpolation functions,while they did not obtain the box dimension of bilinear fractal interpola-tion functions.By using the method of Barnsley and Massopust,Zhang obtained a partial result of the box dimensions of bilinear fractal interpolation surfaces under some special hypotheses.By using a similar method to the classical method in Falconer's book in 1990,as well as the method of Ruan,Su and Yao in 2009,we obtain box dimensions of bilinear fractal interpolation surfaces in a more general case.Basically,our method is to estimate the total sum of oscillations in small area of a function.Our method is more straightforward and more effective than the method by Barnsley and Massopust.In fact,our method can be used to present a more concise proof of box dimensional formula for one dimensional fractal interpolation function with xi=i/N.2.Application of fractal interpolation function.Kantelhardt introduced multifractal detrend-ed fluctuation analysis in 2002.This method has been widely applied to complex systems.In this thesis,we present a new method called multifractal detrended fluctuation analysis based on pseudo-bilinear fractal interpolation functions.We also present another new method called mul-tifractal detrended fluctuation analysis based on recurrent fractal interpolation functions.Using these methods,we analyze the characteristics of metallic glass and gold price respectively.Final-ly,we generalize a new method called mixed multifractal analysis,and we detect the main factors of gold market.