A Study Of Related Properties For Hidden Variable Fractal Interpolation Functions

In1986，Barnsley put forward the concept of Fractal Interpolation Function(FIF)based on the theory of Iterated Function System(IFS). From then on, a newinterpolation method, fractal interpolation method has arisen. Compared withtraditional interpolation methods (such as polynomial interpolation, rationalinterpolation and spline interpolation), fractal interpolation method shows a great dealof advantages in fitting non-smooth, rough curves and irregular surfaces. The methodhas provided a new theoretic tool for data fitting, function approximation, computerapplication and other areas. Moreover, the method also shows great vitality insimulation of natural landscape, topographical geomorphology, signal processing andmany other pratical application areas. In order to improve the flexibility and diversityof fractal interpolation, Barnsley and Massopust introduced the concept of HiddenVariable Fractal Interpolation Function(HVFIF). HVFIF is the projection ofvector-valued FIF on some plane. It is still generated by a class of IFSs with greaterdegrees of freedom. FIF and HVFIF are usually continuous but non-differential, and itis difficult to describe their analysis properties with the classic calculus. Fractionalintegral is a strong tool for researching fractal function and hidden variable fractalinterpolation function. Since FIFs and HVFIFs are generated by IFSs, the freeparameters and functional terms in IFSs have important influences on FIFs, HVFIFsand their fractional integrals．It is important to consider the changes of FIFs，HVFIFsand some related properties with the changes of both the vertical scaling factors andfunctional terms.This dissertation introduces background of fractals and research significance,and simply recalls the related literature in chapter one. In chapter two, we simplyrecall some of the basic concepts, including the dimensions of fractal sets, definitionsof the IFSs, FIFs, HVFIFs and fractional integrals, and the theory and methods of thefractal geometry. In chapter three, the perturbation errors of FIFs, which are caused bythe joint perturbation of both the vertical scaling factors and functional terms in IFS,are investigated. An analytic expression and an upper estimate for the perturbationerrors of FIFs are presented. In addition, the upper bound of the errors for thefractional integrals of the corresponding FIFs is also given. The obtained results showthat the FIFs and their fractional integrals are not sensitive to the slight perturbation ofthe parameters of IFS. The error analysis for HVFIFs and their moments andRiemann-Liouville fractional integrals is made in chapert four. Firstly we investigate the perturbation errors of the corresponding HVFIFs when the free parameters andfunctional terms of the IFS have a small perturbation. An explicit expression and anupper estimate for the perturbation errors are obtained. Then, we discuss theperturbation errors of moments of the HVFIFs, and give an upper bound estimate ofthe errors. Findly, the upper bound of the errors for the fractional integrals of theHVFIFs is also given. In chapter five, we make a conclusion to our studies andprospect the future development of fractals.