Construction And Properties Of The Iterated Function Systems Of Fractal Interpolation

On the basis of the theory of the iterated function systems (IFSs), Barnsley came up with the concept of the fractal interpolation functions(IFSs) in 1986. Contrasted with traditional interpolation methods, fractal interpolation shows special advantages in the process of fitting nonsmooth and irregular phenomena and objects, and has become a powerful tool for the simulation and portrayal of the objects with a certain self-similar which are widespread in nature. Fractal interpolation theory and its applications have been involved in many fields now.In this treatise, based on the basic theory and methods of fractal interpolation, a series of the IFSs of fractal interpolation with more flexibility is constructed. The IFSs cover not only the polynomial functions of order 1 or 2, but a more general form, that is, they consist of some Lipschitz functions, and include arbitrary polynomial functions as special case. The detailed are organized as follows:In chapter I, a brief introduction of the background about the fractal and fractal interpolation is given.In chapter II, a class of iterated function systems of fractal interpolation with more elasticity is constructed on R2. We prove that the attractors of the class of IFSs are the fractal interpolation curves passing through the interpolation points which selected before, and study the continuous dependence of the FIFs with respect to free parameters. The changes of the corresponding FIFs are discussed when this kind of iterated function systems has a small perturbation, and the error estimate is caculated.In chapter III, the smoothness and stability of the FIFs construed are investigated, and the results of the former characteristic are presented. Their stability in a general interpolation case (not required equidistant) is also discussed.In chapter IV, we generalize the fractal interpolation iterated function systems from R2 to R3. We constructe a class of IFSs of the bivariate fractal interpolation and prove that the attractors of this class of IFSs are the fractal interpolation surfaces passing through the interpolation points which selected before. We disscuss the continuous dependence of these FIFs with respect to free parameters. The errors between the FIFs generated by the perturbed IFS and the FIF generated by the per-disturbance IFS are established. Finally, the error estimate of s-FIF to the corresponding continuous function in the sense of two norms is discussed.In chapter V, a class of the IFS of fractal interpolation with function vertical scaling factors is constructed on R3. We prove that the attractors of this class of IFSs are the fractal interpolation surfaces passing through the interpolation points which selected before, and the error estimate between the FIF generated by the perturbed IFS and the FIF generated by the per-disturbance IFS is given. Finally, the sensitivity analysis of the class of IFSs with respect to functions and function vertical scaling factors is made.In chapter Ⅵ, we make a summary of the full text and give a prospect.