Perturbation Properties For Several Types Of Fractal Interpolation Functions

Fractal interpolation functions (FIFs) are generated by a special class of iteratedfunction systems (IFSs), they show a distinct superiority in the fitting of the curvesand surfaces which are continuous everywhere but unsmooth and widely exist innature. First, in this thesis we make a brief introduction to the generation, thedevelopment, and the research status in the domestic and foreign for fractalinterpolation functions. Then, we elaborate the basic concepts and some fundamentaltheorems of the iterated function system and interpolation function. The properties ofseveral types of fractal interpolation functions and their moment integrals are mainlystudied in this thesis. Main contents and main conclutions are as follows:First of all, for any given bivariate continuous function, anα-fractal function inthe three-dimensional space is generated with the iterated function system of fractalinterpolation. The conditions of the construction ofα-fractal function are given. Theerrors between theα-fractal function and the corresponding continuous function inthe senses ofL2α-norm and infinity norm are estimated. The error of their momentsare also studied, and the error upperbound is obtained. The results show that theseerrors have a certain relationship with vertical scaling factors.Secondly, the sensitivity of a class ofα-fractal interpolation functions (α-FIFs)with function vertical scaling factors, including the sensitivity of their moments, arestudied. We show that the IFSs with function vertical scaling factors and theirperturbed IFSs have an unique attractor, respectively. A specific example is presented.It is shown that if a small perturbation occurs in the IFSs generating the class ofα-FIFs, then a small perturbation is also produced in the correspondingα-FIFs andtheir moments. The estimation of the perturbation errors are given.Thirdly, we discuss the sensitivity of a class of fractal interpolation function withfunction hidden variable and their moments. Based on a well-known class of vectorvalued IFSs, we introduce a class of IFSs with function hidden variable and prove thatthey have an unique attractor under certain conditions. Next we investigate therelations between the FIFs and their perturbed FIFs. Furthermore, we compute theerror of moments of the two FIFs, and an estimate of the perturbation errors isobtained. The results show that all perturbed parameters have a certain impact on theerrors.