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Fractal Interpolation is a novel method to construct irregular functions from interpolation data. In this talk, I shall talk on different kinds of Fractal Interpolation Functions and show a few results that I have derived in each kind. The talk will begin with a brief introduction of Fractals. The first result will be on Fractal Interpolation Function (FIF) giving the regularity of the same. Then, I will introduce a special kind of Fractal Interpolation Function called “Super Fractal Interpolation Function” (SFIF) for finer simulation of the objects of the nature or outcomes of scientific experiments that reveal one or more structures embedded into another (i.e. hybrid structures). Further, I will discuss properties of integrability, differentiability of an SFIF and convergence of a Cubic Spline SFIF. To simulate fractal curves and surfaces found in nature or occurring as outcomes of scientific experiments whose graphs exhibit partly self-affine and partly non-self-affine nature, the notions of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) and Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) are employed. The insertion of a new point in a given set of interpolation data is called the problem of node insertion. The effect of insertion of new point on the related IFS and the Coalescence Hidden-variable Fractal Interpolation Function will be discussed. Smoothness and Fractal Dimension of a CHFIF obtained with a node will also discussed. Next, to reconstruct the functions in L2(R) at desired level of resolution, the multiresolution analysis of L2(R) based on CHFIFs will be discussed. After that, smoothness, stability and bounds of fractal dimension of CHFIS will be discussed. In the end, I shall discuss my future plans. |