Fractal interpolation function (FIF) is a function which interpolates a given data set and whose graph is a fractal set. In 1986, M. F. Barnsley introduced the FIF using an iterated function system (IFS). Ever since, many researchers have studied the FIFs and applied them to a lot of scientific fields such as computer graphics, function approximation, signal processing, metallurgy, earth science, surface physics, geography, geology, medical science, chemistry, etc.
Rakotch contraction is the generalization of Banach contraction, which implies that the use of Rakotch’s fixed point theorem enables modeling of more objects than the use of Banach’s fixed point theorem. Moreover, hidden variable recurrent fractal interpolation functions (HVRFIFs) with Hölder function factors are more general than fractal interpolation functions (FIFs), recurrent FIFs and hidden variable FIFs with Lipschitz function factors.
Ro Chung Il, a researcher at the Faculty of Applied Mathematics, has demonstrated that HVRFIFs can be constructed using the Rakotch’s fixed point theorem, and investigated the analytical and geometric properties of those HVRFIFs.
Firstly, he constructed nonlinear hidden variable recurrent fractal interpolation functions with Hölder function factors on the basis of a given data set using Rakotch contractions. Next, he analyzed the smoothness of the HVRFIFs and showed that they are stable for the small perturbations of the given data. Finally, he obtained the lower and upper bounds for their box-counting dimensions.
If further information is needed, please refer to his paper “Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch’s fixed point theorem” in “Applied Mathematics and Computation” (SCI).
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