A New Class of Rational Quadratic Fractal Functions with Positive Shape Preservation

Abstract

Fractal interpolation functions (FIFs) developed through iterated function systems prove more general than their classical counterparts. However, the theory of fractal interpolation functions in the domain of shape preserving interpolation is not fully explored. In this paper, we introduce a new kind of iterated function system (IFS) involving rational functions of the form \(\frac{p_{n}(x)} {q_{n}(x)}\), where p n (x) are quadratic polynomials determined through the interpolation conditions of the corresponding FIF and q n (x) are preassigned quadratic polynomials involving one free shape parameter. The presence of the scaling factors in our rational FIF adds a layer of flexibility to its classical counterpart and provides fractality in the derivative of the interpolant. The uniform convergence of the rational quadratic FIF to the original data generating function is established. Suitable conditions on the rational IFS parameters are developed so that the corresponding rational quadratic fractal interpolant inherits the positivity property of the given data.KeywordsFractalsIterated function systemsFractal interpolation functionsRational quadratic FIFUniform convergencePositivity