Constrained shape preserving rational cubic fractal interpolation functions

Abstract

In this paper, we discuss the construction of $\mathcal {C}^1$-rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The $\mathcal {C}^1$-RCFIF is the fractal design of the traditional rational cubic interpolant of the form ${p_i(\theta )}/{q_i(\theta )}$, where $p_i(\theta )$ and $q_i(\theta )$ are cubic and quadratic polynomials with three tension parameters. We present the error estimate of the approximation of RCFIF with the original function in $\mathcal {C}^k[x_1,x_n]$, $k=1,3$. When the data set is constrained between two piecewise straight lines, we derive the sufficient conditions on the IFS parameters of the RCFIF so that it lies between those two lines. Numerical examples are given to support the theoretical results.