Rational Cubic Fractal Interpolation Functions Research Articles

Constrained univariate and bivariate rational fractal interpolation

We study constrained natures of a new class of univariate and bivariate rational cubic fractal interpolation functions (RCFIFs). We derive the convergence results of the RCFIF towards an original function in In particular, when data lies (i) between two piecewise defined lines (ii) within a rectangle, we derive sufficient condition based on the restrictions of IFS parameters at fewer discretized values so that the corresponding RCFIF preserves the inherent property associated with constrained data. Using transfinite interpolation via blending functions, we extend constrained aspects to rational bivariate RCFIFs that lie above a piecewise plane and within a cuboid.

Constrained shape preserving rational cubic fractal interpolation functions

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.