A New Class of Monotone/Convex Rational Fractal Function

This paper presents a description and analysis of a rational cubic spline FIF (RCSFIF) that has two shape parameters in each subinterval when it is defined implicitly. To be precise, we consider the iterated function system (IFS) with $q_n=\frac{P_n}{Q_n}$, $n \in \mathbb{N}_{N-1}$, where $P_n(x)$ are cubic polynomials to be determined through interpolatory conditions of the corresponding FIF and $Q_n(x)$ are preassigned quadratic polynomials each containing two free shape/rationality parameters. We establish the convergence of the proposed RCSFIF $g$ to the original function $\Phi \in \mathcal{C}^3(I)$ with respect to the uniform norm. We also provide the sufficient conditions for an automatic selection of the rational IFS parameters to preserve monotonicity and convexity of a prescribed set of data points. We consider some examples to illustrate the developed fractal interpolation scheme and its shape preserving aspects.Comment: 18 Pages, 18 Figures. arXiv admin note: text overlap with arXiv:1809.0820

Parameter Identification of Constrained Data by a New Class of Rational Fractal Function

This paper sets a theoretical foundation for the applications of the fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified befittingly so that the graph of the resulting $\mathcal{C}^1$-RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the $\mathcal{C}^1$-RCSFIF. The problem of visualization of constrained data is also addressed when the data is lying above a straight line, the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examplesComment: 16 pages, 9 Figures. Presented by Sangita Jha at International Conference on Mathematics and Computing, Haldia, January 17-21, 201