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

TL;DR

This paper develops a theoretical framework for rational cubic spline fractal interpolation functions with shape parameters, enabling constrained data fitting within specified bounds and addressing visualization constraints.

Contribution

It introduces a new class of rational cubic spline FIFs with shape parameters and conditions for data constraints, expanding the applicability of fractal interpolation.

Findings

Constructed RCSFIFs with quadratic denominator and shape parameters.

Provided conditions for positivity and data constraints.

Demonstrated effectiveness through numerical examples.

Abstract

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 examples