Towards a More General Type of Univariate Constrained Interpolation With Fractal Splines

TL;DR

This paper develops a more general approach to univariate constrained fractal interpolation using rational cubic splines, enabling the interpolating curves to lie above or below given spline functions, with theoretical analysis and illustrative examples.

Contribution

It introduces strategies for parameter selection in rational fractal splines to achieve constrained interpolation relative to linear or quadratic splines, expanding previous methods.

Findings

Successfully achieves constrained interpolation with fractal splines.

Provides approximation error analysis using Peano kernel theorem.

Includes illustrative examples demonstrating the methods.

Abstract

Recently, in [Electronic Transaction on Numerical Analysis, 41 (2014), pp. 420-442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current article is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.