Volume 41, pp. 420-442, 2014.

$\alpha$-fractal rational splines for constrained interpolation

Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand

Abstract

This article is devoted to the development of a constructive approach to constrained interpolation problems from a fractal perspective. A general construction of an $\alpha$-fractal function $s^{\alpha }\in {\mathcal{C}}^p,$ the space of all $p$-times continuously differentiable functions, by a fractal perturbation of a traditional function $s\in {\mathcal{C}}^p$ using a finite sequence of base functions is introduced. The construction of smooth $\alpha$-fractal functions described here allows us to embed shape parameters within the structure of differentiable fractal functions. As a consequence, it provides a unified approach to the fractal generalization of various traditional non-recursive rational splines studied in the field of shape preserving interpolation. In particular, we introduce a class of $\alpha$-fractal rational cubic splines $s^{\alpha }\in {\mathcal{C}}^1$ and investigate its shape preserving aspects. It is shown that $s^{\alpha }$ converges to the original function $\mathrm{\Phi }\in {\mathcal{C}}^2$ with respect to the ${\mathcal{C}}^1$-norm provided that a suitable mild condition is imposed on the scaling vector $\alpha$. Besides adding a layer of flexibility, the constructed smooth $\alpha$-fractal rational spline outperforms its classical non-recursive counterpart in approximating functions with derivatives of varying irregularity. Numerical examples are presented to demonstrate the practical importance of the shape preserving $\alpha$-fractal rational cubic splines.

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Key words

iterated function system, $\alpha$-fractal function, rational cubic spline, convergence, convexity, monotonicity, positivity

AMS subject classifications

28A80, 26A48, 26A51, 65D07, 41A20, 41A29, 41A05

Links to the cited ETNA articles

[26]	Vol. 20 (2005), pp. 64-74 M. A. Navascues: Fractal trigonometric approximation

ETNA articles which cite this article

Vol. 44 (2015), pp. 639-659 Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand: Monotone-Comonotone approximation by fractal cubic splines and polynomials

Vol. 51 (2019), pp. 1-14 Vijender Nallapu: Bernstein fractal approximation and fractal full Müntz theorems