\(\alpha\)-fractal rational splines for constrained interpolation

Viswanathan, Puthan Veedu ; Chand, Arya Kumar Bedabrata

Electronic transactions on numerical analysis, Tome 41 (2014), pp. 420-442

Résumé

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 $\Phi \in \mathcal{C}^2$ 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.

Détail

Classification : 28A80, 26A48, 26A51, 65D07, 41A20, 41A29, 41A05
Keywords: iterated function system, $\alpha$-fractal function, rational cubic spline, convergence, convexity, monotonicity, positivity