Monotone-comonotone approximation by fractal cubic splines and polynomials

Viswanathan, Puthan Veedu ; Chand, Arya Kumar Bedabrata

Electronic transactions on numerical analysis, Tome 44 (2015), pp. 639-659.

Résumé

Summary: We develop cubic fractal interpolation functions $H^{\alpha}$ as continuously differentiable $\alpha$-fractal functions corresponding to the traditional piecewise cubic interpolant $H$. The elements of the iterated function system are identified so that the class of $\alpha$-fractal functions $f^{\alpha}$ reflects the monotonicity and $\mathcal{C}^1$-continuity of the source function $f$. We use this monotonicity preserving fractal perturbation to: (i) prove the existence of piecewise defined fractal polynomials that are comonotone with a continuous function, (ii) obtain some estimates for monotone and comonotone approximation by fractal polynomials. Drawing on the Fritsch-Carlson theory of monotone cubic interpolation and the developed monotonicity preserving fractal perturbation, we describe an algorithm that constructs a class of monotone cubic fractal interpolation functions $H^{\alpha}$ for a prescribed set of monotone data. This new class of monotone interpolants provides a large flexibility in the choice of a differentiable monotone interpolant. Furthermore, the proposed class outperforms its traditional non-recursive counterpart in approximation of monotone functions whose first derivatives have varying irregularity/fractality (smooth to nowhere differentiable).

Détail

Classification : 65D05, 41A29, 41A30, 28A80
Keywords: fractal function, cubic Hermite fractal interpolation function, fractal polynomial, fritsch-Carlson algorithm, comonotonicity