## Monotone-Comonotone approximation by fractal cubic splines and polynomials

Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand

### Abstract

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).

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### Key words

Fractal function, cubic Hermite fractal interpolation function, fractal polynomial, Fritsch-Carlson algorithm, comonotonicity

### AMS subject classifications

65D05, 41A29, 41A30, 28A80

### Links to the cited ETNA articles

 [27] Vol. 20 (2005), pp. 64-74 M. A. Navascues: Fractal trigonometric approximation [37] Vol. 41 (2014), pp. 420-442 Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand: $\alpha$-fractal rational splines for constrained interpolation

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