MONOTONE-COMONOTONE APPROXIMATION BY FRACTAL CUBIC SPLINES AND POLYNOMIALS

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 integral(alpha) reflects the monotonicity and 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).