Volume 44, pp. 639-659, 2015.
Monotone-Comonotone approximation by fractal cubic splines and polynomials
Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand
Abstract
We develop cubic fractal interpolation functions as
continuously differentiable -fractal functions
corresponding to the traditional piecewise cubic interpolant .
The elements of the iterated function system are identified so
that the class of -fractal functions reflects the monotonicity and -continuity of the source function .
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
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).
Full Text (PDF) [548 KB],
BibTeX
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