Approximation using hidden variable fractal interpolation function

Abstract

The notion of hidden variable fractal interpolation provides a method to approximate functions that are self-referential or non-self-referential, and consequently allows great flexibility and diversity for the fractal modeling problem. The current article intends to apply hidden variable fractal interpolation to associate a class of ${\mathbb{R}}^2$-valued continuous fractal functions with a prescribed continuous function. Suitable values of the parameters are identified so that the fractal functions retain positivity and regularity of the germ function. As an application of the developed theory, we obtain positive ${\mathcal{C}}^1$-cubic spline hidden variable fractal interpolation functions corresponding to a prescribed set of positive data, thus initiating a new approach to shape preserving approximation via hidden variable fractal function. Depending on the values of the parameters, these positive interpolants can reflect the self-referentiality or non-self-referentiality of the original data defining function and fractality of its derivative. Therefore, the present scheme outperforms the traditional nonrecursive positivity preserving ${\mathcal{C}}^1$-cubic spline interpolation scheme and its fractal extension studied recently in the literature.