Arya Kumar Bedabrata Chand

Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420-442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

This paper addresses a method to obtain rational cubic fractal functions, which generate surfaces that lie above a plane via blending functions. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are above the plane whenever the given interpolation data along the grid lines are above the plane. Our rational cubic spline FIS is above the plane whenever the corresponding fractal boundary curves are above the plane. We illustrate our interpolation scheme with some numerical examples.

The current article is intended to demonstrate that the theory of fractal functions when applied in conjunction with methods in the classical numerical analysis can supply new solution techniques that supplement and subsume the existing ones. To this end, in the first part of the paper, we review a C1-continuous rational cubic fractal interpolation function (FIF) introduced recently [Viswanathan and Chand, Elec. Trans. Numer. Anal. 41 (2014), pp. 420-442]. We carry out the con- vergence analysis of this univariate rational FIF and determine suitable values of the derivative parameters so that its global smoothness enhances to C2. In the subsequent part of the article, we apply Coons technique of transfinite interpolation in order to construct a new kind of C1-continuous bivariate fractal interpolation surface.