Preserving convexity through rational cubic spline fractal interpolation function

Abstract

We propose a new type of C1-rational cubic spline Fractal Interpolation Function (FIF) for convexity preserving univariate interpolation. The associated Iterated Function System (IFS) involves rational functions of the form Pn(x)Qn(x), where Pn(x) are cubic polynomials determined through the Hermite interpolation conditions of the FIF and Qn(x) are preassigned quadratic polynomials with two shape parameters. The rational cubic spline FIF converges to the original function Φ as rapidly as the rth power of the mesh norm approaches to zero, provided Φ(r) is continuous for r=1 or 2 and certain mild conditions on the scaling factors are imposed. Furthermore, suitable values for the rational IFS parameters are identified so that the property of convexity carries from the data set to the rational cubic FIFs. In contrast to the classical non-recursive convexity preserving interpolation schemes, the present fractal scheme is well suited for the approximation of a convex function Φ whose derivative is continuous but has varying irregularity.