Shape Preserving Rational Cubic Spline Fractal Interpolation

Abstract: Fractal interpolation functions (FIFs) developed through iterated function systems (IFSs) prove more versatile than classical interpolants. However, the applications of FIFs in the domain of `shape preserving interpolation' are not fully addressed so far. Among various techniques available in the classical numerical analysis, rational interpolation schemes are well suited for the shape preservation problems and shape modification analysis. In this paper, the capability of FIFs to generalize smooth classical interpolants, and the effectiveness of rational function models in shape preservation are intertwingly exploited to provide a new solution to the shape preserving interpolation problem in fractal perspective. As a common platform for these two techniques to work together, we introduce rational cubic spline FIFs involving tension parameters for the first time in literature. Suitable conditions on parameters of the associated IFS are developed so that the rational fractal interpolant inherits fundamental shape properties such as monotonicity, convexity, and positivity present in the given data. With some suitable hypotheses on the original function, the convergence analysis of the $\mathcal{C}^1$-rational cubic spline FIF is carried out. Due to the presence of the scaling factors in the rational cubic spline fractal interpolant, our approach generalizes the classical results on the shape preserving rational interpolation by Delbourgo and Gregory [SIAM J. Sci. Stat. Comput., 6 (1985), pp. 967-976]. The effectiveness of the shape preserving interpolation schemes are illustrated with suitably chosen numerical examples and graphs, which support the practical utility of our methods.