  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.

Numerical Analysis