Zipper rational fractal interpolation functions

A classical piecewise interpolant can be redefined by utilizing horizontal contractive maps that project the entire domain onto subintervals, ensuring uniqueness. By applying the concept of a zipper, this traditional spline approach is expanded into a broader category of piecewise interpolants through the use of a binary vector signature. In particular, we extend rational spline with cubic/quadratic functions with shape parameters to generate a new class of zipper rational interpolants. Further, we fractalize these interpolants to generate zipper rational cubic \(\alpha \) $\alpha$ -fractal functions. It is demonstrated that the proposed interpolants achieve uniform convergence to a \(C^1\) $C^1$ -data generating function. We establish appropriate constraints on shape parameters, vertical scalings, and signatures to ensure the creation of shape-preserving zipper rational cubic splines and zipper rational cubic \(\alpha \) $\alpha$ -fractal functions. These theoretical results are substantiated with carefully selected numerical examples.