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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-fractal functions. It is demonstrated that the proposed interpolants achieve uniform convergence to a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>1$$\end{document}-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-fractal functions. These theoretical results are substantiated with carefully selected numerical examples.