Approximation by shape preserving fractal functions with variable scalings

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions fα∈C(I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$\end{document}, where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document} is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}-fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}-fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}-fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in C[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}[0,1]$$\end{document}.