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 continuous on compact interval I. This produces class of $$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$ , where $$\varvec{\alpha }$$ is vector functional components. presence scaling in these helps get wide variety mappings for approximation problems. current article explores the shape-preserving properties -fractal variable scalings, optimal ranges are derived fundamental shapes germ f. We several examples illustrate shape preserving results apply our methodologies Also, it shown that order convergence polynomial original shaped matches approximation. Further, based functions, we analogue Chebyshev alternation theorem. To end, deduce version classical full Müntz theorem $${\mathcal {C}}[0,1]$$ .